isocnv2 - Metamath Proof Explorer

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Theorem isocnv2 7195

Description: Converse law for isomorphism. (Contributed by Mario Carneiro, 30-Jan-2014.)

**Assertion**
Ref	Expression
isocnv2	⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ↔ 𝐻 Isom ^◡𝑅, ^◡𝑆(𝐴, 𝐵))

Proof of Theorem isocnv2 Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ralcom 3282	. . . 4 ⊢ (∀𝑦 ∈ 𝐴 ∀𝑥 ∈ 𝐴 (𝑦𝑅𝑥 ↔ (𝐻‘𝑦)𝑆(𝐻‘𝑥)) ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 ↔ (𝐻‘𝑦)𝑆(𝐻‘𝑥)))
2		vex 3434	. . . . . . 7 ⊢ 𝑥 ∈ V
3		vex 3434	. . . . . . 7 ⊢ 𝑦 ∈ V
4	2, 3	brcnv 5788	. . . . . 6 ⊢ (𝑥^◡𝑅𝑦 ↔ 𝑦𝑅𝑥)
5		fvex 6781	. . . . . . 7 ⊢ (𝐻‘𝑥) ∈ V
6		fvex 6781	. . . . . . 7 ⊢ (𝐻‘𝑦) ∈ V
7	5, 6	brcnv 5788	. . . . . 6 ⊢ ((𝐻‘𝑥)^◡𝑆(𝐻‘𝑦) ↔ (𝐻‘𝑦)𝑆(𝐻‘𝑥))
8	4, 7	bibi12i 339	. . . . 5 ⊢ ((𝑥^◡𝑅𝑦 ↔ (𝐻‘𝑥)^◡𝑆(𝐻‘𝑦)) ↔ (𝑦𝑅𝑥 ↔ (𝐻‘𝑦)𝑆(𝐻‘𝑥)))
9	8	2ralbii 3093	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥^◡𝑅𝑦 ↔ (𝐻‘𝑥)^◡𝑆(𝐻‘𝑦)) ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 ↔ (𝐻‘𝑦)𝑆(𝐻‘𝑥)))
10	1, 9	bitr4i 277	. . 3 ⊢ (∀𝑦 ∈ 𝐴 ∀𝑥 ∈ 𝐴 (𝑦𝑅𝑥 ↔ (𝐻‘𝑦)𝑆(𝐻‘𝑥)) ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥^◡𝑅𝑦 ↔ (𝐻‘𝑥)^◡𝑆(𝐻‘𝑦)))
11	10	anbi2i 622	. 2 ⊢ ((𝐻:𝐴–1-1-onto→𝐵 ∧ ∀𝑦 ∈ 𝐴 ∀𝑥 ∈ 𝐴 (𝑦𝑅𝑥 ↔ (𝐻‘𝑦)𝑆(𝐻‘𝑥))) ↔ (𝐻:𝐴–1-1-onto→𝐵 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥^◡𝑅𝑦 ↔ (𝐻‘𝑥)^◡𝑆(𝐻‘𝑦))))
12		df-isom 6439	. 2 ⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ↔ (𝐻:𝐴–1-1-onto→𝐵 ∧ ∀𝑦 ∈ 𝐴 ∀𝑥 ∈ 𝐴 (𝑦𝑅𝑥 ↔ (𝐻‘𝑦)𝑆(𝐻‘𝑥))))
13		df-isom 6439	. 2 ⊢ (𝐻 Isom ^◡𝑅, ^◡𝑆(𝐴, 𝐵) ↔ (𝐻:𝐴–1-1-onto→𝐵 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥^◡𝑅𝑦 ↔ (𝐻‘𝑥)^◡𝑆(𝐻‘𝑦))))
14	11, 12, 13	3bitr4i 302	1 ⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ↔ 𝐻 Isom ^◡𝑅, ^◡𝑆(𝐴, 𝐵))

Colors of variables: wff setvar class

Syntax hints: ↔ wb 205 ∧ wa 395 ∀wral 3065 class class class wbr 5078 ^◡ccnv 5587 –1-1-onto→wf1o 6429 ‘cfv 6430 Isom wiso 6431

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1801 ax-4 1815 ax-5 1916 ax-6 1974 ax-7 2014 ax-8 2111 ax-9 2119 ax-10 2140 ax-11 2157 ax-12 2174 ax-ext 2710 ax-sep 5226 ax-nul 5233 ax-pr 5355

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3an 1087 df-tru 1544 df-fal 1554 df-ex 1786 df-nf 1790 df-sb 2071 df-mo 2541 df-eu 2570 df-clab 2717 df-cleq 2731 df-clel 2817 df-ral 3070 df-rex 3071 df-rab 3074 df-v 3432 df-dif 3894 df-un 3896 df-in 3898 df-ss 3908 df-nul 4262 df-if 4465 df-sn 4567 df-pr 4569 df-op 4573 df-uni 4845 df-br 5079 df-opab 5141 df-cnv 5596 df-iota 6388 df-fv 6438 df-isom 6439

This theorem is referenced by: infiso 9228 wofib 9265 leiso 14154 gtiso 31012

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