Theorem isocnv3 6582

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		brxp 5147	. . . . . . . 8 ⊢ (𝑥(𝐴 × 𝐴)𝑦 ↔ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴))
2		isocnv3.1	. . . . . . . . . . 11 ⊢ 𝐶 = ((𝐴 × 𝐴) ∖ 𝑅)
3	2	breqi 4659	. . . . . . . . . 10 ⊢ (𝑥𝐶𝑦 ↔ 𝑥((𝐴 × 𝐴) ∖ 𝑅)𝑦)
4		brdif 4705	. . . . . . . . . 10 ⊢ (𝑥((𝐴 × 𝐴) ∖ 𝑅)𝑦 ↔ (𝑥(𝐴 × 𝐴)𝑦 ∧ ¬ 𝑥𝑅𝑦))
5	3, 4	bitri 264	. . . . . . . . 9 ⊢ (𝑥𝐶𝑦 ↔ (𝑥(𝐴 × 𝐴)𝑦 ∧ ¬ 𝑥𝑅𝑦))
6	5	baib 944	. . . . . . . 8 ⊢ (𝑥(𝐴 × 𝐴)𝑦 → (𝑥𝐶𝑦 ↔ ¬ 𝑥𝑅𝑦))
7	1, 6	sylbir 225	. . . . . . 7 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (𝑥𝐶𝑦 ↔ ¬ 𝑥𝑅𝑦))
8	7	adantl 482	. . . . . 6 ⊢ ((𝐻:𝐴–1-1-onto→𝐵 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → (𝑥𝐶𝑦 ↔ ¬ 𝑥𝑅𝑦))
9		f1of 6137	. . . . . . . 8 ⊢ (𝐻:𝐴–1-1-onto→𝐵 → 𝐻:𝐴⟶𝐵)
10		ffvelrn 6357	. . . . . . . . . 10 ⊢ ((𝐻:𝐴⟶𝐵 ∧ 𝑥 ∈ 𝐴) → (𝐻‘𝑥) ∈ 𝐵)
11		ffvelrn 6357	. . . . . . . . . 10 ⊢ ((𝐻:𝐴⟶𝐵 ∧ 𝑦 ∈ 𝐴) → (𝐻‘𝑦) ∈ 𝐵)
12	10, 11	anim12dan 882	. . . . . . . . 9 ⊢ ((𝐻:𝐴⟶𝐵 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → ((𝐻‘𝑥) ∈ 𝐵 ∧ (𝐻‘𝑦) ∈ 𝐵))
13		brxp 5147	. . . . . . . . 9 ⊢ ((𝐻‘𝑥)(𝐵 × 𝐵)(𝐻‘𝑦) ↔ ((𝐻‘𝑥) ∈ 𝐵 ∧ (𝐻‘𝑦) ∈ 𝐵))
14	12, 13	sylibr 224	. . . . . . . 8 ⊢ ((𝐻:𝐴⟶𝐵 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → (𝐻‘𝑥)(𝐵 × 𝐵)(𝐻‘𝑦))
15	9, 14	sylan 488	. . . . . . 7 ⊢ ((𝐻:𝐴–1-1-onto→𝐵 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → (𝐻‘𝑥)(𝐵 × 𝐵)(𝐻‘𝑦))
16		isocnv3.2	. . . . . . . . . 10 ⊢ 𝐷 = ((𝐵 × 𝐵) ∖ 𝑆)
17	16	breqi 4659	. . . . . . . . 9 ⊢ ((𝐻‘𝑥)𝐷(𝐻‘𝑦) ↔ (𝐻‘𝑥)((𝐵 × 𝐵) ∖ 𝑆)(𝐻‘𝑦))
18		brdif 4705	. . . . . . . . 9 ⊢ ((𝐻‘𝑥)((𝐵 × 𝐵) ∖ 𝑆)(𝐻‘𝑦) ↔ ((𝐻‘𝑥)(𝐵 × 𝐵)(𝐻‘𝑦) ∧ ¬ (𝐻‘𝑥)𝑆(𝐻‘𝑦)))
19	17, 18	bitri 264	. . . . . . . 8 ⊢ ((𝐻‘𝑥)𝐷(𝐻‘𝑦) ↔ ((𝐻‘𝑥)(𝐵 × 𝐵)(𝐻‘𝑦) ∧ ¬ (𝐻‘𝑥)𝑆(𝐻‘𝑦)))
20	19	baib 944	. . . . . . 7 ⊢ ((𝐻‘𝑥)(𝐵 × 𝐵)(𝐻‘𝑦) → ((𝐻‘𝑥)𝐷(𝐻‘𝑦) ↔ ¬ (𝐻‘𝑥)𝑆(𝐻‘𝑦)))
21	15, 20	syl 17	. . . . . 6 ⊢ ((𝐻:𝐴–1-1-onto→𝐵 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → ((𝐻‘𝑥)𝐷(𝐻‘𝑦) ↔ ¬ (𝐻‘𝑥)𝑆(𝐻‘𝑦)))
22	8, 21	bibi12d 335	. . . . 5 ⊢ ((𝐻:𝐴–1-1-onto→𝐵 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → ((𝑥𝐶𝑦 ↔ (𝐻‘𝑥)𝐷(𝐻‘𝑦)) ↔ (¬ 𝑥𝑅𝑦 ↔ ¬ (𝐻‘𝑥)𝑆(𝐻‘𝑦))))
23		notbi 309	. . . . 5 ⊢ ((𝑥𝑅𝑦 ↔ (𝐻‘𝑥)𝑆(𝐻‘𝑦)) ↔ (¬ 𝑥𝑅𝑦 ↔ ¬ (𝐻‘𝑥)𝑆(𝐻‘𝑦)))
24	22, 23	syl6rbbr 279	. . . 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 2988	. . 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 669	. 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 5897	. 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 5897	. 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 292	1 ⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ↔ 𝐻 Isom 𝐶, 𝐷 (𝐴, 𝐵))

Syntax hints:  ¬ wn 3   ↔ wb 196   ∧ wa 384   = wceq 1483   ∈ wcel 1990  ∀wral 2912   ∖ cdif 3571   class class class wbr 4653   × cxp 5112  ⟶wf 5884  –1-1-onto→wf1o 5887  ‘cfv 5888   Isom wiso 5889

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-f1o 5895  df-fv 5896  df-isom 5897

This theorem is referenced by:  leiso  13243  gtiso  29478