brpermmodelcnv - Mathbox for Eric Schmidt

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Theorem brpermmodelcnv 46258

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⟶1-1ontoV
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⟶1-1ontoV
2		permmodel.2	. . 3 ⊢ 𝑅 = (^◡𝐹 ∘ E )
3		brpermmodel.3	. . 3 ⊢ 𝐴 ∈ V
4		fvex 6929	. . 3 ⊢ (^◡𝐹‘𝐵) ∈ V
5	1, 2, 3, 4	brpermmodel 46257	. 2 ⊢ (𝐴𝑅(^◡𝐹‘𝐵) ↔ 𝐴 ∈ (𝐹‘(^◡𝐹‘𝐵)))
6		brpermmodel.4	. . . 4 ⊢ 𝐵 ∈ V
7		f1ocnvfv2 7316	. . . 4 ⊢ ((𝐹:V⟶1-1ontoV ∧ 𝐵 ∈ V) → (𝐹‘(^◡𝐹‘𝐵)) = 𝐵)
8	1, 6, 7	mp2an 705	. . 3 ⊢ (𝐹‘(^◡𝐹‘𝐵)) = 𝐵
9	8	eleq2i 2856	. 2 ⊢ (𝐴 ∈ (𝐹‘(^◡𝐹‘𝐵)) ↔ 𝐴 ∈ 𝐵)
10	5, 9	bitri 278	1 ⊢ (𝐴𝑅(^◡𝐹‘𝐵) ↔ 𝐴 ∈ 𝐵)

Colors of variables: wff setvar class

This proof depends on syntax axioms: ↔ wb 209 = wceq 1570 ∈ wcel 2147 Vcvv 3454 class class class wbr 5108 E cep 5557 ^◡ccnv 5657 ∘ ccom 5662 ⟶1-1ontowf1o 6565 ‘cfv 6566

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 1945 ax-6 2002 ax-7 2043 ax-8 2149 ax-9 2157 ax-10 2180 ax-11 2197 ax-12 2216 ax-ext 2735 ax-sep 5255 ax-nul 5266 ax-pr 5398

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 2102 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2839 df-ne 2960 df-ral 3081 df-rex 3091 df-rab 3417 df-v 3456 df-ss 3901 df-dif 4050 df-in 4094 df-un 4154 df-nul 4282 df-if 4486 df-sn 4589 df-pr 4591 df-op 4841 df-uni 4873 df-br 5109 df-opab 5174 df-id 5553 df-eprel 5558 df-xp 5664 df-rel 5665 df-cnv 5666 df-co 5667 df-dm 5668 df-rn 5669 df-res 5670 df-ima 5671 df-iota 6520 df-fun 6568 df-fn 6569 df-f 6570 df-f1 6571 df-fo 6572 df-f1o 6573 df-fv 6574

This theorem is used by: permaxsep 46261 permaxnul 46262 permaxpow 46263 permaxpr 46264 permaxun 46265 permaxinf2lem 46266 permac8prim 46268

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