Theorem permaxext 45432

Description: The Axiom of Extensionality ax-ext 2708 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 3433	. . . . . 6 ⊢ 𝑧 ∈ V
4		vex 3433	. . . . . 6 ⊢ 𝑥 ∈ V
5	1, 2, 3, 4	brpermmodel 45430	. . . . 5 ⊢ (𝑧𝑅𝑥 ↔ 𝑧 ∈ (𝐹‘𝑥))
6		vex 3433	. . . . . 6 ⊢ 𝑦 ∈ V
7	1, 2, 3, 6	brpermmodel 45430	. . . . 5 ⊢ (𝑧𝑅𝑦 ↔ 𝑧 ∈ (𝐹‘𝑦))
8	5, 7	bibi12i 339	. . . 4 ⊢ ((𝑧𝑅𝑥 ↔ 𝑧𝑅𝑦) ↔ (𝑧 ∈ (𝐹‘𝑥) ↔ 𝑧 ∈ (𝐹‘𝑦)))
9	8	albii 1821	. . 3 ⊢ (∀𝑧(𝑧𝑅𝑥 ↔ 𝑧𝑅𝑦) ↔ ∀𝑧(𝑧 ∈ (𝐹‘𝑥) ↔ 𝑧 ∈ (𝐹‘𝑦)))
10		dfcleq 2729	. . 3 ⊢ ((𝐹‘𝑥) = (𝐹‘𝑦) ↔ ∀𝑧(𝑧 ∈ (𝐹‘𝑥) ↔ 𝑧 ∈ (𝐹‘𝑦)))
11	9, 10	bitr4i 278	. 2 ⊢ (∀𝑧(𝑧𝑅𝑥 ↔ 𝑧𝑅𝑦) ↔ (𝐹‘𝑥) = (𝐹‘𝑦))
12		f1of1 6779	. . . . 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 7211	. . . 4 ⊢ ((𝐹:V–1-1→V ∧ (𝑥 ∈ V ∧ 𝑦 ∈ V)) → ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦))
15	13, 14	mpan 691	. . 3 ⊢ ((𝑥 ∈ V ∧ 𝑦 ∈ V) → ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦))
16	15	el2v 3436	. 2 ⊢ ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦)
17	11, 16	sylbi 217	1 ⊢ (∀𝑧(𝑧𝑅𝑥 ↔ 𝑧𝑅𝑦) → 𝑥 = 𝑦)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395  ∀wal 1540   = wceq 1542   ∈ wcel 2114  Vcvv 3429   class class class wbr 5085   E cep 5530  ^◡ccnv 5630   ∘ ccom 5635  –1-1→wf1 6495  –1-1-onto→wf1o 6497  ‘cfv 6498

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-sep 5231  ax-nul 5241  ax-pr 5375

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-ne 2933  df-ral 3052  df-rex 3062  df-rab 3390  df-v 3431  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-br 5086  df-opab 5148  df-id 5526  df-eprel 5531  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-f1o 6505  df-fv 6506

This theorem is referenced by:  nregmodelaxext  45445