brpermmodelcnv - Mathbox for Eric Schmidt

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Theorem brpermmodelcnv 45245

Description: Ordinary membership expressed in terms of the permutation model's membership relation. (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
brpermmodelcnv	⊢ (𝐴𝑅(^◡𝐹‘𝐵) ↔ 𝐴 ∈ 𝐵)

**Proof of Theorem brpermmodelcnv**
Step	Hyp	Ref	Expression
1		permmodel.1	. . 3 ⊢ 𝐹:V–1-1-onto→V
2		permmodel.2	. . 3 ⊢ 𝑅 = (^◡𝐹 ∘ E )
3		brpermmodel.3	. . 3 ⊢ 𝐴 ∈ V
4		fvex 6847	. . 3 ⊢ (^◡𝐹‘𝐵) ∈ V
5	1, 2, 3, 4	brpermmodel 45244	. 2 ⊢ (𝐴𝑅(^◡𝐹‘𝐵) ↔ 𝐴 ∈ (𝐹‘(^◡𝐹‘𝐵)))
6		brpermmodel.4	. . . 4 ⊢ 𝐵 ∈ V
7		f1ocnvfv2 7223	. . . 4 ⊢ ((𝐹:V–1-1-onto→V ∧ 𝐵 ∈ V) → (𝐹‘(^◡𝐹‘𝐵)) = 𝐵)
8	1, 6, 7	mp2an 692	. . 3 ⊢ (𝐹‘(^◡𝐹‘𝐵)) = 𝐵
9	8	eleq2i 2828	. 2 ⊢ (𝐴 ∈ (𝐹‘(^◡𝐹‘𝐵)) ↔ 𝐴 ∈ 𝐵)
10	5, 9	bitri 275	1 ⊢ (𝐴𝑅(^◡𝐹‘𝐵) ↔ 𝐴 ∈ 𝐵)

Colors of variables: wff setvar class

Syntax hints: ↔ wb 206 = wceq 1541 ∈ wcel 2113 Vcvv 3440 class class class wbr 5098 E cep 5523 ^◡ccnv 5623 ∘ ccom 5628 –1-1-onto→wf1o 6491 ‘cfv 6492

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 2184 ax-ext 2708 ax-sep 5241 ax-nul 5251 ax-pr 5377

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 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-ne 2933 df-ral 3052 df-rex 3061 df-rab 3400 df-v 3442 df-dif 3904 df-un 3906 df-in 3908 df-ss 3918 df-nul 4286 df-if 4480 df-sn 4581 df-pr 4583 df-op 4587 df-uni 4864 df-br 5099 df-opab 5161 df-id 5519 df-eprel 5524 df-xp 5630 df-rel 5631 df-cnv 5632 df-co 5633 df-dm 5634 df-rn 5635 df-res 5636 df-ima 5637 df-iota 6448 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500

This theorem is referenced by: permaxsep 45248 permaxnul 45249 permaxpow 45250 permaxpr 45251 permaxun 45252 permaxinf2lem 45253 permac8prim 45255

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