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3277efa | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 | import OAI.SetTheory.PartitionPrinciple.Geometry.GroundNaturalFamilyShift
namespace OAI
section
namespace PartitionPilot.Forcing.TransitiveGround
open SetFormula
universe u
variable (M : TransitiveGround.{u})
theorem groundFunction_enumeration
(pairing : Realize M.sets SetFormula.pairing Fin.elim0)
(unions : Realize M.sets SetFormula.union Fin.elim0)
(powers : Realize M.sets powerSet Fin.elim0) (separation : SeparationSchema M.sets)
(infinite : Realize M.sets infinity Fin.elim0)
{B N f : ZFSet.{u}} (hB : B ∈ M.sets) (hN : N ∈ M.sets) (hf : f ∈ M.sets)
(nonempty : B.Nonempty) (small : M.groundEmbeddable B ZFSet.omega)
(function : ZFSet.IsFunc B N f) :
∃ e ∈ M.sets, ∃ names ∈ M.sets,
ZFSet.IsFunc ZFSet.omega B e ∧ ZFSet.IsFunc ZFSet.omega N names ∧
(∀ v ∈ B, ∃ n ∈ ZFSet.omega, ZFSet.pair n v ∈ e) ∧
(∀ n ∈ ZFSet.omega, ∀ v tau, ZFSet.pair n v ∈ e →
ZFSet.pair n tau ∈ names → ZFSet.pair v tau ∈ f) ∧
(∀ n ∈ ZFSet.omega, ∀ v tau, ZFSet.pair n v ∈ e →
ZFSet.pair v tau ∈ f → ZFSet.pair n tau ∈ names) := by
obtain ⟨e, he, enumeration, onto⟩ := M.groundCountable_enumeration
pairing unions powers separation infinite hB nonempty small
let names := relationCompose ZFSet.omega B N e f
have namesGround := M.relationCompose_mem pairing unions powers separation
(M.omega_mem separation infinite) hB hN he hf
have namesFunction : ZFSet.IsFunc ZFSet.omega N names :=
relationCompose_function_of_subset enumeration function (fun _ member => member)
refine ⟨e, he, names, namesGround, enumeration, namesFunction, onto, ?_, ?_⟩
· intro n hn v tau nv nt
obtain ⟨_, _, w, _, nw, wt⟩ := (pair_mem_relationCompose _ _ _ _ _ n tau).1 nt
obtain ⟨chosen, _, unique⟩ := enumeration.2 n hn
have equal : w = v := (unique w nw).trans (unique v nv).symm
exact equal ▸ wt
· intro n hn v tau nv vt
have bound := ZFSet.pair_mem_prod.1 (function.1 vt)
exact (pair_mem_relationCompose _ _ _ _ _ n tau).2 ⟨hn, bound.2, v, bound.1, nv, vt⟩
end PartitionPilot.Forcing.TransitiveGround
end
end OAI
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