Theorem isocnv2 5791

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 5786	. . 3 ⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → 𝐻:𝐴–1-1-onto→𝐵)
2		f1ofn 5443	. . 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 5786	. . 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 2633	. . . . 5 ⊢ (∀𝑦 ∈ 𝐴 ∀𝑥 ∈ 𝐴 (𝑦𝑅𝑥 ↔ (𝐻‘𝑦)𝑆(𝐻‘𝑥)) ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 ↔ (𝐻‘𝑦)𝑆(𝐻‘𝑥)))
7		vex 2733	. . . . . . . . . 10 ⊢ 𝑥 ∈ V
8		vex 2733	. . . . . . . . . 10 ⊢ 𝑦 ∈ V
9	7, 8	brcnv 4794	. . . . . . . . 9 ⊢ (𝑥◡𝑅𝑦 ↔ 𝑦𝑅𝑥)
10	9	a1i 9	. . . . . . . 8 ⊢ (((𝐻 Fn 𝐴 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝐴) → (𝑥◡𝑅𝑦 ↔ 𝑦𝑅𝑥))
11		funfvex 5513	. . . . . . . . . . 11 ⊢ ((Fun 𝐻 ∧ 𝑥 ∈ dom 𝐻) → (𝐻‘𝑥) ∈ V)
12	11	funfni 5298	. . . . . . . . . 10 ⊢ ((𝐻 Fn 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝐻‘𝑥) ∈ V)
13	12	adantr 274	. . . . . . . . 9 ⊢ (((𝐻 Fn 𝐴 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝐴) → (𝐻‘𝑥) ∈ V)
14		funfvex 5513	. . . . . . . . . . 11 ⊢ ((Fun 𝐻 ∧ 𝑦 ∈ dom 𝐻) → (𝐻‘𝑦) ∈ V)
15	14	funfni 5298	. . . . . . . . . 10 ⊢ ((𝐻 Fn 𝐴 ∧ 𝑦 ∈ 𝐴) → (𝐻‘𝑦) ∈ V)
16	15	adantlr 474	. . . . . . . . 9 ⊢ (((𝐻 Fn 𝐴 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝐴) → (𝐻‘𝑦) ∈ V)
17		brcnvg 4792	. . . . . . . . 9 ⊢ (((𝐻‘𝑥) ∈ V ∧ (𝐻‘𝑦) ∈ V) → ((𝐻‘𝑥)◡𝑆(𝐻‘𝑦) ↔ (𝐻‘𝑦)𝑆(𝐻‘𝑥)))
18	13, 16, 17	syl2anc 409	. . . . . . . 8 ⊢ (((𝐻 Fn 𝐴 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝐴) → ((𝐻‘𝑥)◡𝑆(𝐻‘𝑦) ↔ (𝐻‘𝑦)𝑆(𝐻‘𝑥)))
19	10, 18	bibi12d 234	. . . . . . 7 ⊢ (((𝐻 Fn 𝐴 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝐴) → ((𝑥◡𝑅𝑦 ↔ (𝐻‘𝑥)◡𝑆(𝐻‘𝑦)) ↔ (𝑦𝑅𝑥 ↔ (𝐻‘𝑦)𝑆(𝐻‘𝑥))))
20	19	ralbidva 2466	. . . . . 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 2466	. . . . 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 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 461	. . 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 5207	. . 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 5207	. . 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 699	1 ⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ↔ 𝐻 Isom ◡𝑅, ◡𝑆(𝐴, 𝐵))

Syntax hints:   ∧ wa 103   ↔ wb 104   ∈ wcel 2141  ∀wral 2448  Vcvv 2730   class class class wbr 3989  ^◡ccnv 4610   Fn wfn 5193  –1-1-onto→wf1o 5197  ‘cfv 5198   Isom wiso 5199

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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-sbc 2956  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-br 3990  df-opab 4051  df-id 4278  df-cnv 4619  df-co 4620  df-dm 4621  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-f1o 5205  df-fv 5206  df-isom 5207

This theorem is referenced by:  infisoti  7009