isocnv2 - Metamath Proof Explorer

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Theorem isocnv2 7100

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 3259	. . . 4 ⊢ (∀𝑦 ∈ 𝐴 ∀𝑥 ∈ 𝐴 (𝑦𝑅𝑥 ↔ (𝐻‘𝑦)𝑆(𝐻‘𝑥)) ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 ↔ (𝐻‘𝑦)𝑆(𝐻‘𝑥)))
2		vex 3403	. . . . . . 7 ⊢ 𝑥 ∈ V
3		vex 3403	. . . . . . 7 ⊢ 𝑦 ∈ V
4	2, 3	brcnv 5726	. . . . . 6 ⊢ (𝑥^◡𝑅𝑦 ↔ 𝑦𝑅𝑥)
5		fvex 6690	. . . . . . 7 ⊢ (𝐻‘𝑥) ∈ V
6		fvex 6690	. . . . . . 7 ⊢ (𝐻‘𝑦) ∈ V
7	5, 6	brcnv 5726	. . . . . 6 ⊢ ((𝐻‘𝑥)^◡𝑆(𝐻‘𝑦) ↔ (𝐻‘𝑦)𝑆(𝐻‘𝑥))
8	4, 7	bibi12i 343	. . . . 5 ⊢ ((𝑥^◡𝑅𝑦 ↔ (𝐻‘𝑥)^◡𝑆(𝐻‘𝑦)) ↔ (𝑦𝑅𝑥 ↔ (𝐻‘𝑦)𝑆(𝐻‘𝑥)))
9	8	2ralbii 3082	. . . 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 281	. . 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 626	. 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 6349	. 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 6349	. 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 306	1 ⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ↔ 𝐻 Isom ^◡𝑅, ^◡𝑆(𝐴, 𝐵))

Colors of variables: wff setvar class

Syntax hints: ↔ wb 209 ∧ wa 399 ∀wral 3054 class class class wbr 5031 ^◡ccnv 5525 –1-1-onto→wf1o 6339 ‘cfv 6340 Isom wiso 6341

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2711 ax-sep 5168 ax-nul 5175 ax-pr 5297

This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2541 df-eu 2571 df-clab 2718 df-cleq 2731 df-clel 2812 df-ral 3059 df-rex 3060 df-v 3401 df-sbc 3682 df-dif 3847 df-un 3849 df-in 3851 df-ss 3861 df-nul 4213 df-if 4416 df-sn 4518 df-pr 4520 df-op 4524 df-uni 4798 df-br 5032 df-opab 5094 df-cnv 5534 df-iota 6298 df-fv 6348 df-isom 6349

This theorem is referenced by: infiso 9048 wofib 9085 leiso 13914 gtiso 30611

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