Theorem brpermmodelcnv 45161

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 6844	. . 3 ⊢ (◡𝐹‘𝐵) ∈ V
5	1, 2, 3, 4	brpermmodel 45160	. 2 ⊢ (𝐴𝑅(◡𝐹‘𝐵) ↔ 𝐴 ∈ (𝐹‘(◡𝐹‘𝐵)))
6		brpermmodel.4	. . . 4 ⊢ 𝐵 ∈ V
7		f1ocnvfv2 7220	. . . 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 2825	. 2 ⊢ (𝐴 ∈ (𝐹‘(◡𝐹‘𝐵)) ↔ 𝐴 ∈ 𝐵)
10	5, 9	bitri 275	1 ⊢ (𝐴𝑅(◡𝐹‘𝐵) ↔ 𝐴 ∈ 𝐵)

Syntax hints:   ↔ wb 206   = wceq 1541   ∈ wcel 2113  Vcvv 3437   class class class wbr 5095   E cep 5520  ^◡ccnv 5620   ∘ ccom 5625  –1-1-onto→wf1o 6488  ‘cfv 6489

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 2182  ax-ext 2705  ax-sep 5238  ax-nul 5248  ax-pr 5374

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 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-ne 2930  df-ral 3049  df-rex 3058  df-rab 3397  df-v 3439  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4283  df-if 4477  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4861  df-br 5096  df-opab 5158  df-id 5516  df-eprel 5521  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-res 5633  df-ima 5634  df-iota 6445  df-fun 6491  df-fn 6492  df-f 6493  df-f1 6494  df-fo 6495  df-f1o 6496  df-fv 6497

This theorem is referenced by:  permaxsep  45164  permaxnul  45165  permaxpow  45166  permaxpr  45167  permaxun  45168  permaxinf2lem  45169  permac8prim  45171