Theorem permaxext 45605

Description: The Axiom of Extensionality ax-ext 2741 holds in permutation models. Part of Exercise II.9.2 of [Kunen2] p. 148. (Contributed by Eric Schmidt, 6-Nov-2025.)

Hypotheses

Ref	Expression
permmodel.1	⊢ 𝐹:V–1-1-onto→V
permmodel.2	⊢ 𝑅 = (◡𝐹 ∘ E )

Assertion

Ref	Expression
permaxext	⊢ (∀𝑧(𝑧𝑅𝑥 ↔ 𝑧𝑅𝑦) → 𝑥 = 𝑦)

Distinct variable groups:   𝑥,𝑦,𝑧   𝑧,𝐹

Allowed substitution hints:   𝑅(𝑥,𝑦,𝑧)   𝐹(𝑥,𝑦)

Proof of Theorem permaxext

Step	Hyp	Ref	Expression
1		permmodel.1	. . . . . 6 ⊢ 𝐹:V–1-1-onto→V
2		permmodel.2	. . . . . 6 ⊢ 𝑅 = (◡𝐹 ∘ E )
3		vex 3467	. . . . . 6 ⊢ 𝑧 ∈ V
4		vex 3467	. . . . . 6 ⊢ 𝑥 ∈ V
5	1, 2, 3, 4	brpermmodel 45603	. . . . 5 ⊢ (𝑧𝑅𝑥 ↔ 𝑧 ∈ (𝐹‘𝑥))
6		vex 3467	. . . . . 6 ⊢ 𝑦 ∈ V
7	1, 2, 3, 6	brpermmodel 45603	. . . . 5 ⊢ (𝑧𝑅𝑦 ↔ 𝑧 ∈ (𝐹‘𝑦))
8	5, 7	bibi12i 342	. . . 4 ⊢ ((𝑧𝑅𝑥 ↔ 𝑧𝑅𝑦) ↔ (𝑧 ∈ (𝐹‘𝑥) ↔ 𝑧 ∈ (𝐹‘𝑦)))
9	8	albii 1846	. . 3 ⊢ (∀𝑧(𝑧𝑅𝑥 ↔ 𝑧𝑅𝑦) ↔ ∀𝑧(𝑧 ∈ (𝐹‘𝑥) ↔ 𝑧 ∈ (𝐹‘𝑦)))
10		dfcleq 2762	. . 3 ⊢ ((𝐹‘𝑥) = (𝐹‘𝑦) ↔ ∀𝑧(𝑧 ∈ (𝐹‘𝑥) ↔ 𝑧 ∈ (𝐹‘𝑦)))
11	9, 10	bitr4i 281	. 2 ⊢ (∀𝑧(𝑧𝑅𝑥 ↔ 𝑧𝑅𝑦) ↔ (𝐹‘𝑥) = (𝐹‘𝑦))
12		f1of1 6820	. . . . 5 ⊢ (𝐹:V–1-1-onto→V → 𝐹:V–1-1→V)
13	1, 12	ax-mp 5	. . . 4 ⊢ 𝐹:V–1-1→V
14		f1veqaeq 7255	. . . 4 ⊢ ((𝐹:V–1-1→V ∧ (𝑥 ∈ V ∧ 𝑦 ∈ V)) → ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦))
15	13, 14	mpan 702	. . 3 ⊢ ((𝑥 ∈ V ∧ 𝑦 ∈ V) → ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦))
16	15	el2v 3470	. 2 ⊢ ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦)
17	11, 16	sylbi 220	1 ⊢ (∀𝑧(𝑧𝑅𝑥 ↔ 𝑧𝑅𝑦) → 𝑥 = 𝑦)

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400  ∀wal 1565   = wceq 1567   ∈ wcel 2149  Vcvv 3463   class class class wbr 5113   E cep 5561  ^◡ccnv 5661   ∘ ccom 5666  –1-1→wf1 6534  –1-1-onto→wf1o 6536  ‘cfv 6537

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-sep 5261  ax-nul 5271  ax-pr 5405

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-ne 2965  df-ral 3086  df-rex 3096  df-rab 3424  df-v 3465  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-br 5114  df-opab 5178  df-id 5557  df-eprel 5562  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-f1o 6544  df-fv 6545

This theorem is referenced by:  nregmodelaxext  45618