isocnv2 - Metamath Proof Explorer

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Theorem isocnv2 7063

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 3307	. . . 4 ⊢ (∀𝑦 ∈ 𝐴 ∀𝑥 ∈ 𝐴 (𝑦𝑅𝑥 ↔ (𝐻‘𝑦)𝑆(𝐻‘𝑥)) ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 ↔ (𝐻‘𝑦)𝑆(𝐻‘𝑥)))
2		vex 3444	. . . . . . 7 ⊢ 𝑥 ∈ V
3		vex 3444	. . . . . . 7 ⊢ 𝑦 ∈ V
4	2, 3	brcnv 5717	. . . . . 6 ⊢ (𝑥^◡𝑅𝑦 ↔ 𝑦𝑅𝑥)
5		fvex 6658	. . . . . . 7 ⊢ (𝐻‘𝑥) ∈ V
6		fvex 6658	. . . . . . 7 ⊢ (𝐻‘𝑦) ∈ V
7	5, 6	brcnv 5717	. . . . . 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 3134	. . . 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 625	. 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 6333	. 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 6333	. 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 3106 class class class wbr 5030 ^◡ccnv 5518 –1-1-onto→wf1o 6323 ‘cfv 6324 Isom wiso 6325

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pr 5295

This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ral 3111 df-rex 3112 df-v 3443 df-sbc 3721 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-cnv 5527 df-iota 6283 df-fv 6332 df-isom 6333

This theorem is referenced by: infiso 8956 wofib 8993 leiso 13813 gtiso 30460

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