brpermmodelcnv - Mathbox for Eric Schmidt

	Mathbox for Eric Schmidt		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > brpermmodelcnv		Structured version Visualization version GIF version

Theorem brpermmodelcnv 44998

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 6839	. . 3 ⊢ (^◡𝐹‘𝐵) ∈ V
5	1, 2, 3, 4	brpermmodel 44997	. 2 ⊢ (𝐴𝑅(^◡𝐹‘𝐵) ↔ 𝐴 ∈ (𝐹‘(^◡𝐹‘𝐵)))
6		brpermmodel.4	. . . 4 ⊢ 𝐵 ∈ V
7		f1ocnvfv2 7218	. . . 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 2820	. 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 3438 class class class wbr 5095 E cep 5522 ^◡ccnv 5622 ∘ ccom 5627 –1-1-onto→wf1o 6485 ‘cfv 6486

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 2701 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 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-ne 2926 df-ral 3045 df-rex 3054 df-rab 3397 df-v 3440 df-dif 3908 df-un 3910 df-in 3912 df-ss 3922 df-nul 4287 df-if 4479 df-sn 4580 df-pr 4582 df-op 4586 df-uni 4862 df-br 5096 df-opab 5158 df-id 5518 df-eprel 5523 df-xp 5629 df-rel 5630 df-cnv 5631 df-co 5632 df-dm 5633 df-rn 5634 df-res 5635 df-ima 5636 df-iota 6442 df-fun 6488 df-fn 6489 df-f 6490 df-f1 6491 df-fo 6492 df-f1o 6493 df-fv 6494

This theorem is referenced by: permaxsep 45001 permaxnul 45002 permaxpow 45003 permaxpr 45004 permaxun 45005 permaxinf2lem 45006 permac8prim 45008

W3C validator