Theorem brpermmodel 46257

Description: The membership relation in a permutation model. We use a permutation 𝐹 of the universe to define a relation 𝑅 that serves as the membership relation in our model. The conclusion of this theorem is Definition II.9.1 of [Kunen2] p. 148. All the axioms of ZFC except for Regularity hold in permutation models, and Regularity will be false if 𝐹 is chosen appropriately. Thus, permutation models can be used to show that Regularity does not follow from the other axioms (with the usual proviso that the axioms are consistent). (Contributed by Eric Schmidt, 6-Nov-2025.)

Hypotheses

Ref	Expression
permmodel.1	⊢ 𝐹:V⟶1-1ontoV
permmodel.2	⊢ 𝑅 = (◡𝐹 ∘ E )
brpermmodel.3	⊢ 𝐴 ∈ V
brpermmodel.4	⊢ 𝐵 ∈ V

Assertion

Ref	Expression
brpermmodel	⊢ (𝐴𝑅𝐵 ↔ 𝐴 ∈ (𝐹‘𝐵))

Proof of Theorem brpermmodel

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		epel 5561	. . . 4 ⊢ (𝐴 E 𝑥 ↔ 𝐴 ∈ 𝑥)
2		vex 3458	. . . . 5 ⊢ 𝑥 ∈ V
3		brpermmodel.4	. . . . 5 ⊢ 𝐵 ∈ V
4	2, 3	brcnv 5872	. . . 4 ⊢ (𝑥◡𝐹𝐵 ↔ 𝐵𝐹𝑥)
5	1, 4	anbi12i 640	. . 3 ⊢ ((𝐴 E 𝑥 ∧ 𝑥◡𝐹𝐵) ↔ (𝐴 ∈ 𝑥 ∧ 𝐵𝐹𝑥))
6	5	exbii 1881	. 2 ⊢ (∃𝑥(𝐴 E 𝑥 ∧ 𝑥◡𝐹𝐵) ↔ ∃𝑥(𝐴 ∈ 𝑥 ∧ 𝐵𝐹𝑥))
7		permmodel.2	. . . 4 ⊢ 𝑅 = (◡𝐹 ∘ E )
8	7	breqi 5114	. . 3 ⊢ (𝐴𝑅𝐵 ↔ 𝐴(◡𝐹 ∘ E )𝐵)
9		brpermmodel.3	. . . 4 ⊢ 𝐴 ∈ V
10	9, 3	brco 5860	. . 3 ⊢ (𝐴(◡𝐹 ∘ E )𝐵 ↔ ∃𝑥(𝐴 E 𝑥 ∧ 𝑥◡𝐹𝐵))
11	8, 10	bitri 278	. 2 ⊢ (𝐴𝑅𝐵 ↔ ∃𝑥(𝐴 E 𝑥 ∧ 𝑥◡𝐹𝐵))
12		permmodel.1	. . . . 5 ⊢ 𝐹:V⟶1-1ontoV
13		f1ofn 6855	. . . . 5 ⊢ (𝐹:V⟶1-1ontoV → 𝐹 Fn V)
14	12, 13	ax-mp 5	. . . 4 ⊢ 𝐹 Fn V
15		fneu 6679	. . . 4 ⊢ ((𝐹 Fn V ∧ 𝐵 ∈ V) → ∃!𝑥 𝐵𝐹𝑥)
16	14, 3, 15	mp2an 705	. . 3 ⊢ ∃!𝑥 𝐵𝐹𝑥
17		eleq1 2852	. . . . . . 7 ⊢ (𝑦 = 𝐴 → (𝑦 ∈ 𝑥 ↔ 𝐴 ∈ 𝑥))
18	17	anbi1d 643	. . . . . 6 ⊢ (𝑦 = 𝐴 → ((𝑦 ∈ 𝑥 ∧ 𝐵𝐹𝑥) ↔ (𝐴 ∈ 𝑥 ∧ 𝐵𝐹𝑥)))
19	18	exbidv 1956	. . . . 5 ⊢ (𝑦 = 𝐴 → (∃𝑥(𝑦 ∈ 𝑥 ∧ 𝐵𝐹𝑥) ↔ ∃𝑥(𝐴 ∈ 𝑥 ∧ 𝐵𝐹𝑥)))
20	19	anbi1d 643	. . . 4 ⊢ (𝑦 = 𝐴 → ((∃𝑥(𝑦 ∈ 𝑥 ∧ 𝐵𝐹𝑥) ∧ ∃!𝑥 𝐵𝐹𝑥) ↔ (∃𝑥(𝐴 ∈ 𝑥 ∧ 𝐵𝐹𝑥) ∧ ∃!𝑥 𝐵𝐹𝑥)))
21		fv3 6934	. . . 4 ⊢ (𝐹‘𝐵) = {𝑦 ∣ (∃𝑥(𝑦 ∈ 𝑥 ∧ 𝐵𝐹𝑥) ∧ ∃!𝑥 𝐵𝐹𝑥)}
22	9, 20, 21	elab2 3639	. . 3 ⊢ (𝐴 ∈ (𝐹‘𝐵) ↔ (∃𝑥(𝐴 ∈ 𝑥 ∧ 𝐵𝐹𝑥) ∧ ∃!𝑥 𝐵𝐹𝑥))
23	16, 22	mpbiran2 723	. 2 ⊢ (𝐴 ∈ (𝐹‘𝐵) ↔ ∃𝑥(𝐴 ∈ 𝑥 ∧ 𝐵𝐹𝑥))
24	6, 11, 23	3bitr4i 306	1 ⊢ (𝐴𝑅𝐵 ↔ 𝐴 ∈ (𝐹‘𝐵))

This proof depends on syntax axioms:   ↔ wb 209   ∧ wa 401   = wceq 1570  ∃wex 1812   ∈ wcel 2147  ∃!weu 2596  Vcvv 3454   class class class wbr 5108   E cep 5557  ^◡ccnv 5657   ∘ ccom 5662   Fn wfn 6561  ⟶1-1ontowf1o 6565  ‘cfv 6566

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5255  ax-pr 5398

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-br 5109  df-opab 5174  df-id 5553  df-eprel 5558  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-iota 6520  df-fun 6568  df-fn 6569  df-f 6570  df-f1 6571  df-f1o 6573  df-fv 6574

This theorem is used by:  brpermmodelcnv  46258  permaxext  46259  permaxrep  46260  permaxsep  46261  permaxpow  46263  permaxun  46265  permaxinf2lem  46266  permac8prim  46268  nregmodellem  46270