state stringlengths 0 159k | srcUpToTactic stringlengths 387 167k | nextTactic stringlengths 3 9k | declUpToTactic stringlengths 22 11.5k | declId stringlengths 38 95 | decl stringlengths 16 1.89k | file_tag stringlengths 17 73 |
|---|---|---|---|---|---|---|
case pos
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
ι' : Sort u_5
f✝ : ι → α
s t : Set α
p : Prop
inst✝ : Decidable p
f g : α → β
h : p
⊢ range f ⊆ range f ∪ range g | /-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura
-/
import Mathlib.Data.Set.Basic
#align_import data.set.image from "leanprover-community/mathlib"@"001ffdc42920050657fd45bd2b8bfbec8eaaeb29"
/-!
# Ima... | exact subset_union_left _ _ | theorem range_ite_subset' {p : Prop} [Decidable p] {f g : α → β} :
range (if p then f else g) ⊆ range f ∪ range g := by
by_cases h : p
· rw [if_pos h]
| Mathlib.Data.Set.Image.1112_0.IJFiTzmYGOCpPSd | theorem range_ite_subset' {p : Prop} [Decidable p] {f g : α → β} :
range (if p then f else g) ⊆ range f ∪ range g | Mathlib_Data_Set_Image |
case neg
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
ι' : Sort u_5
f✝ : ι → α
s t : Set α
p : Prop
inst✝ : Decidable p
f g : α → β
h : ¬p
⊢ range (if p then f else g) ⊆ range f ∪ range g | /-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura
-/
import Mathlib.Data.Set.Basic
#align_import data.set.image from "leanprover-community/mathlib"@"001ffdc42920050657fd45bd2b8bfbec8eaaeb29"
/-!
# Ima... | rw [if_neg h] | theorem range_ite_subset' {p : Prop} [Decidable p] {f g : α → β} :
range (if p then f else g) ⊆ range f ∪ range g := by
by_cases h : p
· rw [if_pos h]
exact subset_union_left _ _
· | Mathlib.Data.Set.Image.1112_0.IJFiTzmYGOCpPSd | theorem range_ite_subset' {p : Prop} [Decidable p] {f g : α → β} :
range (if p then f else g) ⊆ range f ∪ range g | Mathlib_Data_Set_Image |
case neg
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
ι' : Sort u_5
f✝ : ι → α
s t : Set α
p : Prop
inst✝ : Decidable p
f g : α → β
h : ¬p
⊢ range g ⊆ range f ∪ range g | /-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura
-/
import Mathlib.Data.Set.Basic
#align_import data.set.image from "leanprover-community/mathlib"@"001ffdc42920050657fd45bd2b8bfbec8eaaeb29"
/-!
# Ima... | exact subset_union_right _ _ | theorem range_ite_subset' {p : Prop} [Decidable p] {f g : α → β} :
range (if p then f else g) ⊆ range f ∪ range g := by
by_cases h : p
· rw [if_pos h]
exact subset_union_left _ _
· rw [if_neg h]
| Mathlib.Data.Set.Image.1112_0.IJFiTzmYGOCpPSd | theorem range_ite_subset' {p : Prop} [Decidable p] {f g : α → β} :
range (if p then f else g) ⊆ range f ∪ range g | Mathlib_Data_Set_Image |
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
ι' : Sort u_5
f✝ : ι → α
s t : Set α
p : α → Prop
inst✝ : DecidablePred p
f g : α → β
⊢ (range fun x => if p x then f x else g x) ⊆ range f ∪ range g | /-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura
-/
import Mathlib.Data.Set.Basic
#align_import data.set.image from "leanprover-community/mathlib"@"001ffdc42920050657fd45bd2b8bfbec8eaaeb29"
/-!
# Ima... | rw [range_subset_iff] | theorem range_ite_subset {p : α → Prop} [DecidablePred p] {f g : α → β} :
(range fun x => if p x then f x else g x) ⊆ range f ∪ range g := by
| Mathlib.Data.Set.Image.1121_0.IJFiTzmYGOCpPSd | theorem range_ite_subset {p : α → Prop} [DecidablePred p] {f g : α → β} :
(range fun x => if p x then f x else g x) ⊆ range f ∪ range g | Mathlib_Data_Set_Image |
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
ι' : Sort u_5
f✝ : ι → α
s t : Set α
p : α → Prop
inst✝ : DecidablePred p
f g : α → β
⊢ ∀ (y : α), (if p y then f y else g y) ∈ range f ∪ range g | /-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura
-/
import Mathlib.Data.Set.Basic
#align_import data.set.image from "leanprover-community/mathlib"@"001ffdc42920050657fd45bd2b8bfbec8eaaeb29"
/-!
# Ima... | intro x | theorem range_ite_subset {p : α → Prop} [DecidablePred p] {f g : α → β} :
(range fun x => if p x then f x else g x) ⊆ range f ∪ range g := by
rw [range_subset_iff]; | Mathlib.Data.Set.Image.1121_0.IJFiTzmYGOCpPSd | theorem range_ite_subset {p : α → Prop} [DecidablePred p] {f g : α → β} :
(range fun x => if p x then f x else g x) ⊆ range f ∪ range g | Mathlib_Data_Set_Image |
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
ι' : Sort u_5
f✝ : ι → α
s t : Set α
p : α → Prop
inst✝ : DecidablePred p
f g : α → β
x : α
⊢ (if p x then f x else g x) ∈ range f ∪ range g | /-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura
-/
import Mathlib.Data.Set.Basic
#align_import data.set.image from "leanprover-community/mathlib"@"001ffdc42920050657fd45bd2b8bfbec8eaaeb29"
/-!
# Ima... | by_cases h : p x | theorem range_ite_subset {p : α → Prop} [DecidablePred p] {f g : α → β} :
(range fun x => if p x then f x else g x) ⊆ range f ∪ range g := by
rw [range_subset_iff]; intro x; | Mathlib.Data.Set.Image.1121_0.IJFiTzmYGOCpPSd | theorem range_ite_subset {p : α → Prop} [DecidablePred p] {f g : α → β} :
(range fun x => if p x then f x else g x) ⊆ range f ∪ range g | Mathlib_Data_Set_Image |
case pos
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
ι' : Sort u_5
f✝ : ι → α
s t : Set α
p : α → Prop
inst✝ : DecidablePred p
f g : α → β
x : α
h : p x
⊢ (if p x then f x else g x) ∈ range f ∪ range g
case neg
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
ι' : Sort u_5
f✝ : ι → α
s t : Set α
p : α → Prop... | /-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura
-/
import Mathlib.Data.Set.Basic
#align_import data.set.image from "leanprover-community/mathlib"@"001ffdc42920050657fd45bd2b8bfbec8eaaeb29"
/-!
# Ima... | simp only [if_pos h, mem_union, mem_range, exists_apply_eq_apply, true_or] | theorem range_ite_subset {p : α → Prop} [DecidablePred p] {f g : α → β} :
(range fun x => if p x then f x else g x) ⊆ range f ∪ range g := by
rw [range_subset_iff]; intro x; by_cases h : p x
| Mathlib.Data.Set.Image.1121_0.IJFiTzmYGOCpPSd | theorem range_ite_subset {p : α → Prop} [DecidablePred p] {f g : α → β} :
(range fun x => if p x then f x else g x) ⊆ range f ∪ range g | Mathlib_Data_Set_Image |
case neg
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
ι' : Sort u_5
f✝ : ι → α
s t : Set α
p : α → Prop
inst✝ : DecidablePred p
f g : α → β
x : α
h : ¬p x
⊢ (if p x then f x else g x) ∈ range f ∪ range g | /-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura
-/
import Mathlib.Data.Set.Basic
#align_import data.set.image from "leanprover-community/mathlib"@"001ffdc42920050657fd45bd2b8bfbec8eaaeb29"
/-!
# Ima... | simp [if_neg h, mem_union, mem_range_self] | theorem range_ite_subset {p : α → Prop} [DecidablePred p] {f g : α → β} :
(range fun x => if p x then f x else g x) ⊆ range f ∪ range g := by
rw [range_subset_iff]; intro x; by_cases h : p x
simp only [if_pos h, mem_union, mem_range, exists_apply_eq_apply, true_or]
| Mathlib.Data.Set.Image.1121_0.IJFiTzmYGOCpPSd | theorem range_ite_subset {p : α → Prop} [DecidablePred p] {f g : α → β} :
(range fun x => if p x then f x else g x) ⊆ range f ∪ range g | Mathlib_Data_Set_Image |
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
ι' : Sort u_5
f : ι → α
s t : Set α
h : Unique ι
⊢ range f = {f default} | /-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura
-/
import Mathlib.Data.Set.Basic
#align_import data.set.image from "leanprover-community/mathlib"@"001ffdc42920050657fd45bd2b8bfbec8eaaeb29"
/-!
# Ima... | ext x | /-- The range of a function from a `Unique` type contains just the
function applied to its single value. -/
theorem range_unique [h : Unique ι] : range f = {f default} := by
| Mathlib.Data.Set.Image.1133_0.IJFiTzmYGOCpPSd | /-- The range of a function from a `Unique` type contains just the
function applied to its single value. -/
theorem range_unique [h : Unique ι] : range f = {f default} | Mathlib_Data_Set_Image |
case h
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
ι' : Sort u_5
f : ι → α
s t : Set α
h : Unique ι
x : α
⊢ x ∈ range f ↔ x ∈ {f default} | /-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura
-/
import Mathlib.Data.Set.Basic
#align_import data.set.image from "leanprover-community/mathlib"@"001ffdc42920050657fd45bd2b8bfbec8eaaeb29"
/-!
# Ima... | rw [mem_range] | /-- The range of a function from a `Unique` type contains just the
function applied to its single value. -/
theorem range_unique [h : Unique ι] : range f = {f default} := by
ext x
| Mathlib.Data.Set.Image.1133_0.IJFiTzmYGOCpPSd | /-- The range of a function from a `Unique` type contains just the
function applied to its single value. -/
theorem range_unique [h : Unique ι] : range f = {f default} | Mathlib_Data_Set_Image |
case h
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
ι' : Sort u_5
f : ι → α
s t : Set α
h : Unique ι
x : α
⊢ (∃ y, f y = x) ↔ x ∈ {f default} | /-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura
-/
import Mathlib.Data.Set.Basic
#align_import data.set.image from "leanprover-community/mathlib"@"001ffdc42920050657fd45bd2b8bfbec8eaaeb29"
/-!
# Ima... | constructor | /-- The range of a function from a `Unique` type contains just the
function applied to its single value. -/
theorem range_unique [h : Unique ι] : range f = {f default} := by
ext x
rw [mem_range]
| Mathlib.Data.Set.Image.1133_0.IJFiTzmYGOCpPSd | /-- The range of a function from a `Unique` type contains just the
function applied to its single value. -/
theorem range_unique [h : Unique ι] : range f = {f default} | Mathlib_Data_Set_Image |
case h.mp
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
ι' : Sort u_5
f : ι → α
s t : Set α
h : Unique ι
x : α
⊢ (∃ y, f y = x) → x ∈ {f default} | /-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura
-/
import Mathlib.Data.Set.Basic
#align_import data.set.image from "leanprover-community/mathlib"@"001ffdc42920050657fd45bd2b8bfbec8eaaeb29"
/-!
# Ima... | rintro ⟨i, hi⟩ | /-- The range of a function from a `Unique` type contains just the
function applied to its single value. -/
theorem range_unique [h : Unique ι] : range f = {f default} := by
ext x
rw [mem_range]
constructor
· | Mathlib.Data.Set.Image.1133_0.IJFiTzmYGOCpPSd | /-- The range of a function from a `Unique` type contains just the
function applied to its single value. -/
theorem range_unique [h : Unique ι] : range f = {f default} | Mathlib_Data_Set_Image |
case h.mp.intro
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
ι' : Sort u_5
f : ι → α
s t : Set α
h : Unique ι
x : α
i : ι
hi : f i = x
⊢ x ∈ {f default} | /-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura
-/
import Mathlib.Data.Set.Basic
#align_import data.set.image from "leanprover-community/mathlib"@"001ffdc42920050657fd45bd2b8bfbec8eaaeb29"
/-!
# Ima... | rw [h.uniq i] at hi | /-- The range of a function from a `Unique` type contains just the
function applied to its single value. -/
theorem range_unique [h : Unique ι] : range f = {f default} := by
ext x
rw [mem_range]
constructor
· rintro ⟨i, hi⟩
| Mathlib.Data.Set.Image.1133_0.IJFiTzmYGOCpPSd | /-- The range of a function from a `Unique` type contains just the
function applied to its single value. -/
theorem range_unique [h : Unique ι] : range f = {f default} | Mathlib_Data_Set_Image |
case h.mp.intro
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
ι' : Sort u_5
f : ι → α
s t : Set α
h : Unique ι
x : α
i : ι
hi : f default = x
⊢ x ∈ {f default} | /-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura
-/
import Mathlib.Data.Set.Basic
#align_import data.set.image from "leanprover-community/mathlib"@"001ffdc42920050657fd45bd2b8bfbec8eaaeb29"
/-!
# Ima... | exact hi ▸ mem_singleton _ | /-- The range of a function from a `Unique` type contains just the
function applied to its single value. -/
theorem range_unique [h : Unique ι] : range f = {f default} := by
ext x
rw [mem_range]
constructor
· rintro ⟨i, hi⟩
rw [h.uniq i] at hi
| Mathlib.Data.Set.Image.1133_0.IJFiTzmYGOCpPSd | /-- The range of a function from a `Unique` type contains just the
function applied to its single value. -/
theorem range_unique [h : Unique ι] : range f = {f default} | Mathlib_Data_Set_Image |
case h.mpr
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
ι' : Sort u_5
f : ι → α
s t : Set α
h : Unique ι
x : α
⊢ x ∈ {f default} → ∃ y, f y = x | /-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura
-/
import Mathlib.Data.Set.Basic
#align_import data.set.image from "leanprover-community/mathlib"@"001ffdc42920050657fd45bd2b8bfbec8eaaeb29"
/-!
# Ima... | exact fun h => ⟨default, h.symm⟩ | /-- The range of a function from a `Unique` type contains just the
function applied to its single value. -/
theorem range_unique [h : Unique ι] : range f = {f default} := by
ext x
rw [mem_range]
constructor
· rintro ⟨i, hi⟩
rw [h.uniq i] at hi
exact hi ▸ mem_singleton _
· | Mathlib.Data.Set.Image.1133_0.IJFiTzmYGOCpPSd | /-- The range of a function from a `Unique` type contains just the
function applied to its single value. -/
theorem range_unique [h : Unique ι] : range f = {f default} | Mathlib_Data_Set_Image |
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
ι' : Sort u_5
f : ι → α
s t : Set α
h : s ⊆ t
⊢ range (inclusion h) = {x | ↑x ∈ s} | /-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura
-/
import Mathlib.Data.Set.Basic
#align_import data.set.image from "leanprover-community/mathlib"@"001ffdc42920050657fd45bd2b8bfbec8eaaeb29"
/-!
# Ima... | ext ⟨x, hx⟩ | @[simp]
theorem range_inclusion (h : s ⊆ t) : range (inclusion h) = { x : t | (x : α) ∈ s } := by
| Mathlib.Data.Set.Image.1154_0.IJFiTzmYGOCpPSd | @[simp]
theorem range_inclusion (h : s ⊆ t) : range (inclusion h) = { x : t | (x : α) ∈ s } | Mathlib_Data_Set_Image |
case h.mk
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
ι' : Sort u_5
f : ι → α
s t : Set α
h : s ⊆ t
x : α
hx : x ∈ t
⊢ { val := x, property := hx } ∈ range (inclusion h) ↔ { val := x, property := hx } ∈ {x | ↑x ∈ s} | /-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura
-/
import Mathlib.Data.Set.Basic
#align_import data.set.image from "leanprover-community/mathlib"@"001ffdc42920050657fd45bd2b8bfbec8eaaeb29"
/-!
# Ima... | apply Iff.intro | @[simp]
theorem range_inclusion (h : s ⊆ t) : range (inclusion h) = { x : t | (x : α) ∈ s } := by
ext ⟨x, hx⟩
-- Porting note: `simp [inclusion]` doesn't solve goal
| Mathlib.Data.Set.Image.1154_0.IJFiTzmYGOCpPSd | @[simp]
theorem range_inclusion (h : s ⊆ t) : range (inclusion h) = { x : t | (x : α) ∈ s } | Mathlib_Data_Set_Image |
case h.mk.mp
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
ι' : Sort u_5
f : ι → α
s t : Set α
h : s ⊆ t
x : α
hx : x ∈ t
⊢ { val := x, property := hx } ∈ range (inclusion h) → { val := x, property := hx } ∈ {x | ↑x ∈ s} | /-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura
-/
import Mathlib.Data.Set.Basic
#align_import data.set.image from "leanprover-community/mathlib"@"001ffdc42920050657fd45bd2b8bfbec8eaaeb29"
/-!
# Ima... | rw [mem_range] | @[simp]
theorem range_inclusion (h : s ⊆ t) : range (inclusion h) = { x : t | (x : α) ∈ s } := by
ext ⟨x, hx⟩
-- Porting note: `simp [inclusion]` doesn't solve goal
apply Iff.intro
· | Mathlib.Data.Set.Image.1154_0.IJFiTzmYGOCpPSd | @[simp]
theorem range_inclusion (h : s ⊆ t) : range (inclusion h) = { x : t | (x : α) ∈ s } | Mathlib_Data_Set_Image |
case h.mk.mp
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
ι' : Sort u_5
f : ι → α
s t : Set α
h : s ⊆ t
x : α
hx : x ∈ t
⊢ (∃ y, inclusion h y = { val := x, property := hx }) → { val := x, property := hx } ∈ {x | ↑x ∈ s} | /-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura
-/
import Mathlib.Data.Set.Basic
#align_import data.set.image from "leanprover-community/mathlib"@"001ffdc42920050657fd45bd2b8bfbec8eaaeb29"
/-!
# Ima... | rintro ⟨a, ha⟩ | @[simp]
theorem range_inclusion (h : s ⊆ t) : range (inclusion h) = { x : t | (x : α) ∈ s } := by
ext ⟨x, hx⟩
-- Porting note: `simp [inclusion]` doesn't solve goal
apply Iff.intro
· rw [mem_range]
| Mathlib.Data.Set.Image.1154_0.IJFiTzmYGOCpPSd | @[simp]
theorem range_inclusion (h : s ⊆ t) : range (inclusion h) = { x : t | (x : α) ∈ s } | Mathlib_Data_Set_Image |
case h.mk.mp.intro
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
ι' : Sort u_5
f : ι → α
s t : Set α
h : s ⊆ t
x : α
hx : x ∈ t
a : ↑s
ha : inclusion h a = { val := x, property := hx }
⊢ { val := x, property := hx } ∈ {x | ↑x ∈ s} | /-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura
-/
import Mathlib.Data.Set.Basic
#align_import data.set.image from "leanprover-community/mathlib"@"001ffdc42920050657fd45bd2b8bfbec8eaaeb29"
/-!
# Ima... | rw [inclusion, Subtype.mk.injEq] at ha | @[simp]
theorem range_inclusion (h : s ⊆ t) : range (inclusion h) = { x : t | (x : α) ∈ s } := by
ext ⟨x, hx⟩
-- Porting note: `simp [inclusion]` doesn't solve goal
apply Iff.intro
· rw [mem_range]
rintro ⟨a, ha⟩
| Mathlib.Data.Set.Image.1154_0.IJFiTzmYGOCpPSd | @[simp]
theorem range_inclusion (h : s ⊆ t) : range (inclusion h) = { x : t | (x : α) ∈ s } | Mathlib_Data_Set_Image |
case h.mk.mp.intro
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
ι' : Sort u_5
f : ι → α
s t : Set α
h : s ⊆ t
x : α
hx : x ∈ t
a : ↑s
ha : ↑a = x
⊢ { val := x, property := hx } ∈ {x | ↑x ∈ s} | /-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura
-/
import Mathlib.Data.Set.Basic
#align_import data.set.image from "leanprover-community/mathlib"@"001ffdc42920050657fd45bd2b8bfbec8eaaeb29"
/-!
# Ima... | rw [mem_setOf, Subtype.coe_mk, ← ha] | @[simp]
theorem range_inclusion (h : s ⊆ t) : range (inclusion h) = { x : t | (x : α) ∈ s } := by
ext ⟨x, hx⟩
-- Porting note: `simp [inclusion]` doesn't solve goal
apply Iff.intro
· rw [mem_range]
rintro ⟨a, ha⟩
rw [inclusion, Subtype.mk.injEq] at ha
| Mathlib.Data.Set.Image.1154_0.IJFiTzmYGOCpPSd | @[simp]
theorem range_inclusion (h : s ⊆ t) : range (inclusion h) = { x : t | (x : α) ∈ s } | Mathlib_Data_Set_Image |
case h.mk.mp.intro
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
ι' : Sort u_5
f : ι → α
s t : Set α
h : s ⊆ t
x : α
hx : x ∈ t
a : ↑s
ha : ↑a = x
⊢ ↑a ∈ s | /-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura
-/
import Mathlib.Data.Set.Basic
#align_import data.set.image from "leanprover-community/mathlib"@"001ffdc42920050657fd45bd2b8bfbec8eaaeb29"
/-!
# Ima... | exact Subtype.coe_prop _ | @[simp]
theorem range_inclusion (h : s ⊆ t) : range (inclusion h) = { x : t | (x : α) ∈ s } := by
ext ⟨x, hx⟩
-- Porting note: `simp [inclusion]` doesn't solve goal
apply Iff.intro
· rw [mem_range]
rintro ⟨a, ha⟩
rw [inclusion, Subtype.mk.injEq] at ha
rw [mem_setOf, Subtype.coe_mk, ← ha]
| Mathlib.Data.Set.Image.1154_0.IJFiTzmYGOCpPSd | @[simp]
theorem range_inclusion (h : s ⊆ t) : range (inclusion h) = { x : t | (x : α) ∈ s } | Mathlib_Data_Set_Image |
case h.mk.mpr
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
ι' : Sort u_5
f : ι → α
s t : Set α
h : s ⊆ t
x : α
hx : x ∈ t
⊢ { val := x, property := hx } ∈ {x | ↑x ∈ s} → { val := x, property := hx } ∈ range (inclusion h) | /-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura
-/
import Mathlib.Data.Set.Basic
#align_import data.set.image from "leanprover-community/mathlib"@"001ffdc42920050657fd45bd2b8bfbec8eaaeb29"
/-!
# Ima... | rw [mem_setOf, Subtype.coe_mk, mem_range] | @[simp]
theorem range_inclusion (h : s ⊆ t) : range (inclusion h) = { x : t | (x : α) ∈ s } := by
ext ⟨x, hx⟩
-- Porting note: `simp [inclusion]` doesn't solve goal
apply Iff.intro
· rw [mem_range]
rintro ⟨a, ha⟩
rw [inclusion, Subtype.mk.injEq] at ha
rw [mem_setOf, Subtype.coe_mk, ← ha]
exact S... | Mathlib.Data.Set.Image.1154_0.IJFiTzmYGOCpPSd | @[simp]
theorem range_inclusion (h : s ⊆ t) : range (inclusion h) = { x : t | (x : α) ∈ s } | Mathlib_Data_Set_Image |
case h.mk.mpr
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
ι' : Sort u_5
f : ι → α
s t : Set α
h : s ⊆ t
x : α
hx : x ∈ t
⊢ x ∈ s → ∃ y, inclusion h y = { val := x, property := hx } | /-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura
-/
import Mathlib.Data.Set.Basic
#align_import data.set.image from "leanprover-community/mathlib"@"001ffdc42920050657fd45bd2b8bfbec8eaaeb29"
/-!
# Ima... | intro hx' | @[simp]
theorem range_inclusion (h : s ⊆ t) : range (inclusion h) = { x : t | (x : α) ∈ s } := by
ext ⟨x, hx⟩
-- Porting note: `simp [inclusion]` doesn't solve goal
apply Iff.intro
· rw [mem_range]
rintro ⟨a, ha⟩
rw [inclusion, Subtype.mk.injEq] at ha
rw [mem_setOf, Subtype.coe_mk, ← ha]
exact S... | Mathlib.Data.Set.Image.1154_0.IJFiTzmYGOCpPSd | @[simp]
theorem range_inclusion (h : s ⊆ t) : range (inclusion h) = { x : t | (x : α) ∈ s } | Mathlib_Data_Set_Image |
case h.mk.mpr
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
ι' : Sort u_5
f : ι → α
s t : Set α
h : s ⊆ t
x : α
hx : x ∈ t
hx' : x ∈ s
⊢ ∃ y, inclusion h y = { val := x, property := hx } | /-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura
-/
import Mathlib.Data.Set.Basic
#align_import data.set.image from "leanprover-community/mathlib"@"001ffdc42920050657fd45bd2b8bfbec8eaaeb29"
/-!
# Ima... | use ⟨x, hx'⟩ | @[simp]
theorem range_inclusion (h : s ⊆ t) : range (inclusion h) = { x : t | (x : α) ∈ s } := by
ext ⟨x, hx⟩
-- Porting note: `simp [inclusion]` doesn't solve goal
apply Iff.intro
· rw [mem_range]
rintro ⟨a, ha⟩
rw [inclusion, Subtype.mk.injEq] at ha
rw [mem_setOf, Subtype.coe_mk, ← ha]
exact S... | Mathlib.Data.Set.Image.1154_0.IJFiTzmYGOCpPSd | @[simp]
theorem range_inclusion (h : s ⊆ t) : range (inclusion h) = { x : t | (x : α) ∈ s } | Mathlib_Data_Set_Image |
case h
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
ι' : Sort u_5
f : ι → α
s t : Set α
h : s ⊆ t
x : α
hx : x ∈ t
hx' : x ∈ s
⊢ inclusion h { val := x, property := hx' } = { val := x, property := hx } | /-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura
-/
import Mathlib.Data.Set.Basic
#align_import data.set.image from "leanprover-community/mathlib"@"001ffdc42920050657fd45bd2b8bfbec8eaaeb29"
/-!
# Ima... | trivial | @[simp]
theorem range_inclusion (h : s ⊆ t) : range (inclusion h) = { x : t | (x : α) ∈ s } := by
ext ⟨x, hx⟩
-- Porting note: `simp [inclusion]` doesn't solve goal
apply Iff.intro
· rw [mem_range]
rintro ⟨a, ha⟩
rw [inclusion, Subtype.mk.injEq] at ha
rw [mem_setOf, Subtype.coe_mk, ← ha]
exact S... | Mathlib.Data.Set.Image.1154_0.IJFiTzmYGOCpPSd | @[simp]
theorem range_inclusion (h : s ⊆ t) : range (inclusion h) = { x : t | (x : α) ∈ s } | Mathlib_Data_Set_Image |
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
ι' : Sort u_5
f✝ : ι → α
s t : Set α
f : α → β
⊢ f ∘ rangeSplitting f = Subtype.val | /-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura
-/
import Mathlib.Data.Set.Basic
#align_import data.set.image from "leanprover-community/mathlib"@"001ffdc42920050657fd45bd2b8bfbec8eaaeb29"
/-!
# Ima... | ext | @[simp]
theorem comp_rangeSplitting (f : α → β) : f ∘ rangeSplitting f = (↑) := by
| Mathlib.Data.Set.Image.1181_0.IJFiTzmYGOCpPSd | @[simp]
theorem comp_rangeSplitting (f : α → β) : f ∘ rangeSplitting f = (↑) | Mathlib_Data_Set_Image |
case h
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
ι' : Sort u_5
f✝ : ι → α
s t : Set α
f : α → β
x✝ : ↑(range f)
⊢ (f ∘ rangeSplitting f) x✝ = ↑x✝ | /-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura
-/
import Mathlib.Data.Set.Basic
#align_import data.set.image from "leanprover-community/mathlib"@"001ffdc42920050657fd45bd2b8bfbec8eaaeb29"
/-!
# Ima... | simp only [Function.comp_apply] | @[simp]
theorem comp_rangeSplitting (f : α → β) : f ∘ rangeSplitting f = (↑) := by
ext
| Mathlib.Data.Set.Image.1181_0.IJFiTzmYGOCpPSd | @[simp]
theorem comp_rangeSplitting (f : α → β) : f ∘ rangeSplitting f = (↑) | Mathlib_Data_Set_Image |
case h
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
ι' : Sort u_5
f✝ : ι → α
s t : Set α
f : α → β
x✝ : ↑(range f)
⊢ f (rangeSplitting f x✝) = ↑x✝ | /-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura
-/
import Mathlib.Data.Set.Basic
#align_import data.set.image from "leanprover-community/mathlib"@"001ffdc42920050657fd45bd2b8bfbec8eaaeb29"
/-!
# Ima... | apply apply_rangeSplitting | @[simp]
theorem comp_rangeSplitting (f : α → β) : f ∘ rangeSplitting f = (↑) := by
ext
simp only [Function.comp_apply]
| Mathlib.Data.Set.Image.1181_0.IJFiTzmYGOCpPSd | @[simp]
theorem comp_rangeSplitting (f : α → β) : f ∘ rangeSplitting f = (↑) | Mathlib_Data_Set_Image |
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
ι' : Sort u_5
f✝ : ι → α
s t : Set α
f : α → β
x : ↑(range f)
⊢ rangeFactorization f (rangeSplitting f x) = x | /-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura
-/
import Mathlib.Data.Set.Basic
#align_import data.set.image from "leanprover-community/mathlib"@"001ffdc42920050657fd45bd2b8bfbec8eaaeb29"
/-!
# Ima... | apply Subtype.ext | theorem leftInverse_rangeSplitting (f : α → β) :
LeftInverse (rangeFactorization f) (rangeSplitting f) := fun x => by
| Mathlib.Data.Set.Image.1189_0.IJFiTzmYGOCpPSd | theorem leftInverse_rangeSplitting (f : α → β) :
LeftInverse (rangeFactorization f) (rangeSplitting f) | Mathlib_Data_Set_Image |
case a
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
ι' : Sort u_5
f✝ : ι → α
s t : Set α
f : α → β
x : ↑(range f)
⊢ ↑(rangeFactorization f (rangeSplitting f x)) = ↑x | /-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura
-/
import Mathlib.Data.Set.Basic
#align_import data.set.image from "leanprover-community/mathlib"@"001ffdc42920050657fd45bd2b8bfbec8eaaeb29"
/-!
# Ima... | simp only [rangeFactorization_coe] | theorem leftInverse_rangeSplitting (f : α → β) :
LeftInverse (rangeFactorization f) (rangeSplitting f) := fun x => by
apply Subtype.ext -- Porting note: why doesn't `ext` find this lemma?
| Mathlib.Data.Set.Image.1189_0.IJFiTzmYGOCpPSd | theorem leftInverse_rangeSplitting (f : α → β) :
LeftInverse (rangeFactorization f) (rangeSplitting f) | Mathlib_Data_Set_Image |
case a
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
ι' : Sort u_5
f✝ : ι → α
s t : Set α
f : α → β
x : ↑(range f)
⊢ f (rangeSplitting f x) = ↑x | /-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura
-/
import Mathlib.Data.Set.Basic
#align_import data.set.image from "leanprover-community/mathlib"@"001ffdc42920050657fd45bd2b8bfbec8eaaeb29"
/-!
# Ima... | apply apply_rangeSplitting | theorem leftInverse_rangeSplitting (f : α → β) :
LeftInverse (rangeFactorization f) (rangeSplitting f) := fun x => by
apply Subtype.ext -- Porting note: why doesn't `ext` find this lemma?
simp only [rangeFactorization_coe]
| Mathlib.Data.Set.Image.1189_0.IJFiTzmYGOCpPSd | theorem leftInverse_rangeSplitting (f : α → β) :
LeftInverse (rangeFactorization f) (rangeSplitting f) | Mathlib_Data_Set_Image |
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
ι' : Sort u_5
s✝ : Set α
f : α → β
hf : Surjective f
s : Set β
hs : Set.Subsingleton (f ⁻¹' s)
fx : β
hx : fx ∈ s
fy : β
hy : fy ∈ s
⊢ fx = fy | /-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura
-/
import Mathlib.Data.Set.Basic
#align_import data.set.image from "leanprover-community/mathlib"@"001ffdc42920050657fd45bd2b8bfbec8eaaeb29"
/-!
# Ima... | rcases hf fx, hf fy with ⟨⟨x, rfl⟩, ⟨y, rfl⟩⟩ | /-- If the preimage of a set under a surjective map is a subsingleton,
the set is a subsingleton. -/
theorem subsingleton_of_preimage {f : α → β} (hf : Function.Surjective f) (s : Set β)
(hs : (f ⁻¹' s).Subsingleton) : s.Subsingleton := fun fx hx fy hy => by
| Mathlib.Data.Set.Image.1260_0.IJFiTzmYGOCpPSd | /-- If the preimage of a set under a surjective map is a subsingleton,
the set is a subsingleton. -/
theorem subsingleton_of_preimage {f : α → β} (hf : Function.Surjective f) (s : Set β)
(hs : (f ⁻¹' s).Subsingleton) : s.Subsingleton | Mathlib_Data_Set_Image |
case intro.intro
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
ι' : Sort u_5
s✝ : Set α
f : α → β
hf : Surjective f
s : Set β
hs : Set.Subsingleton (f ⁻¹' s)
x : α
hx : f x ∈ s
y : α
hy : f y ∈ s
⊢ f x = f y | /-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura
-/
import Mathlib.Data.Set.Basic
#align_import data.set.image from "leanprover-community/mathlib"@"001ffdc42920050657fd45bd2b8bfbec8eaaeb29"
/-!
# Ima... | exact congr_arg f (hs hx hy) | /-- If the preimage of a set under a surjective map is a subsingleton,
the set is a subsingleton. -/
theorem subsingleton_of_preimage {f : α → β} (hf : Function.Surjective f) (s : Set β)
(hs : (f ⁻¹' s).Subsingleton) : s.Subsingleton := fun fx hx fy hy => by
rcases hf fx, hf fy with ⟨⟨x, rfl⟩, ⟨y, rfl⟩⟩
| Mathlib.Data.Set.Image.1260_0.IJFiTzmYGOCpPSd | /-- If the preimage of a set under a surjective map is a subsingleton,
the set is a subsingleton. -/
theorem subsingleton_of_preimage {f : α → β} (hf : Function.Surjective f) (s : Set β)
(hs : (f ⁻¹' s).Subsingleton) : s.Subsingleton | Mathlib_Data_Set_Image |
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
ι' : Sort u_5
s✝ : Set α
s : Set β
hs : Set.Nontrivial s
f : α → β
hf : Surjective f
⊢ Set.Nontrivial (f ⁻¹' s) | /-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura
-/
import Mathlib.Data.Set.Basic
#align_import data.set.image from "leanprover-community/mathlib"@"001ffdc42920050657fd45bd2b8bfbec8eaaeb29"
/-!
# Ima... | rcases hs with ⟨fx, hx, fy, hy, hxy⟩ | /-- The preimage of a nontrivial set under a surjective map is nontrivial. -/
theorem Nontrivial.preimage {s : Set β} (hs : s.Nontrivial) {f : α → β}
(hf : Function.Surjective f) : (f ⁻¹' s).Nontrivial := by
| Mathlib.Data.Set.Image.1272_0.IJFiTzmYGOCpPSd | /-- The preimage of a nontrivial set under a surjective map is nontrivial. -/
theorem Nontrivial.preimage {s : Set β} (hs : s.Nontrivial) {f : α → β}
(hf : Function.Surjective f) : (f ⁻¹' s).Nontrivial | Mathlib_Data_Set_Image |
case intro.intro.intro.intro
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
ι' : Sort u_5
s✝ : Set α
s : Set β
f : α → β
hf : Surjective f
fx : β
hx : fx ∈ s
fy : β
hy : fy ∈ s
hxy : fx ≠ fy
⊢ Set.Nontrivial (f ⁻¹' s) | /-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura
-/
import Mathlib.Data.Set.Basic
#align_import data.set.image from "leanprover-community/mathlib"@"001ffdc42920050657fd45bd2b8bfbec8eaaeb29"
/-!
# Ima... | rcases hf fx, hf fy with ⟨⟨x, rfl⟩, ⟨y, rfl⟩⟩ | /-- The preimage of a nontrivial set under a surjective map is nontrivial. -/
theorem Nontrivial.preimage {s : Set β} (hs : s.Nontrivial) {f : α → β}
(hf : Function.Surjective f) : (f ⁻¹' s).Nontrivial := by
rcases hs with ⟨fx, hx, fy, hy, hxy⟩
| Mathlib.Data.Set.Image.1272_0.IJFiTzmYGOCpPSd | /-- The preimage of a nontrivial set under a surjective map is nontrivial. -/
theorem Nontrivial.preimage {s : Set β} (hs : s.Nontrivial) {f : α → β}
(hf : Function.Surjective f) : (f ⁻¹' s).Nontrivial | Mathlib_Data_Set_Image |
case intro.intro.intro.intro.intro.intro
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
ι' : Sort u_5
s✝ : Set α
s : Set β
f : α → β
hf : Surjective f
x : α
hx : f x ∈ s
y : α
hy : f y ∈ s
hxy : f x ≠ f y
⊢ Set.Nontrivial (f ⁻¹' s) | /-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura
-/
import Mathlib.Data.Set.Basic
#align_import data.set.image from "leanprover-community/mathlib"@"001ffdc42920050657fd45bd2b8bfbec8eaaeb29"
/-!
# Ima... | exact ⟨x, hx, y, hy, mt (congr_arg f) hxy⟩ | /-- The preimage of a nontrivial set under a surjective map is nontrivial. -/
theorem Nontrivial.preimage {s : Set β} (hs : s.Nontrivial) {f : α → β}
(hf : Function.Surjective f) : (f ⁻¹' s).Nontrivial := by
rcases hs with ⟨fx, hx, fy, hy, hxy⟩
rcases hf fx, hf fy with ⟨⟨x, rfl⟩, ⟨y, rfl⟩⟩
| Mathlib.Data.Set.Image.1272_0.IJFiTzmYGOCpPSd | /-- The preimage of a nontrivial set under a surjective map is nontrivial. -/
theorem Nontrivial.preimage {s : Set β} (hs : s.Nontrivial) {f : α → β}
(hf : Function.Surjective f) : (f ⁻¹' s).Nontrivial | Mathlib_Data_Set_Image |
α : Type u_2
β : Type u_3
ι : Sort u_1
f : α → β
hf : Injective f
⊢ Surjective (preimage f) | /-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura
-/
import Mathlib.Data.Set.Basic
#align_import data.set.image from "leanprover-community/mathlib"@"001ffdc42920050657fd45bd2b8bfbec8eaaeb29"
/-!
# Ima... | intro s | theorem Injective.preimage_surjective (hf : Injective f) : Surjective (preimage f) := by
| Mathlib.Data.Set.Image.1317_0.IJFiTzmYGOCpPSd | theorem Injective.preimage_surjective (hf : Injective f) : Surjective (preimage f) | Mathlib_Data_Set_Image |
α : Type u_2
β : Type u_3
ι : Sort u_1
f : α → β
hf : Injective f
s : Set α
⊢ ∃ a, f ⁻¹' a = s | /-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura
-/
import Mathlib.Data.Set.Basic
#align_import data.set.image from "leanprover-community/mathlib"@"001ffdc42920050657fd45bd2b8bfbec8eaaeb29"
/-!
# Ima... | use f '' s | theorem Injective.preimage_surjective (hf : Injective f) : Surjective (preimage f) := by
intro s
| Mathlib.Data.Set.Image.1317_0.IJFiTzmYGOCpPSd | theorem Injective.preimage_surjective (hf : Injective f) : Surjective (preimage f) | Mathlib_Data_Set_Image |
case h
α : Type u_2
β : Type u_3
ι : Sort u_1
f : α → β
hf : Injective f
s : Set α
⊢ f ⁻¹' (f '' s) = s | /-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura
-/
import Mathlib.Data.Set.Basic
#align_import data.set.image from "leanprover-community/mathlib"@"001ffdc42920050657fd45bd2b8bfbec8eaaeb29"
/-!
# Ima... | rw [hf.preimage_image] | theorem Injective.preimage_surjective (hf : Injective f) : Surjective (preimage f) := by
intro s
use f '' s
| Mathlib.Data.Set.Image.1317_0.IJFiTzmYGOCpPSd | theorem Injective.preimage_surjective (hf : Injective f) : Surjective (preimage f) | Mathlib_Data_Set_Image |
α : Type u_2
β : Type u_3
ι : Sort u_1
f : α → β
hf : Surjective f
⊢ Surjective (image f) | /-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura
-/
import Mathlib.Data.Set.Basic
#align_import data.set.image from "leanprover-community/mathlib"@"001ffdc42920050657fd45bd2b8bfbec8eaaeb29"
/-!
# Ima... | intro s | theorem Surjective.image_surjective (hf : Surjective f) : Surjective (image f) := by
| Mathlib.Data.Set.Image.1332_0.IJFiTzmYGOCpPSd | theorem Surjective.image_surjective (hf : Surjective f) : Surjective (image f) | Mathlib_Data_Set_Image |
α : Type u_2
β : Type u_3
ι : Sort u_1
f : α → β
hf : Surjective f
s : Set β
⊢ ∃ a, f '' a = s | /-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura
-/
import Mathlib.Data.Set.Basic
#align_import data.set.image from "leanprover-community/mathlib"@"001ffdc42920050657fd45bd2b8bfbec8eaaeb29"
/-!
# Ima... | use f ⁻¹' s | theorem Surjective.image_surjective (hf : Surjective f) : Surjective (image f) := by
intro s
| Mathlib.Data.Set.Image.1332_0.IJFiTzmYGOCpPSd | theorem Surjective.image_surjective (hf : Surjective f) : Surjective (image f) | Mathlib_Data_Set_Image |
case h
α : Type u_2
β : Type u_3
ι : Sort u_1
f : α → β
hf : Surjective f
s : Set β
⊢ f '' (f ⁻¹' s) = s | /-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura
-/
import Mathlib.Data.Set.Basic
#align_import data.set.image from "leanprover-community/mathlib"@"001ffdc42920050657fd45bd2b8bfbec8eaaeb29"
/-!
# Ima... | rw [hf.image_preimage] | theorem Surjective.image_surjective (hf : Surjective f) : Surjective (image f) := by
intro s
use f ⁻¹' s
| Mathlib.Data.Set.Image.1332_0.IJFiTzmYGOCpPSd | theorem Surjective.image_surjective (hf : Surjective f) : Surjective (image f) | Mathlib_Data_Set_Image |
α : Type u_2
β : Type u_3
ι : Sort u_1
f : α → β
hf : Surjective f
s : Set β
⊢ Set.Nonempty (f ⁻¹' s) ↔ Set.Nonempty s | /-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura
-/
import Mathlib.Data.Set.Basic
#align_import data.set.image from "leanprover-community/mathlib"@"001ffdc42920050657fd45bd2b8bfbec8eaaeb29"
/-!
# Ima... | rw [← nonempty_image_iff, hf.image_preimage] | @[simp]
theorem Surjective.nonempty_preimage (hf : Surjective f) {s : Set β} :
(f ⁻¹' s).Nonempty ↔ s.Nonempty := by | Mathlib.Data.Set.Image.1338_0.IJFiTzmYGOCpPSd | @[simp]
theorem Surjective.nonempty_preimage (hf : Surjective f) {s : Set β} :
(f ⁻¹' s).Nonempty ↔ s.Nonempty | Mathlib_Data_Set_Image |
α : Type u_2
β : Type u_3
ι : Sort u_1
f : α → β
hf : Injective f
⊢ Injective (image f) | /-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura
-/
import Mathlib.Data.Set.Basic
#align_import data.set.image from "leanprover-community/mathlib"@"001ffdc42920050657fd45bd2b8bfbec8eaaeb29"
/-!
# Ima... | intro s t h | theorem Injective.image_injective (hf : Injective f) : Injective (image f) := by
| Mathlib.Data.Set.Image.1343_0.IJFiTzmYGOCpPSd | theorem Injective.image_injective (hf : Injective f) : Injective (image f) | Mathlib_Data_Set_Image |
α : Type u_2
β : Type u_3
ι : Sort u_1
f : α → β
hf : Injective f
s t : Set α
h : f '' s = f '' t
⊢ s = t | /-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura
-/
import Mathlib.Data.Set.Basic
#align_import data.set.image from "leanprover-community/mathlib"@"001ffdc42920050657fd45bd2b8bfbec8eaaeb29"
/-!
# Ima... | rw [← preimage_image_eq s hf, ← preimage_image_eq t hf, h] | theorem Injective.image_injective (hf : Injective f) : Injective (image f) := by
intro s t h
| Mathlib.Data.Set.Image.1343_0.IJFiTzmYGOCpPSd | theorem Injective.image_injective (hf : Injective f) : Injective (image f) | Mathlib_Data_Set_Image |
α : Type u_3
β : Type u_2
ι : Sort u_1
f : α → β
s t : Set β
hf : Surjective f
⊢ f ⁻¹' s ⊆ f ⁻¹' t ↔ s ⊆ t | /-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura
-/
import Mathlib.Data.Set.Basic
#align_import data.set.image from "leanprover-community/mathlib"@"001ffdc42920050657fd45bd2b8bfbec8eaaeb29"
/-!
# Ima... | apply Set.preimage_subset_preimage_iff | theorem Surjective.preimage_subset_preimage_iff {s t : Set β} (hf : Surjective f) :
f ⁻¹' s ⊆ f ⁻¹' t ↔ s ⊆ t := by
| Mathlib.Data.Set.Image.1348_0.IJFiTzmYGOCpPSd | theorem Surjective.preimage_subset_preimage_iff {s t : Set β} (hf : Surjective f) :
f ⁻¹' s ⊆ f ⁻¹' t ↔ s ⊆ t | Mathlib_Data_Set_Image |
case hs
α : Type u_3
β : Type u_2
ι : Sort u_1
f : α → β
s t : Set β
hf : Surjective f
⊢ s ⊆ range f | /-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura
-/
import Mathlib.Data.Set.Basic
#align_import data.set.image from "leanprover-community/mathlib"@"001ffdc42920050657fd45bd2b8bfbec8eaaeb29"
/-!
# Ima... | rw [hf.range_eq] | theorem Surjective.preimage_subset_preimage_iff {s t : Set β} (hf : Surjective f) :
f ⁻¹' s ⊆ f ⁻¹' t ↔ s ⊆ t := by
apply Set.preimage_subset_preimage_iff
| Mathlib.Data.Set.Image.1348_0.IJFiTzmYGOCpPSd | theorem Surjective.preimage_subset_preimage_iff {s t : Set β} (hf : Surjective f) :
f ⁻¹' s ⊆ f ⁻¹' t ↔ s ⊆ t | Mathlib_Data_Set_Image |
case hs
α : Type u_3
β : Type u_2
ι : Sort u_1
f : α → β
s t : Set β
hf : Surjective f
⊢ s ⊆ univ | /-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura
-/
import Mathlib.Data.Set.Basic
#align_import data.set.image from "leanprover-community/mathlib"@"001ffdc42920050657fd45bd2b8bfbec8eaaeb29"
/-!
# Ima... | apply subset_univ | theorem Surjective.preimage_subset_preimage_iff {s t : Set β} (hf : Surjective f) :
f ⁻¹' s ⊆ f ⁻¹' t ↔ s ⊆ t := by
apply Set.preimage_subset_preimage_iff
rw [hf.range_eq]
| Mathlib.Data.Set.Image.1348_0.IJFiTzmYGOCpPSd | theorem Surjective.preimage_subset_preimage_iff {s t : Set β} (hf : Surjective f) :
f ⁻¹' s ⊆ f ⁻¹' t ↔ s ⊆ t | Mathlib_Data_Set_Image |
α : Type u_2
β : Type u_3
ι : Sort u_1
f : α → β
hf : Injective f
s : Set α
⊢ (f '' s)ᶜ = f '' sᶜ ∪ (range f)ᶜ | /-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura
-/
import Mathlib.Data.Set.Basic
#align_import data.set.image from "leanprover-community/mathlib"@"001ffdc42920050657fd45bd2b8bfbec8eaaeb29"
/-!
# Ima... | ext y | theorem Injective.compl_image_eq (hf : Injective f) (s : Set α) :
(f '' s)ᶜ = f '' sᶜ ∪ (range f)ᶜ := by
| Mathlib.Data.Set.Image.1370_0.IJFiTzmYGOCpPSd | theorem Injective.compl_image_eq (hf : Injective f) (s : Set α) :
(f '' s)ᶜ = f '' sᶜ ∪ (range f)ᶜ | Mathlib_Data_Set_Image |
case h
α : Type u_2
β : Type u_3
ι : Sort u_1
f : α → β
hf : Injective f
s : Set α
y : β
⊢ y ∈ (f '' s)ᶜ ↔ y ∈ f '' sᶜ ∪ (range f)ᶜ | /-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura
-/
import Mathlib.Data.Set.Basic
#align_import data.set.image from "leanprover-community/mathlib"@"001ffdc42920050657fd45bd2b8bfbec8eaaeb29"
/-!
# Ima... | rcases em (y ∈ range f) with (⟨x, rfl⟩ | hx) | theorem Injective.compl_image_eq (hf : Injective f) (s : Set α) :
(f '' s)ᶜ = f '' sᶜ ∪ (range f)ᶜ := by
ext y
| Mathlib.Data.Set.Image.1370_0.IJFiTzmYGOCpPSd | theorem Injective.compl_image_eq (hf : Injective f) (s : Set α) :
(f '' s)ᶜ = f '' sᶜ ∪ (range f)ᶜ | Mathlib_Data_Set_Image |
case h.inl.intro
α : Type u_2
β : Type u_3
ι : Sort u_1
f : α → β
hf : Injective f
s : Set α
x : α
⊢ f x ∈ (f '' s)ᶜ ↔ f x ∈ f '' sᶜ ∪ (range f)ᶜ | /-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura
-/
import Mathlib.Data.Set.Basic
#align_import data.set.image from "leanprover-community/mathlib"@"001ffdc42920050657fd45bd2b8bfbec8eaaeb29"
/-!
# Ima... | simp [hf.eq_iff] | theorem Injective.compl_image_eq (hf : Injective f) (s : Set α) :
(f '' s)ᶜ = f '' sᶜ ∪ (range f)ᶜ := by
ext y
rcases em (y ∈ range f) with (⟨x, rfl⟩ | hx)
· | Mathlib.Data.Set.Image.1370_0.IJFiTzmYGOCpPSd | theorem Injective.compl_image_eq (hf : Injective f) (s : Set α) :
(f '' s)ᶜ = f '' sᶜ ∪ (range f)ᶜ | Mathlib_Data_Set_Image |
case h.inr
α : Type u_2
β : Type u_3
ι : Sort u_1
f : α → β
hf : Injective f
s : Set α
y : β
hx : y ∉ range f
⊢ y ∈ (f '' s)ᶜ ↔ y ∈ f '' sᶜ ∪ (range f)ᶜ | /-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura
-/
import Mathlib.Data.Set.Basic
#align_import data.set.image from "leanprover-community/mathlib"@"001ffdc42920050657fd45bd2b8bfbec8eaaeb29"
/-!
# Ima... | rw [mem_range, not_exists] at hx | theorem Injective.compl_image_eq (hf : Injective f) (s : Set α) :
(f '' s)ᶜ = f '' sᶜ ∪ (range f)ᶜ := by
ext y
rcases em (y ∈ range f) with (⟨x, rfl⟩ | hx)
· simp [hf.eq_iff]
· | Mathlib.Data.Set.Image.1370_0.IJFiTzmYGOCpPSd | theorem Injective.compl_image_eq (hf : Injective f) (s : Set α) :
(f '' s)ᶜ = f '' sᶜ ∪ (range f)ᶜ | Mathlib_Data_Set_Image |
case h.inr
α : Type u_2
β : Type u_3
ι : Sort u_1
f : α → β
hf : Injective f
s : Set α
y : β
hx : ∀ (x : α), ¬f x = y
⊢ y ∈ (f '' s)ᶜ ↔ y ∈ f '' sᶜ ∪ (range f)ᶜ | /-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura
-/
import Mathlib.Data.Set.Basic
#align_import data.set.image from "leanprover-community/mathlib"@"001ffdc42920050657fd45bd2b8bfbec8eaaeb29"
/-!
# Ima... | simp [hx] | theorem Injective.compl_image_eq (hf : Injective f) (s : Set α) :
(f '' s)ᶜ = f '' sᶜ ∪ (range f)ᶜ := by
ext y
rcases em (y ∈ range f) with (⟨x, rfl⟩ | hx)
· simp [hf.eq_iff]
· rw [mem_range, not_exists] at hx
| Mathlib.Data.Set.Image.1370_0.IJFiTzmYGOCpPSd | theorem Injective.compl_image_eq (hf : Injective f) (s : Set α) :
(f '' s)ᶜ = f '' sᶜ ∪ (range f)ᶜ | Mathlib_Data_Set_Image |
α : Type u_2
β : Type u_3
ι : Sort u_1
f : α → β
g : β → α
h : LeftInverse g f
s : Set α
⊢ g '' (f '' s) = s | /-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura
-/
import Mathlib.Data.Set.Basic
#align_import data.set.image from "leanprover-community/mathlib"@"001ffdc42920050657fd45bd2b8bfbec8eaaeb29"
/-!
# Ima... | rw [← image_comp, h.comp_eq_id, image_id] | theorem LeftInverse.image_image {g : β → α} (h : LeftInverse g f) (s : Set α) : g '' (f '' s) = s :=
by | Mathlib.Data.Set.Image.1379_0.IJFiTzmYGOCpPSd | theorem LeftInverse.image_image {g : β → α} (h : LeftInverse g f) (s : Set α) : g '' (f '' s) = s | Mathlib_Data_Set_Image |
α : Type u_2
β : Type u_3
ι : Sort u_1
f : α → β
g : β → α
h : LeftInverse g f
s : Set α
⊢ f ⁻¹' (g ⁻¹' s) = s | /-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura
-/
import Mathlib.Data.Set.Basic
#align_import data.set.image from "leanprover-community/mathlib"@"001ffdc42920050657fd45bd2b8bfbec8eaaeb29"
/-!
# Ima... | rw [← preimage_comp, h.comp_eq_id, preimage_id] | theorem LeftInverse.preimage_preimage {g : β → α} (h : LeftInverse g f) (s : Set α) :
f ⁻¹' (g ⁻¹' s) = s := by | Mathlib.Data.Set.Image.1383_0.IJFiTzmYGOCpPSd | theorem LeftInverse.preimage_preimage {g : β → α} (h : LeftInverse g f) (s : Set α) :
f ⁻¹' (g ⁻¹' s) = s | Mathlib_Data_Set_Image |
α : Type u_1
s t : Set α
h : t ⊆ s
⊢ val '' {x | ↑x ∈ t} = t | /-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura
-/
import Mathlib.Data.Set.Basic
#align_import data.set.image from "leanprover-community/mathlib"@"001ffdc42920050657fd45bd2b8bfbec8eaaeb29"
/-!
# Ima... | ext x | @[simp]
theorem coe_image_of_subset {s t : Set α} (h : t ⊆ s) : (↑) '' { x : ↥s | ↑x ∈ t } = t := by
| Mathlib.Data.Set.Image.1415_0.IJFiTzmYGOCpPSd | @[simp]
theorem coe_image_of_subset {s t : Set α} (h : t ⊆ s) : (↑) '' { x : ↥s | ↑x ∈ t } = t | Mathlib_Data_Set_Image |
case h
α : Type u_1
s t : Set α
h : t ⊆ s
x : α
⊢ x ∈ val '' {x | ↑x ∈ t} ↔ x ∈ t | /-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura
-/
import Mathlib.Data.Set.Basic
#align_import data.set.image from "leanprover-community/mathlib"@"001ffdc42920050657fd45bd2b8bfbec8eaaeb29"
/-!
# Ima... | rw [mem_image] | @[simp]
theorem coe_image_of_subset {s t : Set α} (h : t ⊆ s) : (↑) '' { x : ↥s | ↑x ∈ t } = t := by
ext x
| Mathlib.Data.Set.Image.1415_0.IJFiTzmYGOCpPSd | @[simp]
theorem coe_image_of_subset {s t : Set α} (h : t ⊆ s) : (↑) '' { x : ↥s | ↑x ∈ t } = t | Mathlib_Data_Set_Image |
case h
α : Type u_1
s t : Set α
h : t ⊆ s
x : α
⊢ (∃ x_1 ∈ {x | ↑x ∈ t}, ↑x_1 = x) ↔ x ∈ t | /-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura
-/
import Mathlib.Data.Set.Basic
#align_import data.set.image from "leanprover-community/mathlib"@"001ffdc42920050657fd45bd2b8bfbec8eaaeb29"
/-!
# Ima... | exact ⟨fun ⟨_, hx', hx⟩ => hx ▸ hx', fun hx => ⟨⟨x, h hx⟩, hx, rfl⟩⟩ | @[simp]
theorem coe_image_of_subset {s t : Set α} (h : t ⊆ s) : (↑) '' { x : ↥s | ↑x ∈ t } = t := by
ext x
rw [mem_image]
| Mathlib.Data.Set.Image.1415_0.IJFiTzmYGOCpPSd | @[simp]
theorem coe_image_of_subset {s t : Set α} (h : t ⊆ s) : (↑) '' { x : ↥s | ↑x ∈ t } = t | Mathlib_Data_Set_Image |
α : Type u_1
s : Set α
⊢ range val = s | /-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura
-/
import Mathlib.Data.Set.Basic
#align_import data.set.image from "leanprover-community/mathlib"@"001ffdc42920050657fd45bd2b8bfbec8eaaeb29"
/-!
# Ima... | rw [← image_univ] | theorem range_coe {s : Set α} : range ((↑) : s → α) = s := by
| Mathlib.Data.Set.Image.1422_0.IJFiTzmYGOCpPSd | theorem range_coe {s : Set α} : range ((↑) : s → α) = s | Mathlib_Data_Set_Image |
α : Type u_1
s : Set α
⊢ val '' univ = s | /-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura
-/
import Mathlib.Data.Set.Basic
#align_import data.set.image from "leanprover-community/mathlib"@"001ffdc42920050657fd45bd2b8bfbec8eaaeb29"
/-!
# Ima... | simp [-image_univ, coe_image] | theorem range_coe {s : Set α} : range ((↑) : s → α) = s := by
rw [← image_univ]
| Mathlib.Data.Set.Image.1422_0.IJFiTzmYGOCpPSd | theorem range_coe {s : Set α} : range ((↑) : s → α) = s | Mathlib_Data_Set_Image |
α : Type u_1
s : Set α
⊢ val ⁻¹' s = univ | /-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura
-/
import Mathlib.Data.Set.Basic
#align_import data.set.image from "leanprover-community/mathlib"@"001ffdc42920050657fd45bd2b8bfbec8eaaeb29"
/-!
# Ima... | rw [← preimage_range, range_coe] | @[simp]
theorem coe_preimage_self (s : Set α) : ((↑) : s → α) ⁻¹' s = univ := by
| Mathlib.Data.Set.Image.1442_0.IJFiTzmYGOCpPSd | @[simp]
theorem coe_preimage_self (s : Set α) : ((↑) : s → α) ⁻¹' s = univ | Mathlib_Data_Set_Image |
α : Type u_1
s : Set α
t : Set ↑s
x : α
x✝ : x ∈ val '' t
y : { x // x ∈ s }
left✝ : y ∈ t
yvaleq : ↑y = x
⊢ x ∈ s | /-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura
-/
import Mathlib.Data.Set.Basic
#align_import data.set.image from "leanprover-community/mathlib"@"001ffdc42920050657fd45bd2b8bfbec8eaaeb29"
/-!
# Ima... | rw [← yvaleq] | theorem coe_image_subset (s : Set α) (t : Set s) : ((↑) : s → α) '' t ⊆ s :=
fun x ⟨y, _, yvaleq⟩ => by
| Mathlib.Data.Set.Image.1451_0.IJFiTzmYGOCpPSd | theorem coe_image_subset (s : Set α) (t : Set s) : ((↑) : s → α) '' t ⊆ s | Mathlib_Data_Set_Image |
α : Type u_1
s : Set α
t : Set ↑s
x : α
x✝ : x ∈ val '' t
y : { x // x ∈ s }
left✝ : y ∈ t
yvaleq : ↑y = x
⊢ ↑y ∈ s | /-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura
-/
import Mathlib.Data.Set.Basic
#align_import data.set.image from "leanprover-community/mathlib"@"001ffdc42920050657fd45bd2b8bfbec8eaaeb29"
/-!
# Ima... | exact y.property | theorem coe_image_subset (s : Set α) (t : Set s) : ((↑) : s → α) '' t ⊆ s :=
fun x ⟨y, _, yvaleq⟩ => by
rw [← yvaleq]; | Mathlib.Data.Set.Image.1451_0.IJFiTzmYGOCpPSd | theorem coe_image_subset (s : Set α) (t : Set s) : ((↑) : s → α) '' t ⊆ s | Mathlib_Data_Set_Image |
α : Type u_1
s t u : Set α
⊢ val ⁻¹' t = val ⁻¹' u ↔ t ∩ s = u ∩ s | /-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura
-/
import Mathlib.Data.Set.Basic
#align_import data.set.image from "leanprover-community/mathlib"@"001ffdc42920050657fd45bd2b8bfbec8eaaeb29"
/-!
# Ima... | rw [← image_preimage_coe, ← image_preimage_coe, coe_injective.image_injective.eq_iff] | theorem preimage_coe_eq_preimage_coe_iff {s t u : Set α} :
((↑) : s → α) ⁻¹' t = ((↑) : s → α) ⁻¹' u ↔ t ∩ s = u ∩ s := by
| Mathlib.Data.Set.Image.1469_0.IJFiTzmYGOCpPSd | theorem preimage_coe_eq_preimage_coe_iff {s t u : Set α} :
((↑) : s → α) ⁻¹' t = ((↑) : s → α) ⁻¹' u ↔ t ∩ s = u ∩ s | Mathlib_Data_Set_Image |
α : Type u_1
s t : Set α
⊢ val ⁻¹' (t ∩ s) = val ⁻¹' t | /-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura
-/
import Mathlib.Data.Set.Basic
#align_import data.set.image from "leanprover-community/mathlib"@"001ffdc42920050657fd45bd2b8bfbec8eaaeb29"
/-!
# Ima... | rw [preimage_coe_eq_preimage_coe_iff, inter_assoc, inter_self] | theorem preimage_coe_inter_self (s t : Set α) :
((↑) : s → α) ⁻¹' (t ∩ s) = ((↑) : s → α) ⁻¹' t := by
| Mathlib.Data.Set.Image.1476_0.IJFiTzmYGOCpPSd | theorem preimage_coe_inter_self (s t : Set α) :
((↑) : s → α) ⁻¹' (t ∩ s) = ((↑) : s → α) ⁻¹' t | Mathlib_Data_Set_Image |
α : Type u_1
t : Set α
p : Set α → Prop
⊢ (∃ s, p (val '' s)) ↔ ∃ s ⊆ t, p s | /-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura
-/
import Mathlib.Data.Set.Basic
#align_import data.set.image from "leanprover-community/mathlib"@"001ffdc42920050657fd45bd2b8bfbec8eaaeb29"
/-!
# Ima... | rw [← exists_subset_range_and_iff, range_coe] | theorem exists_set_subtype {t : Set α} (p : Set α → Prop) :
(∃ s : Set t, p (((↑) : t → α) '' s)) ↔ ∃ s : Set α, s ⊆ t ∧ p s := by
| Mathlib.Data.Set.Image.1486_0.IJFiTzmYGOCpPSd | theorem exists_set_subtype {t : Set α} (p : Set α → Prop) :
(∃ s : Set t, p (((↑) : t → α) '' s)) ↔ ∃ s : Set α, s ⊆ t ∧ p s | Mathlib_Data_Set_Image |
α : Type u_1
t : Set α
p : Set α → Prop
⊢ (∀ (s : Set ↑t), p (val '' s)) ↔ ∀ s ⊆ t, p s | /-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura
-/
import Mathlib.Data.Set.Basic
#align_import data.set.image from "leanprover-community/mathlib"@"001ffdc42920050657fd45bd2b8bfbec8eaaeb29"
/-!
# Ima... | rw [← forall_subset_range_iff, range_coe] | theorem forall_set_subtype {t : Set α} (p : Set α → Prop) :
(∀ s : Set t, p (((↑) : t → α) '' s)) ↔ ∀ s : Set α, s ⊆ t → p s := by
| Mathlib.Data.Set.Image.1491_0.IJFiTzmYGOCpPSd | theorem forall_set_subtype {t : Set α} (p : Set α → Prop) :
(∀ s : Set t, p (((↑) : t → α) '' s)) ↔ ∀ s : Set α, s ⊆ t → p s | Mathlib_Data_Set_Image |
α : Type u_1
s t : Set α
⊢ Set.Nonempty (val ⁻¹' t) ↔ Set.Nonempty (s ∩ t) | /-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura
-/
import Mathlib.Data.Set.Basic
#align_import data.set.image from "leanprover-community/mathlib"@"001ffdc42920050657fd45bd2b8bfbec8eaaeb29"
/-!
# Ima... | rw [inter_comm, ← image_preimage_coe, nonempty_image_iff] | theorem preimage_coe_nonempty {s t : Set α} :
(((↑) : s → α) ⁻¹' t).Nonempty ↔ (s ∩ t).Nonempty := by
| Mathlib.Data.Set.Image.1495_0.IJFiTzmYGOCpPSd | theorem preimage_coe_nonempty {s t : Set α} :
(((↑) : s → α) ⁻¹' t).Nonempty ↔ (s ∩ t).Nonempty | Mathlib_Data_Set_Image |
α : Type u_1
s t : Set α
⊢ val ⁻¹' t = ∅ ↔ s ∩ t = ∅ | /-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura
-/
import Mathlib.Data.Set.Basic
#align_import data.set.image from "leanprover-community/mathlib"@"001ffdc42920050657fd45bd2b8bfbec8eaaeb29"
/-!
# Ima... | simp [← not_nonempty_iff_eq_empty, preimage_coe_nonempty] | theorem preimage_coe_eq_empty {s t : Set α} : ((↑) : s → α) ⁻¹' t = ∅ ↔ s ∩ t = ∅ := by
| Mathlib.Data.Set.Image.1500_0.IJFiTzmYGOCpPSd | theorem preimage_coe_eq_empty {s t : Set α} : ((↑) : s → α) ⁻¹' t = ∅ ↔ s ∩ t = ∅ | Mathlib_Data_Set_Image |
α : Type u_1
β : Type u_2
f : Option α → β
⊢ Injective f ↔ Injective (f ∘ some) ∧ f none ∉ range (f ∘ some) | /-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura
-/
import Mathlib.Data.Set.Basic
#align_import data.set.image from "leanprover-community/mathlib"@"001ffdc42920050657fd45bd2b8bfbec8eaaeb29"
/-!
# Ima... | simp only [mem_range, not_exists, (· ∘ ·)] | theorem injective_iff {α β} {f : Option α → β} :
Injective f ↔ Injective (f ∘ some) ∧ f none ∉ range (f ∘ some) := by
| Mathlib.Data.Set.Image.1525_0.IJFiTzmYGOCpPSd | theorem injective_iff {α β} {f : Option α → β} :
Injective f ↔ Injective (f ∘ some) ∧ f none ∉ range (f ∘ some) | Mathlib_Data_Set_Image |
α : Type u_1
β : Type u_2
f : Option α → β
⊢ Injective f ↔ (Injective fun x => f (some x)) ∧ ∀ (x : α), ¬f (some x) = f none | /-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura
-/
import Mathlib.Data.Set.Basic
#align_import data.set.image from "leanprover-community/mathlib"@"001ffdc42920050657fd45bd2b8bfbec8eaaeb29"
/-!
# Ima... | refine'
⟨fun hf => ⟨hf.comp (Option.some_injective _), fun x => hf.ne <| Option.some_ne_none _⟩, _⟩ | theorem injective_iff {α β} {f : Option α → β} :
Injective f ↔ Injective (f ∘ some) ∧ f none ∉ range (f ∘ some) := by
simp only [mem_range, not_exists, (· ∘ ·)]
| Mathlib.Data.Set.Image.1525_0.IJFiTzmYGOCpPSd | theorem injective_iff {α β} {f : Option α → β} :
Injective f ↔ Injective (f ∘ some) ∧ f none ∉ range (f ∘ some) | Mathlib_Data_Set_Image |
α : Type u_1
β : Type u_2
f : Option α → β
⊢ ((Injective fun x => f (some x)) ∧ ∀ (x : α), ¬f (some x) = f none) → Injective f | /-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura
-/
import Mathlib.Data.Set.Basic
#align_import data.set.image from "leanprover-community/mathlib"@"001ffdc42920050657fd45bd2b8bfbec8eaaeb29"
/-!
# Ima... | rintro ⟨h_some, h_none⟩ (_ | a) (_ | b) hab | theorem injective_iff {α β} {f : Option α → β} :
Injective f ↔ Injective (f ∘ some) ∧ f none ∉ range (f ∘ some) := by
simp only [mem_range, not_exists, (· ∘ ·)]
refine'
⟨fun hf => ⟨hf.comp (Option.some_injective _), fun x => hf.ne <| Option.some_ne_none _⟩, _⟩
| Mathlib.Data.Set.Image.1525_0.IJFiTzmYGOCpPSd | theorem injective_iff {α β} {f : Option α → β} :
Injective f ↔ Injective (f ∘ some) ∧ f none ∉ range (f ∘ some) | Mathlib_Data_Set_Image |
case intro.none.none
α : Type u_1
β : Type u_2
f : Option α → β
h_some : Injective fun x => f (some x)
h_none : ∀ (x : α), ¬f (some x) = f none
hab : f none = f none
⊢ none = none
case intro.none.some
α : Type u_1
β : Type u_2
f : Option α → β
h_some : Injective fun x => f (some x)
h_none : ∀ (x : α), ¬f (some x) = f n... | /-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura
-/
import Mathlib.Data.Set.Basic
#align_import data.set.image from "leanprover-community/mathlib"@"001ffdc42920050657fd45bd2b8bfbec8eaaeb29"
/-!
# Ima... | exacts [rfl, (h_none _ hab.symm).elim, (h_none _ hab).elim, congr_arg some (h_some hab)] | theorem injective_iff {α β} {f : Option α → β} :
Injective f ↔ Injective (f ∘ some) ∧ f none ∉ range (f ∘ some) := by
simp only [mem_range, not_exists, (· ∘ ·)]
refine'
⟨fun hf => ⟨hf.comp (Option.some_injective _), fun x => hf.ne <| Option.some_ne_none _⟩, _⟩
rintro ⟨h_some, h_none⟩ (_ | a) (_ | b) hab
... | Mathlib.Data.Set.Image.1525_0.IJFiTzmYGOCpPSd | theorem injective_iff {α β} {f : Option α → β} :
Injective f ↔ Injective (f ∘ some) ∧ f none ∉ range (f ∘ some) | Mathlib_Data_Set_Image |
α : Type u
β : Type v
f : α → β
⊢ Injective (preimage f) ↔ Surjective f | /-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura
-/
import Mathlib.Data.Set.Basic
#align_import data.set.image from "leanprover-community/mathlib"@"001ffdc42920050657fd45bd2b8bfbec8eaaeb29"
/-!
# Ima... | refine' ⟨fun h y => _, Surjective.preimage_injective⟩ | @[simp]
theorem preimage_injective : Injective (preimage f) ↔ Surjective f := by
| Mathlib.Data.Set.Image.1561_0.IJFiTzmYGOCpPSd | @[simp]
theorem preimage_injective : Injective (preimage f) ↔ Surjective f | Mathlib_Data_Set_Image |
α : Type u
β : Type v
f : α → β
h : Injective (preimage f)
y : β
⊢ ∃ a, f a = y | /-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura
-/
import Mathlib.Data.Set.Basic
#align_import data.set.image from "leanprover-community/mathlib"@"001ffdc42920050657fd45bd2b8bfbec8eaaeb29"
/-!
# Ima... | obtain ⟨x, hx⟩ : (f ⁻¹' {y}).Nonempty := by
rw [h.nonempty_apply_iff preimage_empty]
apply singleton_nonempty | @[simp]
theorem preimage_injective : Injective (preimage f) ↔ Surjective f := by
refine' ⟨fun h y => _, Surjective.preimage_injective⟩
| Mathlib.Data.Set.Image.1561_0.IJFiTzmYGOCpPSd | @[simp]
theorem preimage_injective : Injective (preimage f) ↔ Surjective f | Mathlib_Data_Set_Image |
α : Type u
β : Type v
f : α → β
h : Injective (preimage f)
y : β
⊢ Set.Nonempty (f ⁻¹' {y}) | /-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura
-/
import Mathlib.Data.Set.Basic
#align_import data.set.image from "leanprover-community/mathlib"@"001ffdc42920050657fd45bd2b8bfbec8eaaeb29"
/-!
# Ima... | rw [h.nonempty_apply_iff preimage_empty] | @[simp]
theorem preimage_injective : Injective (preimage f) ↔ Surjective f := by
refine' ⟨fun h y => _, Surjective.preimage_injective⟩
obtain ⟨x, hx⟩ : (f ⁻¹' {y}).Nonempty := by
| Mathlib.Data.Set.Image.1561_0.IJFiTzmYGOCpPSd | @[simp]
theorem preimage_injective : Injective (preimage f) ↔ Surjective f | Mathlib_Data_Set_Image |
α : Type u
β : Type v
f : α → β
h : Injective (preimage f)
y : β
⊢ Set.Nonempty {y} | /-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura
-/
import Mathlib.Data.Set.Basic
#align_import data.set.image from "leanprover-community/mathlib"@"001ffdc42920050657fd45bd2b8bfbec8eaaeb29"
/-!
# Ima... | apply singleton_nonempty | @[simp]
theorem preimage_injective : Injective (preimage f) ↔ Surjective f := by
refine' ⟨fun h y => _, Surjective.preimage_injective⟩
obtain ⟨x, hx⟩ : (f ⁻¹' {y}).Nonempty := by
rw [h.nonempty_apply_iff preimage_empty]
| Mathlib.Data.Set.Image.1561_0.IJFiTzmYGOCpPSd | @[simp]
theorem preimage_injective : Injective (preimage f) ↔ Surjective f | Mathlib_Data_Set_Image |
case intro
α : Type u
β : Type v
f : α → β
h : Injective (preimage f)
y : β
x : α
hx : x ∈ f ⁻¹' {y}
⊢ ∃ a, f a = y | /-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura
-/
import Mathlib.Data.Set.Basic
#align_import data.set.image from "leanprover-community/mathlib"@"001ffdc42920050657fd45bd2b8bfbec8eaaeb29"
/-!
# Ima... | exact ⟨x, hx⟩ | @[simp]
theorem preimage_injective : Injective (preimage f) ↔ Surjective f := by
refine' ⟨fun h y => _, Surjective.preimage_injective⟩
obtain ⟨x, hx⟩ : (f ⁻¹' {y}).Nonempty := by
rw [h.nonempty_apply_iff preimage_empty]
apply singleton_nonempty
| Mathlib.Data.Set.Image.1561_0.IJFiTzmYGOCpPSd | @[simp]
theorem preimage_injective : Injective (preimage f) ↔ Surjective f | Mathlib_Data_Set_Image |
α : Type u
β : Type v
f : α → β
⊢ Surjective (preimage f) ↔ Injective f | /-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura
-/
import Mathlib.Data.Set.Basic
#align_import data.set.image from "leanprover-community/mathlib"@"001ffdc42920050657fd45bd2b8bfbec8eaaeb29"
/-!
# Ima... | refine' ⟨fun h x x' hx => _, Injective.preimage_surjective⟩ | @[simp]
theorem preimage_surjective : Surjective (preimage f) ↔ Injective f := by
| Mathlib.Data.Set.Image.1570_0.IJFiTzmYGOCpPSd | @[simp]
theorem preimage_surjective : Surjective (preimage f) ↔ Injective f | Mathlib_Data_Set_Image |
α : Type u
β : Type v
f : α → β
h : Surjective (preimage f)
x x' : α
hx : f x = f x'
⊢ x = x' | /-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura
-/
import Mathlib.Data.Set.Basic
#align_import data.set.image from "leanprover-community/mathlib"@"001ffdc42920050657fd45bd2b8bfbec8eaaeb29"
/-!
# Ima... | cases' h {x} with s hs | @[simp]
theorem preimage_surjective : Surjective (preimage f) ↔ Injective f := by
refine' ⟨fun h x x' hx => _, Injective.preimage_surjective⟩
| Mathlib.Data.Set.Image.1570_0.IJFiTzmYGOCpPSd | @[simp]
theorem preimage_surjective : Surjective (preimage f) ↔ Injective f | Mathlib_Data_Set_Image |
case intro
α : Type u
β : Type v
f : α → β
h : Surjective (preimage f)
x x' : α
hx : f x = f x'
s : Set β
hs : f ⁻¹' s = {x}
⊢ x = x' | /-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura
-/
import Mathlib.Data.Set.Basic
#align_import data.set.image from "leanprover-community/mathlib"@"001ffdc42920050657fd45bd2b8bfbec8eaaeb29"
/-!
# Ima... | have := mem_singleton x | @[simp]
theorem preimage_surjective : Surjective (preimage f) ↔ Injective f := by
refine' ⟨fun h x x' hx => _, Injective.preimage_surjective⟩
cases' h {x} with s hs; | Mathlib.Data.Set.Image.1570_0.IJFiTzmYGOCpPSd | @[simp]
theorem preimage_surjective : Surjective (preimage f) ↔ Injective f | Mathlib_Data_Set_Image |
case intro
α : Type u
β : Type v
f : α → β
h : Surjective (preimage f)
x x' : α
hx : f x = f x'
s : Set β
hs : f ⁻¹' s = {x}
this : x ∈ {x}
⊢ x = x' | /-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura
-/
import Mathlib.Data.Set.Basic
#align_import data.set.image from "leanprover-community/mathlib"@"001ffdc42920050657fd45bd2b8bfbec8eaaeb29"
/-!
# Ima... | rwa [← hs, mem_preimage, hx, ← mem_preimage, hs, mem_singleton_iff, eq_comm] at this | @[simp]
theorem preimage_surjective : Surjective (preimage f) ↔ Injective f := by
refine' ⟨fun h x x' hx => _, Injective.preimage_surjective⟩
cases' h {x} with s hs; have := mem_singleton x
| Mathlib.Data.Set.Image.1570_0.IJFiTzmYGOCpPSd | @[simp]
theorem preimage_surjective : Surjective (preimage f) ↔ Injective f | Mathlib_Data_Set_Image |
α : Type u
β : Type v
f : α → β
⊢ Surjective (image f) ↔ Surjective f | /-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura
-/
import Mathlib.Data.Set.Basic
#align_import data.set.image from "leanprover-community/mathlib"@"001ffdc42920050657fd45bd2b8bfbec8eaaeb29"
/-!
# Ima... | refine' ⟨fun h y => _, Surjective.image_surjective⟩ | @[simp]
theorem image_surjective : Surjective (image f) ↔ Surjective f := by
| Mathlib.Data.Set.Image.1577_0.IJFiTzmYGOCpPSd | @[simp]
theorem image_surjective : Surjective (image f) ↔ Surjective f | Mathlib_Data_Set_Image |
α : Type u
β : Type v
f : α → β
h : Surjective (image f)
y : β
⊢ ∃ a, f a = y | /-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura
-/
import Mathlib.Data.Set.Basic
#align_import data.set.image from "leanprover-community/mathlib"@"001ffdc42920050657fd45bd2b8bfbec8eaaeb29"
/-!
# Ima... | cases' h {y} with s hs | @[simp]
theorem image_surjective : Surjective (image f) ↔ Surjective f := by
refine' ⟨fun h y => _, Surjective.image_surjective⟩
| Mathlib.Data.Set.Image.1577_0.IJFiTzmYGOCpPSd | @[simp]
theorem image_surjective : Surjective (image f) ↔ Surjective f | Mathlib_Data_Set_Image |
case intro
α : Type u
β : Type v
f : α → β
h : Surjective (image f)
y : β
s : Set α
hs : f '' s = {y}
⊢ ∃ a, f a = y | /-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura
-/
import Mathlib.Data.Set.Basic
#align_import data.set.image from "leanprover-community/mathlib"@"001ffdc42920050657fd45bd2b8bfbec8eaaeb29"
/-!
# Ima... | have := mem_singleton y | @[simp]
theorem image_surjective : Surjective (image f) ↔ Surjective f := by
refine' ⟨fun h y => _, Surjective.image_surjective⟩
cases' h {y} with s hs
| Mathlib.Data.Set.Image.1577_0.IJFiTzmYGOCpPSd | @[simp]
theorem image_surjective : Surjective (image f) ↔ Surjective f | Mathlib_Data_Set_Image |
case intro
α : Type u
β : Type v
f : α → β
h : Surjective (image f)
y : β
s : Set α
hs : f '' s = {y}
this : y ∈ {y}
⊢ ∃ a, f a = y | /-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura
-/
import Mathlib.Data.Set.Basic
#align_import data.set.image from "leanprover-community/mathlib"@"001ffdc42920050657fd45bd2b8bfbec8eaaeb29"
/-!
# Ima... | rw [← hs] at this | @[simp]
theorem image_surjective : Surjective (image f) ↔ Surjective f := by
refine' ⟨fun h y => _, Surjective.image_surjective⟩
cases' h {y} with s hs
have := mem_singleton y; | Mathlib.Data.Set.Image.1577_0.IJFiTzmYGOCpPSd | @[simp]
theorem image_surjective : Surjective (image f) ↔ Surjective f | Mathlib_Data_Set_Image |
case intro
α : Type u
β : Type v
f : α → β
h : Surjective (image f)
y : β
s : Set α
hs : f '' s = {y}
this : y ∈ f '' s
⊢ ∃ a, f a = y | /-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura
-/
import Mathlib.Data.Set.Basic
#align_import data.set.image from "leanprover-community/mathlib"@"001ffdc42920050657fd45bd2b8bfbec8eaaeb29"
/-!
# Ima... | rcases this with ⟨x, _, hx⟩ | @[simp]
theorem image_surjective : Surjective (image f) ↔ Surjective f := by
refine' ⟨fun h y => _, Surjective.image_surjective⟩
cases' h {y} with s hs
have := mem_singleton y; rw [← hs] at this; | Mathlib.Data.Set.Image.1577_0.IJFiTzmYGOCpPSd | @[simp]
theorem image_surjective : Surjective (image f) ↔ Surjective f | Mathlib_Data_Set_Image |
case intro.intro.intro
α : Type u
β : Type v
f : α → β
h : Surjective (image f)
y : β
s : Set α
hs : f '' s = {y}
x : α
left✝ : x ∈ s
hx : f x = y
⊢ ∃ a, f a = y | /-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura
-/
import Mathlib.Data.Set.Basic
#align_import data.set.image from "leanprover-community/mathlib"@"001ffdc42920050657fd45bd2b8bfbec8eaaeb29"
/-!
# Ima... | exact ⟨x, hx⟩ | @[simp]
theorem image_surjective : Surjective (image f) ↔ Surjective f := by
refine' ⟨fun h y => _, Surjective.image_surjective⟩
cases' h {y} with s hs
have := mem_singleton y; rw [← hs] at this; rcases this with ⟨x, _, hx⟩
| Mathlib.Data.Set.Image.1577_0.IJFiTzmYGOCpPSd | @[simp]
theorem image_surjective : Surjective (image f) ↔ Surjective f | Mathlib_Data_Set_Image |
α : Type u
β : Type v
f : α → β
⊢ Injective (image f) ↔ Injective f | /-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura
-/
import Mathlib.Data.Set.Basic
#align_import data.set.image from "leanprover-community/mathlib"@"001ffdc42920050657fd45bd2b8bfbec8eaaeb29"
/-!
# Ima... | refine' ⟨fun h x x' hx => _, Injective.image_injective⟩ | @[simp]
theorem image_injective : Injective (image f) ↔ Injective f := by
| Mathlib.Data.Set.Image.1585_0.IJFiTzmYGOCpPSd | @[simp]
theorem image_injective : Injective (image f) ↔ Injective f | Mathlib_Data_Set_Image |
α : Type u
β : Type v
f : α → β
h : Injective (image f)
x x' : α
hx : f x = f x'
⊢ x = x' | /-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura
-/
import Mathlib.Data.Set.Basic
#align_import data.set.image from "leanprover-community/mathlib"@"001ffdc42920050657fd45bd2b8bfbec8eaaeb29"
/-!
# Ima... | rw [← singleton_eq_singleton_iff] | @[simp]
theorem image_injective : Injective (image f) ↔ Injective f := by
refine' ⟨fun h x x' hx => _, Injective.image_injective⟩
| Mathlib.Data.Set.Image.1585_0.IJFiTzmYGOCpPSd | @[simp]
theorem image_injective : Injective (image f) ↔ Injective f | Mathlib_Data_Set_Image |
α : Type u
β : Type v
f : α → β
h : Injective (image f)
x x' : α
hx : f x = f x'
⊢ {x} = {x'} | /-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura
-/
import Mathlib.Data.Set.Basic
#align_import data.set.image from "leanprover-community/mathlib"@"001ffdc42920050657fd45bd2b8bfbec8eaaeb29"
/-!
# Ima... | apply h | @[simp]
theorem image_injective : Injective (image f) ↔ Injective f := by
refine' ⟨fun h x x' hx => _, Injective.image_injective⟩
rw [← singleton_eq_singleton_iff]; | Mathlib.Data.Set.Image.1585_0.IJFiTzmYGOCpPSd | @[simp]
theorem image_injective : Injective (image f) ↔ Injective f | Mathlib_Data_Set_Image |
case a
α : Type u
β : Type v
f : α → β
h : Injective (image f)
x x' : α
hx : f x = f x'
⊢ f '' {x} = f '' {x'} | /-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura
-/
import Mathlib.Data.Set.Basic
#align_import data.set.image from "leanprover-community/mathlib"@"001ffdc42920050657fd45bd2b8bfbec8eaaeb29"
/-!
# Ima... | rw [image_singleton, image_singleton, hx] | @[simp]
theorem image_injective : Injective (image f) ↔ Injective f := by
refine' ⟨fun h x x' hx => _, Injective.image_injective⟩
rw [← singleton_eq_singleton_iff]; apply h
| Mathlib.Data.Set.Image.1585_0.IJFiTzmYGOCpPSd | @[simp]
theorem image_injective : Injective (image f) ↔ Injective f | Mathlib_Data_Set_Image |
α : Type u
β : Type v
f✝ f : α → β
hf : Bijective f
s : Set β
t : Set α
⊢ f ⁻¹' s = t ↔ s = f '' t | /-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura
-/
import Mathlib.Data.Set.Basic
#align_import data.set.image from "leanprover-community/mathlib"@"001ffdc42920050657fd45bd2b8bfbec8eaaeb29"
/-!
# Ima... | rw [← image_eq_image hf.1, hf.2.image_preimage] | theorem preimage_eq_iff_eq_image {f : α → β} (hf : Bijective f) {s t} : f ⁻¹' s = t ↔ s = f '' t :=
by | Mathlib.Data.Set.Image.1592_0.IJFiTzmYGOCpPSd | theorem preimage_eq_iff_eq_image {f : α → β} (hf : Bijective f) {s t} : f ⁻¹' s = t ↔ s = f '' t | Mathlib_Data_Set_Image |
α : Type u
β : Type v
f✝ f : α → β
hf : Bijective f
s : Set α
t : Set β
⊢ s = f ⁻¹' t ↔ f '' s = t | /-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura
-/
import Mathlib.Data.Set.Basic
#align_import data.set.image from "leanprover-community/mathlib"@"001ffdc42920050657fd45bd2b8bfbec8eaaeb29"
/-!
# Ima... | rw [← image_eq_image hf.1, hf.2.image_preimage] | theorem eq_preimage_iff_image_eq {f : α → β} (hf : Bijective f) {s t} : s = f ⁻¹' t ↔ f '' s = t :=
by | Mathlib.Data.Set.Image.1596_0.IJFiTzmYGOCpPSd | theorem eq_preimage_iff_image_eq {f : α → β} (hf : Bijective f) {s t} : s = f ⁻¹' t ↔ f '' s = t | Mathlib_Data_Set_Image |
α : Type u_1
β : Type u_2
γ : Type u_3
f✝ : α → β
s✝ t✝ : Set α
f : β → α
g : γ → α
s : Set β
t : Set γ
h : ∀ b ∈ s, ∀ c ∈ t, f b ≠ g c
⊢ f '' s ⊓ g '' t ≤ ⊥ | /-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura
-/
import Mathlib.Data.Set.Basic
#align_import data.set.image from "leanprover-community/mathlib"@"001ffdc42920050657fd45bd2b8bfbec8eaaeb29"
/-!
# Ima... | rintro a ⟨⟨b, hb, eq⟩, c, hc, rfl⟩ | theorem disjoint_image_image {f : β → α} {g : γ → α} {s : Set β} {t : Set γ}
(h : ∀ b ∈ s, ∀ c ∈ t, f b ≠ g c) : Disjoint (f '' s) (g '' t) :=
disjoint_iff_inf_le.mpr <| by | Mathlib.Data.Set.Image.1616_0.IJFiTzmYGOCpPSd | theorem disjoint_image_image {f : β → α} {g : γ → α} {s : Set β} {t : Set γ}
(h : ∀ b ∈ s, ∀ c ∈ t, f b ≠ g c) : Disjoint (f '' s) (g '' t) | Mathlib_Data_Set_Image |
case intro.intro.intro.intro.intro
α : Type u_1
β : Type u_2
γ : Type u_3
f✝ : α → β
s✝ t✝ : Set α
f : β → α
g : γ → α
s : Set β
t : Set γ
h : ∀ b ∈ s, ∀ c ∈ t, f b ≠ g c
b : β
hb : b ∈ s
c : γ
hc : c ∈ t
eq : f b = g c
⊢ g c ∈ ⊥ | /-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura
-/
import Mathlib.Data.Set.Basic
#align_import data.set.image from "leanprover-community/mathlib"@"001ffdc42920050657fd45bd2b8bfbec8eaaeb29"
/-!
# Ima... | exact h b hb c hc eq | theorem disjoint_image_image {f : β → α} {g : γ → α} {s : Set β} {t : Set γ}
(h : ∀ b ∈ s, ∀ c ∈ t, f b ≠ g c) : Disjoint (f '' s) (g '' t) :=
disjoint_iff_inf_le.mpr <| by rintro a ⟨⟨b, hb, eq⟩, c, hc, rfl⟩; | Mathlib.Data.Set.Image.1616_0.IJFiTzmYGOCpPSd | theorem disjoint_image_image {f : β → α} {g : γ → α} {s : Set β} {t : Set γ}
(h : ∀ b ∈ s, ∀ c ∈ t, f b ≠ g c) : Disjoint (f '' s) (g '' t) | Mathlib_Data_Set_Image |
α : Type u_1
β : Type u_2
γ : Type u_3
f : α → β
s✝ t✝ : Set α
hf : Surjective f
s t : Set β
h : Disjoint (f ⁻¹' s) (f ⁻¹' t)
⊢ Disjoint s t | /-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura
-/
import Mathlib.Data.Set.Basic
#align_import data.set.image from "leanprover-community/mathlib"@"001ffdc42920050657fd45bd2b8bfbec8eaaeb29"
/-!
# Ima... | rw [disjoint_iff_inter_eq_empty, ← image_preimage_eq (_ ∩ _) hf, preimage_inter, h.inter_eq,
image_empty] | theorem _root_.Disjoint.of_preimage (hf : Surjective f) {s t : Set β}
(h : Disjoint (f ⁻¹' s) (f ⁻¹' t)) : Disjoint s t := by
| Mathlib.Data.Set.Image.1636_0.IJFiTzmYGOCpPSd | theorem _root_.Disjoint.of_preimage (hf : Surjective f) {s t : Set β}
(h : Disjoint (f ⁻¹' s) (f ⁻¹' t)) : Disjoint s t | Mathlib_Data_Set_Image |
α : Type u_1
β : Type u_2
γ : Type u_3
f : α → β
s✝ t : Set α
s : Set β
h : Disjoint s (range f)
⊢ f ⁻¹' s = ∅ | /-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura
-/
import Mathlib.Data.Set.Basic
#align_import data.set.image from "leanprover-community/mathlib"@"001ffdc42920050657fd45bd2b8bfbec8eaaeb29"
/-!
# Ima... | simpa using h.preimage f | theorem preimage_eq_empty {s : Set β} (h : Disjoint s (range f)) :
f ⁻¹' s = ∅ :=
by | Mathlib.Data.Set.Image.1648_0.IJFiTzmYGOCpPSd | theorem preimage_eq_empty {s : Set β} (h : Disjoint s (range f)) :
f ⁻¹' s = ∅ | Mathlib_Data_Set_Image |
α : Type u_1
β : Type u_2
γ : Type u_3
f : α → β
s✝ t : Set α
s : Set β
h : f ⁻¹' s = ∅
⊢ Disjoint s (range f) | /-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura
-/
import Mathlib.Data.Set.Basic
#align_import data.set.image from "leanprover-community/mathlib"@"001ffdc42920050657fd45bd2b8bfbec8eaaeb29"
/-!
# Ima... | simp only [eq_empty_iff_forall_not_mem, disjoint_iff_inter_eq_empty, not_exists, mem_inter_iff,
not_and, mem_range, mem_preimage] at h ⊢ | theorem preimage_eq_empty_iff {s : Set β} : f ⁻¹' s = ∅ ↔ Disjoint s (range f) :=
⟨fun h => by
| Mathlib.Data.Set.Image.1653_0.IJFiTzmYGOCpPSd | theorem preimage_eq_empty_iff {s : Set β} : f ⁻¹' s = ∅ ↔ Disjoint s (range f) | Mathlib_Data_Set_Image |
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