Theorem permaxext 45542

Description: The Axiom of Extensionality ax-ext 2733 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 3457	. . . . . 6 ⊢ 𝑧 ∈ V
4		vex 3457	. . . . . 6 ⊢ 𝑥 ∈ V
5	1, 2, 3, 4	brpermmodel 45540	. . . . 5 ⊢ (𝑧𝑅𝑥 ↔ 𝑧 ∈ (𝐹‘𝑥))
6		vex 3457	. . . . . 6 ⊢ 𝑦 ∈ V
7	1, 2, 3, 6	brpermmodel 45540	. . . . 5 ⊢ (𝑧𝑅𝑦 ↔ 𝑧 ∈ (𝐹‘𝑦))
8	5, 7	bibi12i 341	. . . 4 ⊢ ((𝑧𝑅𝑥 ↔ 𝑧𝑅𝑦) ↔ (𝑧 ∈ (𝐹‘𝑥) ↔ 𝑧 ∈ (𝐹‘𝑦)))
9	8	albii 1838	. . 3 ⊢ (∀𝑧(𝑧𝑅𝑥 ↔ 𝑧𝑅𝑦) ↔ ∀𝑧(𝑧 ∈ (𝐹‘𝑥) ↔ 𝑧 ∈ (𝐹‘𝑦)))
10		dfcleq 2754	. . 3 ⊢ ((𝐹‘𝑥) = (𝐹‘𝑦) ↔ ∀𝑧(𝑧 ∈ (𝐹‘𝑥) ↔ 𝑧 ∈ (𝐹‘𝑦)))
11	9, 10	bitr4i 280	. 2 ⊢ (∀𝑧(𝑧𝑅𝑥 ↔ 𝑧𝑅𝑦) ↔ (𝐹‘𝑥) = (𝐹‘𝑦))
12		f1of1 6800	. . . . 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 7235	. . . 4 ⊢ ((𝐹:V–1-1→V ∧ (𝑥 ∈ V ∧ 𝑦 ∈ V)) → ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦))
15	13, 14	mpan 700	. . 3 ⊢ ((𝑥 ∈ V ∧ 𝑦 ∈ V) → ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦))
16	15	el2v 3460	. 2 ⊢ ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦)
17	11, 16	sylbi 219	1 ⊢ (∀𝑧(𝑧𝑅𝑥 ↔ 𝑧𝑅𝑦) → 𝑥 = 𝑦)

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 399  ∀wal 1557   = wceq 1559   ∈ wcel 2141  Vcvv 3453   class class class wbr 5097   E cep 5542  ^◡ccnv 5642   ∘ ccom 5647  –1-1→wf1 6513  –1-1-onto→wf1o 6515  ‘cfv 6516

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-sep 5243  ax-nul 5253  ax-pr 5387

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-ne 2957  df-ral 3076  df-rex 3086  df-rab 3414  df-v 3455  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-if 4478  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4863  df-br 5098  df-opab 5160  df-id 5538  df-eprel 5543  df-xp 5649  df-rel 5650  df-cnv 5651  df-co 5652  df-dm 5653  df-iota 6472  df-fun 6518  df-fn 6519  df-f 6520  df-f1 6521  df-f1o 6523  df-fv 6524

This theorem is referenced by:  nregmodelaxext  45555