Theorem isocnv2 5472

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		isof1o 5467	. . 3 ⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → 𝐻:𝐴–1-1-onto→𝐵)
2		f1ofn 5147	. . 3 ⊢ (𝐻:𝐴–1-1-onto→𝐵 → 𝐻 Fn 𝐴)
3	1, 2	syl 14	. 2 ⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → 𝐻 Fn 𝐴)
4		isof1o 5467	. . 3 ⊢ (𝐻 Isom ◡𝑅, ◡𝑆(𝐴, 𝐵) → 𝐻:𝐴–1-1-onto→𝐵)
5	4, 2	syl 14	. 2 ⊢ (𝐻 Isom ◡𝑅, ◡𝑆(𝐴, 𝐵) → 𝐻 Fn 𝐴)
6		vex 2604	. . . . . . . . . 10 ⊢ 𝑥 ∈ V
7		vex 2604	. . . . . . . . . 10 ⊢ 𝑦 ∈ V
8	6, 7	brcnv 4536	. . . . . . . . 9 ⊢ (𝑥◡𝑅𝑦 ↔ 𝑦𝑅𝑥)
9	8	a1i 9	. . . . . . . 8 ⊢ (((𝐻 Fn 𝐴 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝐴) → (𝑥◡𝑅𝑦 ↔ 𝑦𝑅𝑥))
10		funfvex 5212	. . . . . . . . . . 11 ⊢ ((Fun 𝐻 ∧ 𝑥 ∈ dom 𝐻) → (𝐻‘𝑥) ∈ V)
11	10	funfni 5019	. . . . . . . . . 10 ⊢ ((𝐻 Fn 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝐻‘𝑥) ∈ V)
12	11	adantr 270	. . . . . . . . 9 ⊢ (((𝐻 Fn 𝐴 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝐴) → (𝐻‘𝑥) ∈ V)
13		funfvex 5212	. . . . . . . . . . 11 ⊢ ((Fun 𝐻 ∧ 𝑦 ∈ dom 𝐻) → (𝐻‘𝑦) ∈ V)
14	13	funfni 5019	. . . . . . . . . 10 ⊢ ((𝐻 Fn 𝐴 ∧ 𝑦 ∈ 𝐴) → (𝐻‘𝑦) ∈ V)
15	14	adantlr 460	. . . . . . . . 9 ⊢ (((𝐻 Fn 𝐴 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝐴) → (𝐻‘𝑦) ∈ V)
16		brcnvg 4534	. . . . . . . . 9 ⊢ (((𝐻‘𝑥) ∈ V ∧ (𝐻‘𝑦) ∈ V) → ((𝐻‘𝑥)◡𝑆(𝐻‘𝑦) ↔ (𝐻‘𝑦)𝑆(𝐻‘𝑥)))
17	12, 15, 16	syl2anc 403	. . . . . . . 8 ⊢ (((𝐻 Fn 𝐴 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝐴) → ((𝐻‘𝑥)◡𝑆(𝐻‘𝑦) ↔ (𝐻‘𝑦)𝑆(𝐻‘𝑥)))
18	9, 17	bibi12d 233	. . . . . . 7 ⊢ (((𝐻 Fn 𝐴 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝐴) → ((𝑥◡𝑅𝑦 ↔ (𝐻‘𝑥)◡𝑆(𝐻‘𝑦)) ↔ (𝑦𝑅𝑥 ↔ (𝐻‘𝑦)𝑆(𝐻‘𝑥))))
19	18	ralbidva 2364	. . . . . 6 ⊢ ((𝐻 Fn 𝐴 ∧ 𝑥 ∈ 𝐴) → (∀𝑦 ∈ 𝐴 (𝑥◡𝑅𝑦 ↔ (𝐻‘𝑥)◡𝑆(𝐻‘𝑦)) ↔ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 ↔ (𝐻‘𝑦)𝑆(𝐻‘𝑥))))
20	19	ralbidva 2364	. . . . 5 ⊢ (𝐻 Fn 𝐴 → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥◡𝑅𝑦 ↔ (𝐻‘𝑥)◡𝑆(𝐻‘𝑦)) ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 ↔ (𝐻‘𝑦)𝑆(𝐻‘𝑥))))
21		ralcom 2517	. . . . 5 ⊢ (∀𝑦 ∈ 𝐴 ∀𝑥 ∈ 𝐴 (𝑦𝑅𝑥 ↔ (𝐻‘𝑦)𝑆(𝐻‘𝑥)) ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 ↔ (𝐻‘𝑦)𝑆(𝐻‘𝑥)))
22	20, 21	syl6rbbr 197	. . . 4 ⊢ (𝐻 Fn 𝐴 → (∀𝑦 ∈ 𝐴 ∀𝑥 ∈ 𝐴 (𝑦𝑅𝑥 ↔ (𝐻‘𝑦)𝑆(𝐻‘𝑥)) ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥◡𝑅𝑦 ↔ (𝐻‘𝑥)◡𝑆(𝐻‘𝑦))))
23	22	anbi2d 451	. . 3 ⊢ (𝐻 Fn 𝐴 → ((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)))))
24		df-isom 4931	. . 3 ⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ↔ (𝐻:𝐴–1-1-onto→𝐵 ∧ ∀𝑦 ∈ 𝐴 ∀𝑥 ∈ 𝐴 (𝑦𝑅𝑥 ↔ (𝐻‘𝑦)𝑆(𝐻‘𝑥))))
25		df-isom 4931	. . 3 ⊢ (𝐻 Isom ◡𝑅, ◡𝑆(𝐴, 𝐵) ↔ (𝐻:𝐴–1-1-onto→𝐵 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥◡𝑅𝑦 ↔ (𝐻‘𝑥)◡𝑆(𝐻‘𝑦))))
26	23, 24, 25	3bitr4g 221	. 2 ⊢ (𝐻 Fn 𝐴 → (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ↔ 𝐻 Isom ◡𝑅, ◡𝑆(𝐴, 𝐵)))
27	3, 5, 26	pm5.21nii 652	1 ⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ↔ 𝐻 Isom ◡𝑅, ◡𝑆(𝐴, 𝐵))

Syntax hints:   ∧ wa 102   ↔ wb 103   ∈ wcel 1433  ∀wral 2348  Vcvv 2601   class class class wbr 3785  ^◡ccnv 4362   Fn wfn 4917  –1-1-onto→wf1o 4921  ‘cfv 4922   Isom wiso 4923

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-pow 3948  ax-pr 3964

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-rex 2354  df-v 2603  df-sbc 2816  df-un 2977  df-in 2979  df-ss 2986  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-br 3786  df-opab 3840  df-id 4048  df-cnv 4371  df-co 4372  df-dm 4373  df-iota 4887  df-fun 4924  df-fn 4925  df-f 4926  df-f1 4927  df-f1o 4929  df-fv 4930  df-isom 4931

This theorem is referenced by:  infisoti  6445