brpermmodelcnv - Mathbox for Eric Schmidt

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Theorem brpermmodelcnv 44987

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 6873	. . 3 ⊢ (^◡𝐹‘𝐵) ∈ V
5	1, 2, 3, 4	brpermmodel 44986	. 2 ⊢ (𝐴𝑅(^◡𝐹‘𝐵) ↔ 𝐴 ∈ (𝐹‘(^◡𝐹‘𝐵)))
6		brpermmodel.4	. . . 4 ⊢ 𝐵 ∈ V
7		f1ocnvfv2 7254	. . . 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 2821	. 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 1540 ∈ wcel 2109 Vcvv 3450 class class class wbr 5109 E cep 5539 ^◡ccnv 5639 ∘ ccom 5644 –1-1-onto→wf1o 6512 ‘cfv 6513

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2702 ax-sep 5253 ax-nul 5263 ax-pr 5389

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2534 df-eu 2563 df-clab 2709 df-cleq 2722 df-clel 2804 df-ne 2927 df-ral 3046 df-rex 3055 df-rab 3409 df-v 3452 df-dif 3919 df-un 3921 df-in 3923 df-ss 3933 df-nul 4299 df-if 4491 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4874 df-br 5110 df-opab 5172 df-id 5535 df-eprel 5540 df-xp 5646 df-rel 5647 df-cnv 5648 df-co 5649 df-dm 5650 df-rn 5651 df-res 5652 df-ima 5653 df-iota 6466 df-fun 6515 df-fn 6516 df-f 6517 df-f1 6518 df-fo 6519 df-f1o 6520 df-fv 6521

This theorem is referenced by: permaxsep 44990 permaxnul 44991 permaxpow 44992 permaxpr 44993 permaxun 44994 permaxinf2lem 44995 permac8prim 44997

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