Theorem isocnv3 7276

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 320	. . . . 5 ⊢ ((𝑥𝑅𝑦 ↔ (𝐻‘𝑥)𝑆(𝐻‘𝑦)) ↔ (¬ 𝑥𝑅𝑦 ↔ ¬ (𝐻‘𝑥)𝑆(𝐻‘𝑦)))
2		brxp 5667	. . . . . . . 8 ⊢ (𝑥(𝐴 × 𝐴)𝑦 ↔ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴))
3		isocnv3.1	. . . . . . . . . . 11 ⊢ 𝐶 = ((𝐴 × 𝐴) ∖ 𝑅)
4	3	breqi 5078	. . . . . . . . . 10 ⊢ (𝑥𝐶𝑦 ↔ 𝑥((𝐴 × 𝐴) ∖ 𝑅)𝑦)
5		brdif 5125	. . . . . . . . . 10 ⊢ (𝑥((𝐴 × 𝐴) ∖ 𝑅)𝑦 ↔ (𝑥(𝐴 × 𝐴)𝑦 ∧ ¬ 𝑥𝑅𝑦))
6	4, 5	bitri 276	. . . . . . . . 9 ⊢ (𝑥𝐶𝑦 ↔ (𝑥(𝐴 × 𝐴)𝑦 ∧ ¬ 𝑥𝑅𝑦))
7	6	baib 540	. . . . . . . 8 ⊢ (𝑥(𝐴 × 𝐴)𝑦 → (𝑥𝐶𝑦 ↔ ¬ 𝑥𝑅𝑦))
8	2, 7	sylbir 236	. . . . . . 7 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (𝑥𝐶𝑦 ↔ ¬ 𝑥𝑅𝑦))
9	8	adantl 482	. . . . . 6 ⊢ ((𝐻:𝐴–1-1-onto→𝐵 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → (𝑥𝐶𝑦 ↔ ¬ 𝑥𝑅𝑦))
10		f1of 6767	. . . . . . . 8 ⊢ (𝐻:𝐴–1-1-onto→𝐵 → 𝐻:𝐴⟶𝐵)
11		ffvelcdm 7022	. . . . . . . . . 10 ⊢ ((𝐻:𝐴⟶𝐵 ∧ 𝑥 ∈ 𝐴) → (𝐻‘𝑥) ∈ 𝐵)
12		ffvelcdm 7022	. . . . . . . . . 10 ⊢ ((𝐻:𝐴⟶𝐵 ∧ 𝑦 ∈ 𝐴) → (𝐻‘𝑦) ∈ 𝐵)
13	11, 12	anim12dan 625	. . . . . . . . 9 ⊢ ((𝐻:𝐴⟶𝐵 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → ((𝐻‘𝑥) ∈ 𝐵 ∧ (𝐻‘𝑦) ∈ 𝐵))
14		brxp 5667	. . . . . . . . 9 ⊢ ((𝐻‘𝑥)(𝐵 × 𝐵)(𝐻‘𝑦) ↔ ((𝐻‘𝑥) ∈ 𝐵 ∧ (𝐻‘𝑦) ∈ 𝐵))
15	13, 14	sylibr 235	. . . . . . . 8 ⊢ ((𝐻:𝐴⟶𝐵 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → (𝐻‘𝑥)(𝐵 × 𝐵)(𝐻‘𝑦))
16	10, 15	sylan 586	. . . . . . 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 5078	. . . . . . . . 9 ⊢ ((𝐻‘𝑥)𝐷(𝐻‘𝑦) ↔ (𝐻‘𝑥)((𝐵 × 𝐵) ∖ 𝑆)(𝐻‘𝑦))
19		brdif 5125	. . . . . . . . 9 ⊢ ((𝐻‘𝑥)((𝐵 × 𝐵) ∖ 𝑆)(𝐻‘𝑦) ↔ ((𝐻‘𝑥)(𝐵 × 𝐵)(𝐻‘𝑦) ∧ ¬ (𝐻‘𝑥)𝑆(𝐻‘𝑦)))
20	18, 19	bitri 276	. . . . . . . 8 ⊢ ((𝐻‘𝑥)𝐷(𝐻‘𝑦) ↔ ((𝐻‘𝑥)(𝐵 × 𝐵)(𝐻‘𝑦) ∧ ¬ (𝐻‘𝑥)𝑆(𝐻‘𝑦)))
21	20	baib 540	. . . . . . 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 346	. . . . 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 291	. . . 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 3201	. . 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 579	. 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 6494	. 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 6494	. 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 304	1 ⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ↔ 𝐻 Isom 𝐶, 𝐷 (𝐴, 𝐵))

Syntax hints:  ¬ wn 3   ↔ wb 207   ∧ wa 396   = wceq 1547   ∈ wcel 2119  ∀wral 3053   ∖ cdif 3880   class class class wbr 5072   × cxp 5616  ⟶wf 6481  –1-1-onto→wf1o 6484  ‘cfv 6485   Isom wiso 6486

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-12 2189  ax-ext 2711  ax-sep 5218  ax-nul 5228  ax-pr 5362

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-ne 2935  df-ral 3054  df-rex 3064  df-rab 3392  df-v 3433  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4262  df-if 4455  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-br 5073  df-opab 5135  df-id 5513  df-xp 5624  df-rel 5625  df-cnv 5626  df-co 5627  df-dm 5628  df-rn 5629  df-iota 6441  df-fun 6487  df-fn 6488  df-f 6489  df-f1 6490  df-f1o 6492  df-fv 6493  df-isom 6494

This theorem is referenced by:  leiso  14412  gtiso  32793