isocnv2 - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > isocnv2		Structured version Visualization version GIF version

Theorem isocnv2 6744

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 3236	. . . 4 ⊢ (∀𝑦 ∈ 𝐴 ∀𝑥 ∈ 𝐴 (𝑦𝑅𝑥 ↔ (𝐻‘𝑦)𝑆(𝐻‘𝑥)) ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 ↔ (𝐻‘𝑦)𝑆(𝐻‘𝑥)))
2		vex 3343	. . . . . . 7 ⊢ 𝑥 ∈ V
3		vex 3343	. . . . . . 7 ⊢ 𝑦 ∈ V
4	2, 3	brcnv 5460	. . . . . 6 ⊢ (𝑥^◡𝑅𝑦 ↔ 𝑦𝑅𝑥)
5		fvex 6362	. . . . . . 7 ⊢ (𝐻‘𝑥) ∈ V
6		fvex 6362	. . . . . . 7 ⊢ (𝐻‘𝑦) ∈ V
7	5, 6	brcnv 5460	. . . . . 6 ⊢ ((𝐻‘𝑥)^◡𝑆(𝐻‘𝑦) ↔ (𝐻‘𝑦)𝑆(𝐻‘𝑥))
8	4, 7	bibi12i 328	. . . . 5 ⊢ ((𝑥^◡𝑅𝑦 ↔ (𝐻‘𝑥)^◡𝑆(𝐻‘𝑦)) ↔ (𝑦𝑅𝑥 ↔ (𝐻‘𝑦)𝑆(𝐻‘𝑥)))
9	8	2ralbii 3119	. . . 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 267	. . 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 732	. 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 6058	. 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 6058	. 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 292	1 ⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ↔ 𝐻 Isom ^◡𝑅, ^◡𝑆(𝐴, 𝐵))

Colors of variables: wff setvar class

Syntax hints: ↔ wb 196 ∧ wa 383 ∀wral 3050 class class class wbr 4804 ^◡ccnv 5265 –1-1-onto→wf1o 6048 ‘cfv 6049 Isom wiso 6050

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1871 ax-4 1886 ax-5 1988 ax-6 2054 ax-7 2090 ax-9 2148 ax-10 2168 ax-11 2183 ax-12 2196 ax-13 2391 ax-ext 2740 ax-sep 4933 ax-nul 4941 ax-pr 5055

This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3an 1074 df-tru 1635 df-ex 1854 df-nf 1859 df-sb 2047 df-eu 2611 df-mo 2612 df-clab 2747 df-cleq 2753 df-clel 2756 df-nfc 2891 df-ral 3055 df-rex 3056 df-rab 3059 df-v 3342 df-sbc 3577 df-dif 3718 df-un 3720 df-in 3722 df-ss 3729 df-nul 4059 df-if 4231 df-sn 4322 df-pr 4324 df-op 4328 df-uni 4589 df-br 4805 df-opab 4865 df-cnv 5274 df-iota 6012 df-fv 6057 df-isom 6058

This theorem is referenced by: infiso 8578 wofib 8615 leiso 13435 gtiso 29787

W3C validator