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e3e0a0d | 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 | import OAI.SetTheory.PartitionPrinciple.Definability.GroundFlatCopyGraphs
namespace OAI
section
namespace PartitionPilot.Forcing.BoundedFormula
open SetFormula
universe u
def relationImageEqualityFormula {n} (output domain codomain graph : Fin n) : BoundedFormula n :=
.conj (.allMem output (.conj (.mem 0 codomain.succ)
(imageMemberFormula 0 domain.succ graph.succ)))
(.allMem codomain (imp (imageMemberFormula 0 domain.succ graph.succ) (.mem 0 output.succ)))
theorem truth_relationImageEqualityFormula {n} (output domain codomain graph : Fin n)
(env : Fin n → ZFSet.{u}) :
Truth (relationImageEqualityFormula output domain codomain graph) env ↔
env output = relationImage (env domain) (env codomain) (env graph) := by
simp only [relationImageEqualityFormula, Truth, truth_imageMemberFormula, truth_imp]
change ((∀ y ∈ env output, y ∈ env codomain ∧ ∃ x ∈ env domain,
ZFSet.pair x y ∈ env graph) ∧
(∀ y ∈ env codomain, (∃ x ∈ env domain, ZFSet.pair x y ∈ env graph) → y ∈ env output)) ↔ _
constructor
· rintro ⟨forward, backward⟩
apply ZFSet.ext
intro y
rw [mem_relationImage]
exact ⟨forward y, fun h => backward y h.1 h.2⟩
· intro equal
constructor
· intro y hy
exact (mem_relationImage _ _ _ y).1 (equal ▸ hy)
· intro y hy member
exact equal ▸ (mem_relationImage _ _ _ y).2 ⟨hy, member⟩
end PartitionPilot.Forcing.BoundedFormula
end
end OAI
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