Theorem isocnv2 5713

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 5708	. . 3 ⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → 𝐻:𝐴–1-1-onto→𝐵)
2		f1ofn 5368	. . 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 5708	. . 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 2689	. . . . . . . . . 10 ⊢ 𝑥 ∈ V
7		vex 2689	. . . . . . . . . 10 ⊢ 𝑦 ∈ V
8	6, 7	brcnv 4722	. . . . . . . . 9 ⊢ (𝑥◡𝑅𝑦 ↔ 𝑦𝑅𝑥)
9	8	a1i 9	. . . . . . . 8 ⊢ (((𝐻 Fn 𝐴 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝐴) → (𝑥◡𝑅𝑦 ↔ 𝑦𝑅𝑥))
10		funfvex 5438	. . . . . . . . . . 11 ⊢ ((Fun 𝐻 ∧ 𝑥 ∈ dom 𝐻) → (𝐻‘𝑥) ∈ V)
11	10	funfni 5223	. . . . . . . . . 10 ⊢ ((𝐻 Fn 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝐻‘𝑥) ∈ V)
12	11	adantr 274	. . . . . . . . 9 ⊢ (((𝐻 Fn 𝐴 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝐴) → (𝐻‘𝑥) ∈ V)
13		funfvex 5438	. . . . . . . . . . 11 ⊢ ((Fun 𝐻 ∧ 𝑦 ∈ dom 𝐻) → (𝐻‘𝑦) ∈ V)
14	13	funfni 5223	. . . . . . . . . 10 ⊢ ((𝐻 Fn 𝐴 ∧ 𝑦 ∈ 𝐴) → (𝐻‘𝑦) ∈ V)
15	14	adantlr 468	. . . . . . . . 9 ⊢ (((𝐻 Fn 𝐴 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝐴) → (𝐻‘𝑦) ∈ V)
16		brcnvg 4720	. . . . . . . . 9 ⊢ (((𝐻‘𝑥) ∈ V ∧ (𝐻‘𝑦) ∈ V) → ((𝐻‘𝑥)◡𝑆(𝐻‘𝑦) ↔ (𝐻‘𝑦)𝑆(𝐻‘𝑥)))
17	12, 15, 16	syl2anc 408	. . . . . . . 8 ⊢ (((𝐻 Fn 𝐴 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝐴) → ((𝐻‘𝑥)◡𝑆(𝐻‘𝑦) ↔ (𝐻‘𝑦)𝑆(𝐻‘𝑥)))
18	9, 17	bibi12d 234	. . . . . . 7 ⊢ (((𝐻 Fn 𝐴 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝐴) → ((𝑥◡𝑅𝑦 ↔ (𝐻‘𝑥)◡𝑆(𝐻‘𝑦)) ↔ (𝑦𝑅𝑥 ↔ (𝐻‘𝑦)𝑆(𝐻‘𝑥))))
19	18	ralbidva 2433	. . . . . 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 2433	. . . . 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 2594	. . . . 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 198	. . . 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 459	. . 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 5132	. . 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 5132	. . 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 222	. 2 ⊢ (𝐻 Fn 𝐴 → (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ↔ 𝐻 Isom ◡𝑅, ◡𝑆(𝐴, 𝐵)))
27	3, 5, 26	pm5.21nii 693	1 ⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ↔ 𝐻 Isom ◡𝑅, ◡𝑆(𝐴, 𝐵))

Syntax hints:   ∧ wa 103   ↔ wb 104   ∈ wcel 1480  ∀wral 2416  Vcvv 2686   class class class wbr 3929  ^◡ccnv 4538   Fn wfn 5118  –1-1-onto→wf1o 5122  ‘cfv 5123   Isom wiso 5124

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-v 2688  df-sbc 2910  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-br 3930  df-opab 3990  df-id 4215  df-cnv 4547  df-co 4548  df-dm 4549  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-f1o 5130  df-fv 5131  df-isom 5132

This theorem is referenced by:  infisoti  6919