brpermmodelcnv - Mathbox for Eric Schmidt

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Theorem brpermmodelcnv 45360

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 6855	. . 3 ⊢ (^◡𝐹‘𝐵) ∈ V
5	1, 2, 3, 4	brpermmodel 45359	. 2 ⊢ (𝐴𝑅(^◡𝐹‘𝐵) ↔ 𝐴 ∈ (𝐹‘(^◡𝐹‘𝐵)))
6		brpermmodel.4	. . . 4 ⊢ 𝐵 ∈ V
7		f1ocnvfv2 7233	. . . 4 ⊢ ((𝐹:V–1-1-onto→V ∧ 𝐵 ∈ V) → (𝐹‘(^◡𝐹‘𝐵)) = 𝐵)
8	1, 6, 7	mp2an 693	. . 3 ⊢ (𝐹‘(^◡𝐹‘𝐵)) = 𝐵
9	8	eleq2i 2829	. 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 1542 ∈ wcel 2114 Vcvv 3442 class class class wbr 5100 E cep 5531 ^◡ccnv 5631 ∘ ccom 5636 –1-1-onto→wf1o 6499 ‘cfv 6500

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-sep 5243 ax-nul 5253 ax-pr 5379

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-ne 2934 df-ral 3053 df-rex 3063 df-rab 3402 df-v 3444 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-nul 4288 df-if 4482 df-sn 4583 df-pr 4585 df-op 4589 df-uni 4866 df-br 5101 df-opab 5163 df-id 5527 df-eprel 5532 df-xp 5638 df-rel 5639 df-cnv 5640 df-co 5641 df-dm 5642 df-rn 5643 df-res 5644 df-ima 5645 df-iota 6456 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508

This theorem is referenced by: permaxsep 45363 permaxnul 45364 permaxpow 45365 permaxpr 45366 permaxun 45367 permaxinf2lem 45368 permac8prim 45370

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