Theorem isocnv2 5807

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 5802	. . 3 ⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → 𝐻:𝐴–1-1-onto→𝐵)
2		f1ofn 5458	. . 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 5802	. . 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		ralcom 2640	. . . . 5 ⊢ (∀𝑦 ∈ 𝐴 ∀𝑥 ∈ 𝐴 (𝑦𝑅𝑥 ↔ (𝐻‘𝑦)𝑆(𝐻‘𝑥)) ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 ↔ (𝐻‘𝑦)𝑆(𝐻‘𝑥)))
7		vex 2740	. . . . . . . . . 10 ⊢ 𝑥 ∈ V
8		vex 2740	. . . . . . . . . 10 ⊢ 𝑦 ∈ V
9	7, 8	brcnv 4806	. . . . . . . . 9 ⊢ (𝑥◡𝑅𝑦 ↔ 𝑦𝑅𝑥)
10	9	a1i 9	. . . . . . . 8 ⊢ (((𝐻 Fn 𝐴 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝐴) → (𝑥◡𝑅𝑦 ↔ 𝑦𝑅𝑥))
11		funfvex 5528	. . . . . . . . . . 11 ⊢ ((Fun 𝐻 ∧ 𝑥 ∈ dom 𝐻) → (𝐻‘𝑥) ∈ V)
12	11	funfni 5312	. . . . . . . . . 10 ⊢ ((𝐻 Fn 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝐻‘𝑥) ∈ V)
13	12	adantr 276	. . . . . . . . 9 ⊢ (((𝐻 Fn 𝐴 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝐴) → (𝐻‘𝑥) ∈ V)
14		funfvex 5528	. . . . . . . . . . 11 ⊢ ((Fun 𝐻 ∧ 𝑦 ∈ dom 𝐻) → (𝐻‘𝑦) ∈ V)
15	14	funfni 5312	. . . . . . . . . 10 ⊢ ((𝐻 Fn 𝐴 ∧ 𝑦 ∈ 𝐴) → (𝐻‘𝑦) ∈ V)
16	15	adantlr 477	. . . . . . . . 9 ⊢ (((𝐻 Fn 𝐴 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝐴) → (𝐻‘𝑦) ∈ V)
17		brcnvg 4804	. . . . . . . . 9 ⊢ (((𝐻‘𝑥) ∈ V ∧ (𝐻‘𝑦) ∈ V) → ((𝐻‘𝑥)◡𝑆(𝐻‘𝑦) ↔ (𝐻‘𝑦)𝑆(𝐻‘𝑥)))
18	13, 16, 17	syl2anc 411	. . . . . . . 8 ⊢ (((𝐻 Fn 𝐴 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝐴) → ((𝐻‘𝑥)◡𝑆(𝐻‘𝑦) ↔ (𝐻‘𝑦)𝑆(𝐻‘𝑥)))
19	10, 18	bibi12d 235	. . . . . . 7 ⊢ (((𝐻 Fn 𝐴 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝐴) → ((𝑥◡𝑅𝑦 ↔ (𝐻‘𝑥)◡𝑆(𝐻‘𝑦)) ↔ (𝑦𝑅𝑥 ↔ (𝐻‘𝑦)𝑆(𝐻‘𝑥))))
20	19	ralbidva 2473	. . . . . 6 ⊢ ((𝐻 Fn 𝐴 ∧ 𝑥 ∈ 𝐴) → (∀𝑦 ∈ 𝐴 (𝑥◡𝑅𝑦 ↔ (𝐻‘𝑥)◡𝑆(𝐻‘𝑦)) ↔ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 ↔ (𝐻‘𝑦)𝑆(𝐻‘𝑥))))
21	20	ralbidva 2473	. . . . 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))))
22	6, 21	bitr4id 199	. . . 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 464	. . 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 5221	. . 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 5221	. . 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 223	. 2 ⊢ (𝐻 Fn 𝐴 → (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ↔ 𝐻 Isom ◡𝑅, ◡𝑆(𝐴, 𝐵)))
27	3, 5, 26	pm5.21nii 704	1 ⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ↔ 𝐻 Isom ◡𝑅, ◡𝑆(𝐴, 𝐵))

Syntax hints:   ∧ wa 104   ↔ wb 105   ∈ wcel 2148  ∀wral 2455  Vcvv 2737   class class class wbr 4000  ^◡ccnv 4622   Fn wfn 5207  –1-1-onto→wf1o 5211  ‘cfv 5212   Isom wiso 5213

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4118  ax-pow 4171  ax-pr 4206

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2739  df-sbc 2963  df-un 3133  df-in 3135  df-ss 3142  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-br 4001  df-opab 4062  df-id 4290  df-cnv 4631  df-co 4632  df-dm 4633  df-iota 5174  df-fun 5214  df-fn 5215  df-f 5216  df-f1 5217  df-f1o 5219  df-fv 5220  df-isom 5221

This theorem is referenced by:  infisoti  7025