Theorem isocnv3 7278

Description: Complementation law for isomorphism. (Contributed by Mario Carneiro, 9-Sep-2015.)

Hypotheses

Ref	Expression
isocnv3.1	⊢ 𝐶 = ((𝐴 × 𝐴) ∖ 𝑅)
isocnv3.2	⊢ 𝐷 = ((𝐵 × 𝐵) ∖ 𝑆)

Assertion

Ref	Expression
isocnv3	⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ↔ 𝐻 Isom 𝐶, 𝐷 (𝐴, 𝐵))

Proof of Theorem isocnv3

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		notbi 319	. . . . 5 ⊢ ((𝑥𝑅𝑦 ↔ (𝐻‘𝑥)𝑆(𝐻‘𝑦)) ↔ (¬ 𝑥𝑅𝑦 ↔ ¬ (𝐻‘𝑥)𝑆(𝐻‘𝑦)))
2		brxp 5673	. . . . . . . 8 ⊢ (𝑥(𝐴 × 𝐴)𝑦 ↔ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴))
3		isocnv3.1	. . . . . . . . . . 11 ⊢ 𝐶 = ((𝐴 × 𝐴) ∖ 𝑅)
4	3	breqi 5104	. . . . . . . . . 10 ⊢ (𝑥𝐶𝑦 ↔ 𝑥((𝐴 × 𝐴) ∖ 𝑅)𝑦)
5		brdif 5151	. . . . . . . . . 10 ⊢ (𝑥((𝐴 × 𝐴) ∖ 𝑅)𝑦 ↔ (𝑥(𝐴 × 𝐴)𝑦 ∧ ¬ 𝑥𝑅𝑦))
6	4, 5	bitri 275	. . . . . . . . 9 ⊢ (𝑥𝐶𝑦 ↔ (𝑥(𝐴 × 𝐴)𝑦 ∧ ¬ 𝑥𝑅𝑦))
7	6	baib 535	. . . . . . . 8 ⊢ (𝑥(𝐴 × 𝐴)𝑦 → (𝑥𝐶𝑦 ↔ ¬ 𝑥𝑅𝑦))
8	2, 7	sylbir 235	. . . . . . 7 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (𝑥𝐶𝑦 ↔ ¬ 𝑥𝑅𝑦))
9	8	adantl 481	. . . . . 6 ⊢ ((𝐻:𝐴–1-1-onto→𝐵 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → (𝑥𝐶𝑦 ↔ ¬ 𝑥𝑅𝑦))
10		f1of 6774	. . . . . . . 8 ⊢ (𝐻:𝐴–1-1-onto→𝐵 → 𝐻:𝐴⟶𝐵)
11		ffvelcdm 7026	. . . . . . . . . 10 ⊢ ((𝐻:𝐴⟶𝐵 ∧ 𝑥 ∈ 𝐴) → (𝐻‘𝑥) ∈ 𝐵)
12		ffvelcdm 7026	. . . . . . . . . 10 ⊢ ((𝐻:𝐴⟶𝐵 ∧ 𝑦 ∈ 𝐴) → (𝐻‘𝑦) ∈ 𝐵)
13	11, 12	anim12dan 619	. . . . . . . . 9 ⊢ ((𝐻:𝐴⟶𝐵 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → ((𝐻‘𝑥) ∈ 𝐵 ∧ (𝐻‘𝑦) ∈ 𝐵))
14		brxp 5673	. . . . . . . . 9 ⊢ ((𝐻‘𝑥)(𝐵 × 𝐵)(𝐻‘𝑦) ↔ ((𝐻‘𝑥) ∈ 𝐵 ∧ (𝐻‘𝑦) ∈ 𝐵))
15	13, 14	sylibr 234	. . . . . . . 8 ⊢ ((𝐻:𝐴⟶𝐵 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → (𝐻‘𝑥)(𝐵 × 𝐵)(𝐻‘𝑦))
16	10, 15	sylan 580	. . . . . . 7 ⊢ ((𝐻:𝐴–1-1-onto→𝐵 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → (𝐻‘𝑥)(𝐵 × 𝐵)(𝐻‘𝑦))
17		isocnv3.2	. . . . . . . . . 10 ⊢ 𝐷 = ((𝐵 × 𝐵) ∖ 𝑆)
18	17	breqi 5104	. . . . . . . . 9 ⊢ ((𝐻‘𝑥)𝐷(𝐻‘𝑦) ↔ (𝐻‘𝑥)((𝐵 × 𝐵) ∖ 𝑆)(𝐻‘𝑦))
19		brdif 5151	. . . . . . . . 9 ⊢ ((𝐻‘𝑥)((𝐵 × 𝐵) ∖ 𝑆)(𝐻‘𝑦) ↔ ((𝐻‘𝑥)(𝐵 × 𝐵)(𝐻‘𝑦) ∧ ¬ (𝐻‘𝑥)𝑆(𝐻‘𝑦)))
20	18, 19	bitri 275	. . . . . . . 8 ⊢ ((𝐻‘𝑥)𝐷(𝐻‘𝑦) ↔ ((𝐻‘𝑥)(𝐵 × 𝐵)(𝐻‘𝑦) ∧ ¬ (𝐻‘𝑥)𝑆(𝐻‘𝑦)))
21	20	baib 535	. . . . . . 7 ⊢ ((𝐻‘𝑥)(𝐵 × 𝐵)(𝐻‘𝑦) → ((𝐻‘𝑥)𝐷(𝐻‘𝑦) ↔ ¬ (𝐻‘𝑥)𝑆(𝐻‘𝑦)))
22	16, 21	syl 17	. . . . . 6 ⊢ ((𝐻:𝐴–1-1-onto→𝐵 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → ((𝐻‘𝑥)𝐷(𝐻‘𝑦) ↔ ¬ (𝐻‘𝑥)𝑆(𝐻‘𝑦)))
23	9, 22	bibi12d 345	. . . . 5 ⊢ ((𝐻:𝐴–1-1-onto→𝐵 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → ((𝑥𝐶𝑦 ↔ (𝐻‘𝑥)𝐷(𝐻‘𝑦)) ↔ (¬ 𝑥𝑅𝑦 ↔ ¬ (𝐻‘𝑥)𝑆(𝐻‘𝑦))))
24	1, 23	bitr4id 290	. . . 4 ⊢ ((𝐻:𝐴–1-1-onto→𝐵 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → ((𝑥𝑅𝑦 ↔ (𝐻‘𝑥)𝑆(𝐻‘𝑦)) ↔ (𝑥𝐶𝑦 ↔ (𝐻‘𝑥)𝐷(𝐻‘𝑦))))
25	24	2ralbidva 3198	. . 3 ⊢ (𝐻:𝐴–1-1-onto→𝐵 → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 ↔ (𝐻‘𝑥)𝑆(𝐻‘𝑦)) ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝐶𝑦 ↔ (𝐻‘𝑥)𝐷(𝐻‘𝑦))))
26	25	pm5.32i 574	. 2 ⊢ ((𝐻:𝐴–1-1-onto→𝐵 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 ↔ (𝐻‘𝑥)𝑆(𝐻‘𝑦))) ↔ (𝐻:𝐴–1-1-onto→𝐵 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝐶𝑦 ↔ (𝐻‘𝑥)𝐷(𝐻‘𝑦))))
27		df-isom 6501	. 2 ⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ↔ (𝐻:𝐴–1-1-onto→𝐵 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 ↔ (𝐻‘𝑥)𝑆(𝐻‘𝑦))))
28		df-isom 6501	. 2 ⊢ (𝐻 Isom 𝐶, 𝐷 (𝐴, 𝐵) ↔ (𝐻:𝐴–1-1-onto→𝐵 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝐶𝑦 ↔ (𝐻‘𝑥)𝐷(𝐻‘𝑦))))
29	26, 27, 28	3bitr4i 303	1 ⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ↔ 𝐻 Isom 𝐶, 𝐷 (𝐴, 𝐵))

Syntax hints:  ¬ wn 3   ↔ wb 206   ∧ wa 395   = wceq 1541   ∈ wcel 2113  ∀wral 3051   ∖ cdif 3898   class class class wbr 5098   × cxp 5622  ⟶wf 6488  –1-1-onto→wf1o 6491  ‘cfv 6492   Isom wiso 6493

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-12 2184  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pr 5377

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-ne 2933  df-ral 3052  df-rex 3061  df-rab 3400  df-v 3442  df-dif 3904  df-un 3906  df-ss 3918  df-nul 4286  df-if 4480  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-br 5099  df-opab 5161  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-f1o 6499  df-fv 6500  df-isom 6501

This theorem is referenced by:  leiso  14382  gtiso  32780