isocnv2 - Metamath Proof Explorer

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Theorem isocnv2 7086

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 3356	. . . 4 ⊢ (∀𝑦 ∈ 𝐴 ∀𝑥 ∈ 𝐴 (𝑦𝑅𝑥 ↔ (𝐻‘𝑦)𝑆(𝐻‘𝑥)) ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 ↔ (𝐻‘𝑦)𝑆(𝐻‘𝑥)))
2		vex 3499	. . . . . . 7 ⊢ 𝑥 ∈ V
3		vex 3499	. . . . . . 7 ⊢ 𝑦 ∈ V
4	2, 3	brcnv 5755	. . . . . 6 ⊢ (𝑥^◡𝑅𝑦 ↔ 𝑦𝑅𝑥)
5		fvex 6685	. . . . . . 7 ⊢ (𝐻‘𝑥) ∈ V
6		fvex 6685	. . . . . . 7 ⊢ (𝐻‘𝑦) ∈ V
7	5, 6	brcnv 5755	. . . . . 6 ⊢ ((𝐻‘𝑥)^◡𝑆(𝐻‘𝑦) ↔ (𝐻‘𝑦)𝑆(𝐻‘𝑥))
8	4, 7	bibi12i 342	. . . . 5 ⊢ ((𝑥^◡𝑅𝑦 ↔ (𝐻‘𝑥)^◡𝑆(𝐻‘𝑦)) ↔ (𝑦𝑅𝑥 ↔ (𝐻‘𝑦)𝑆(𝐻‘𝑥)))
9	8	2ralbii 3168	. . . 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 280	. . 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 624	. 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 6366	. 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 6366	. 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 305	1 ⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ↔ 𝐻 Isom ^◡𝑅, ^◡𝑆(𝐴, 𝐵))

Colors of variables: wff setvar class

Syntax hints: ↔ wb 208 ∧ wa 398 ∀wral 3140 class class class wbr 5068 ^◡ccnv 5556 –1-1-onto→wf1o 6356 ‘cfv 6357 Isom wiso 6358

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 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pr 5332

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-sbc 3775 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-br 5069 df-opab 5131 df-cnv 5565 df-iota 6316 df-fv 6365 df-isom 6366

This theorem is referenced by: infiso 8974 wofib 9011 leiso 13820 gtiso 30438

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