Theorem brpermmodel 45160

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 5524	. . . 4 ⊢ (𝐴 E 𝑥 ↔ 𝐴 ∈ 𝑥)
2		vex 3441	. . . . 5 ⊢ 𝑥 ∈ V
3		brpermmodel.4	. . . . 5 ⊢ 𝐵 ∈ V
4	2, 3	brcnv 5828	. . . 4 ⊢ (𝑥◡𝐹𝐵 ↔ 𝐵𝐹𝑥)
5	1, 4	anbi12i 628	. . 3 ⊢ ((𝐴 E 𝑥 ∧ 𝑥◡𝐹𝐵) ↔ (𝐴 ∈ 𝑥 ∧ 𝐵𝐹𝑥))
6	5	exbii 1849	. 2 ⊢ (∃𝑥(𝐴 E 𝑥 ∧ 𝑥◡𝐹𝐵) ↔ ∃𝑥(𝐴 ∈ 𝑥 ∧ 𝐵𝐹𝑥))
7		permmodel.2	. . . 4 ⊢ 𝑅 = (◡𝐹 ∘ E )
8	7	breqi 5101	. . 3 ⊢ (𝐴𝑅𝐵 ↔ 𝐴(◡𝐹 ∘ E )𝐵)
9		brpermmodel.3	. . . 4 ⊢ 𝐴 ∈ V
10	9, 3	brco 5816	. . 3 ⊢ (𝐴(◡𝐹 ∘ E )𝐵 ↔ ∃𝑥(𝐴 E 𝑥 ∧ 𝑥◡𝐹𝐵))
11	8, 10	bitri 275	. 2 ⊢ (𝐴𝑅𝐵 ↔ ∃𝑥(𝐴 E 𝑥 ∧ 𝑥◡𝐹𝐵))
12		permmodel.1	. . . . 5 ⊢ 𝐹:V–1-1-onto→V
13		f1ofn 6772	. . . . 5 ⊢ (𝐹:V–1-1-onto→V → 𝐹 Fn V)
14	12, 13	ax-mp 5	. . . 4 ⊢ 𝐹 Fn V
15		fneu 6599	. . . 4 ⊢ ((𝐹 Fn V ∧ 𝐵 ∈ V) → ∃!𝑥 𝐵𝐹𝑥)
16	14, 3, 15	mp2an 692	. . 3 ⊢ ∃!𝑥 𝐵𝐹𝑥
17		eleq1 2821	. . . . . . 7 ⊢ (𝑦 = 𝐴 → (𝑦 ∈ 𝑥 ↔ 𝐴 ∈ 𝑥))
18	17	anbi1d 631	. . . . . 6 ⊢ (𝑦 = 𝐴 → ((𝑦 ∈ 𝑥 ∧ 𝐵𝐹𝑥) ↔ (𝐴 ∈ 𝑥 ∧ 𝐵𝐹𝑥)))
19	18	exbidv 1922	. . . . 5 ⊢ (𝑦 = 𝐴 → (∃𝑥(𝑦 ∈ 𝑥 ∧ 𝐵𝐹𝑥) ↔ ∃𝑥(𝐴 ∈ 𝑥 ∧ 𝐵𝐹𝑥)))
20	19	anbi1d 631	. . . 4 ⊢ (𝑦 = 𝐴 → ((∃𝑥(𝑦 ∈ 𝑥 ∧ 𝐵𝐹𝑥) ∧ ∃!𝑥 𝐵𝐹𝑥) ↔ (∃𝑥(𝐴 ∈ 𝑥 ∧ 𝐵𝐹𝑥) ∧ ∃!𝑥 𝐵𝐹𝑥)))
21		fv3 6849	. . . 4 ⊢ (𝐹‘𝐵) = {𝑦 ∣ (∃𝑥(𝑦 ∈ 𝑥 ∧ 𝐵𝐹𝑥) ∧ ∃!𝑥 𝐵𝐹𝑥)}
22	9, 20, 21	elab2 3634	. . 3 ⊢ (𝐴 ∈ (𝐹‘𝐵) ↔ (∃𝑥(𝐴 ∈ 𝑥 ∧ 𝐵𝐹𝑥) ∧ ∃!𝑥 𝐵𝐹𝑥))
23	16, 22	mpbiran2 710	. 2 ⊢ (𝐴 ∈ (𝐹‘𝐵) ↔ ∃𝑥(𝐴 ∈ 𝑥 ∧ 𝐵𝐹𝑥))
24	6, 11, 23	3bitr4i 303	1 ⊢ (𝐴𝑅𝐵 ↔ 𝐴 ∈ (𝐹‘𝐵))

Syntax hints:   ↔ wb 206   ∧ wa 395   = wceq 1541  ∃wex 1780   ∈ wcel 2113  ∃!weu 2565  Vcvv 3437   class class class wbr 5095   E cep 5520  ^◡ccnv 5620   ∘ ccom 5625   Fn wfn 6484  –1-1-onto→wf1o 6488  ‘cfv 6489

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-sep 5238  ax-nul 5248  ax-pr 5374

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-ne 2930  df-ral 3049  df-rex 3058  df-rab 3397  df-v 3439  df-dif 3901  df-un 3903  df-ss 3915  df-nul 4283  df-if 4477  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4861  df-br 5096  df-opab 5158  df-id 5516  df-eprel 5521  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-iota 6445  df-fun 6491  df-fn 6492  df-f 6493  df-f1 6494  df-f1o 6496  df-fv 6497

This theorem is referenced by:  brpermmodelcnv  45161  permaxext  45162  permaxrep  45163  permaxsep  45164  permaxpow  45166  permaxun  45168  permaxinf2lem  45169  permac8prim  45171  nregmodellem  45173