Theorem brpermmodel 45971

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–1-1-onto→V
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 5554	. . . 4 ⊢ (𝐴 E 𝑥 ↔ 𝐴 ∈ 𝑥)
2		vex 3455	. . . . 5 ⊢ 𝑥 ∈ V
3		brpermmodel.4	. . . . 5 ⊢ 𝐵 ∈ V
4	2, 3	brcnv 5860	. . . 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 5109	. . 3 ⊢ (𝐴𝑅𝐵 ↔ 𝐴(◡𝐹 ∘ E )𝐵)
9		brpermmodel.3	. . . 4 ⊢ 𝐴 ∈ V
10	9, 3	brco 5848	. . 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–1-1-onto→V
13		f1ofn 6823	. . . . 5 ⊢ (𝐹:V–1-1-onto→V → 𝐹 Fn V)
14	12, 13	ax-mp 5	. . . 4 ⊢ 𝐹 Fn V
15		fneu 6647	. . . 4 ⊢ ((𝐹 Fn V ∧ 𝐵 ∈ V) → ∃!𝑥 𝐵𝐹𝑥)
16	14, 3, 15	mp2an 705	. . 3 ⊢ ∃!𝑥 𝐵𝐹𝑥
17		eleq1 2849	. . . . . . 7 ⊢ (𝑦 = 𝐴 → (𝑦 ∈ 𝑥 ↔ 𝐴 ∈ 𝑥))
18	17	anbi1d 643	. . . . . 6 ⊢ (𝑦 = 𝐴 → ((𝑦 ∈ 𝑥 ∧ 𝐵𝐹𝑥) ↔ (𝐴 ∈ 𝑥 ∧ 𝐵𝐹𝑥)))
19	18	exbidv 1954	. . . . 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 6901	. . . 4 ⊢ (𝐹‘𝐵) = {𝑦 ∣ (∃𝑥(𝑦 ∈ 𝑥 ∧ 𝐵𝐹𝑥) ∧ ∃!𝑥 𝐵𝐹𝑥)}
22	9, 20, 21	elab2 3636	. . 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 2145  ∃!weu 2594  Vcvv 3451   class class class wbr 5103   E cep 5550  ^◡ccnv 5650   ∘ ccom 5655   Fn wfn 6532  –1-1-onto→wf1o 6536  ‘cfv 6537

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 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-sep 5249  ax-pr 5391

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 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-ne 2957  df-ral 3078  df-rex 3088  df-rab 3414  df-v 3453  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-id 5546  df-eprel 5551  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-f1o 6544  df-fv 6545

This theorem is used by:  brpermmodelcnv  45972  permaxext  45973  permaxrep  45974  permaxsep  45975  permaxpow  45977  permaxun  45979  permaxinf2lem  45980  permac8prim  45982  nregmodellem  45984