Theorem brpermmodel 45447

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 5521	. . . 4 ⊢ (𝐴 E 𝑥 ↔ 𝐴 ∈ 𝑥)
2		vex 3435	. . . . 5 ⊢ 𝑥 ∈ V
3		brpermmodel.4	. . . . 5 ⊢ 𝐵 ∈ V
4	2, 3	brcnv 5824	. . . 4 ⊢ (𝑥◡𝐹𝐵 ↔ 𝐵𝐹𝑥)
5	1, 4	anbi12i 634	. . 3 ⊢ ((𝐴 E 𝑥 ∧ 𝑥◡𝐹𝐵) ↔ (𝐴 ∈ 𝑥 ∧ 𝐵𝐹𝑥))
6	5	exbii 1855	. 2 ⊢ (∃𝑥(𝐴 E 𝑥 ∧ 𝑥◡𝐹𝐵) ↔ ∃𝑥(𝐴 ∈ 𝑥 ∧ 𝐵𝐹𝑥))
7		permmodel.2	. . . 4 ⊢ 𝑅 = (◡𝐹 ∘ E )
8	7	breqi 5078	. . 3 ⊢ (𝐴𝑅𝐵 ↔ 𝐴(◡𝐹 ∘ E )𝐵)
9		brpermmodel.3	. . . 4 ⊢ 𝐴 ∈ V
10	9, 3	brco 5812	. . 3 ⊢ (𝐴(◡𝐹 ∘ E )𝐵 ↔ ∃𝑥(𝐴 E 𝑥 ∧ 𝑥◡𝐹𝐵))
11	8, 10	bitri 276	. 2 ⊢ (𝐴𝑅𝐵 ↔ ∃𝑥(𝐴 E 𝑥 ∧ 𝑥◡𝐹𝐵))
12		permmodel.1	. . . . 5 ⊢ 𝐹:V–1-1-onto→V
13		f1ofn 6768	. . . . 5 ⊢ (𝐹:V–1-1-onto→V → 𝐹 Fn V)
14	12, 13	ax-mp 5	. . . 4 ⊢ 𝐹 Fn V
15		fneu 6595	. . . 4 ⊢ ((𝐹 Fn V ∧ 𝐵 ∈ V) → ∃!𝑥 𝐵𝐹𝑥)
16	14, 3, 15	mp2an 698	. . 3 ⊢ ∃!𝑥 𝐵𝐹𝑥
17		eleq1 2827	. . . . . . 7 ⊢ (𝑦 = 𝐴 → (𝑦 ∈ 𝑥 ↔ 𝐴 ∈ 𝑥))
18	17	anbi1d 637	. . . . . 6 ⊢ (𝑦 = 𝐴 → ((𝑦 ∈ 𝑥 ∧ 𝐵𝐹𝑥) ↔ (𝐴 ∈ 𝑥 ∧ 𝐵𝐹𝑥)))
19	18	exbidv 1928	. . . . 5 ⊢ (𝑦 = 𝐴 → (∃𝑥(𝑦 ∈ 𝑥 ∧ 𝐵𝐹𝑥) ↔ ∃𝑥(𝐴 ∈ 𝑥 ∧ 𝐵𝐹𝑥)))
20	19	anbi1d 637	. . . 4 ⊢ (𝑦 = 𝐴 → ((∃𝑥(𝑦 ∈ 𝑥 ∧ 𝐵𝐹𝑥) ∧ ∃!𝑥 𝐵𝐹𝑥) ↔ (∃𝑥(𝐴 ∈ 𝑥 ∧ 𝐵𝐹𝑥) ∧ ∃!𝑥 𝐵𝐹𝑥)))
21		fv3 6845	. . . 4 ⊢ (𝐹‘𝐵) = {𝑦 ∣ (∃𝑥(𝑦 ∈ 𝑥 ∧ 𝐵𝐹𝑥) ∧ ∃!𝑥 𝐵𝐹𝑥)}
22	9, 20, 21	elab2 3620	. . 3 ⊢ (𝐴 ∈ (𝐹‘𝐵) ↔ (∃𝑥(𝐴 ∈ 𝑥 ∧ 𝐵𝐹𝑥) ∧ ∃!𝑥 𝐵𝐹𝑥))
23	16, 22	mpbiran2 716	. 2 ⊢ (𝐴 ∈ (𝐹‘𝐵) ↔ ∃𝑥(𝐴 ∈ 𝑥 ∧ 𝐵𝐹𝑥))
24	6, 11, 23	3bitr4i 304	1 ⊢ (𝐴𝑅𝐵 ↔ 𝐴 ∈ (𝐹‘𝐵))

Syntax hints:   ↔ wb 207   ∧ wa 396   = wceq 1547  ∃wex 1786   ∈ wcel 2119  ∃!weu 2572  Vcvv 3431   class class class wbr 5072   E cep 5517  ^◡ccnv 5617   ∘ ccom 5622   Fn wfn 6480  –1-1-onto→wf1o 6484  ‘cfv 6485

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-sep 5218  ax-pr 5362

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-ne 2935  df-ral 3054  df-rex 3064  df-rab 3392  df-v 3433  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4262  df-if 4455  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-br 5073  df-opab 5135  df-id 5513  df-eprel 5518  df-xp 5624  df-rel 5625  df-cnv 5626  df-co 5627  df-dm 5628  df-iota 6441  df-fun 6487  df-fn 6488  df-f 6489  df-f1 6490  df-f1o 6492  df-fv 6493

This theorem is referenced by:  brpermmodelcnv  45448  permaxext  45449  permaxrep  45450  permaxsep  45451  permaxpow  45453  permaxun  45455  permaxinf2lem  45456  permac8prim  45458  nregmodellem  45460