Theorem isocnv3 7340

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 322	. . . . 5 ⊢ ((𝑥𝑅𝑦 ↔ (𝐻‘𝑥)𝑆(𝐻‘𝑦)) ↔ (¬ 𝑥𝑅𝑦 ↔ ¬ (𝐻‘𝑥)𝑆(𝐻‘𝑦)))
2		brxp 5700	. . . . . . . 8 ⊢ (𝑥(𝐴 × 𝐴)𝑦 ↔ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴))
3		isocnv3.1	. . . . . . . . . . 11 ⊢ 𝐶 = ((𝐴 × 𝐴) ∖ 𝑅)
4	3	breqi 5109	. . . . . . . . . 10 ⊢ (𝑥𝐶𝑦 ↔ 𝑥((𝐴 × 𝐴) ∖ 𝑅)𝑦)
5		brdif 5158	. . . . . . . . . 10 ⊢ (𝑥((𝐴 × 𝐴) ∖ 𝑅)𝑦 ↔ (𝑥(𝐴 × 𝐴)𝑦 ∧ ¬ 𝑥𝑅𝑦))
6	4, 5	bitri 278	. . . . . . . . 9 ⊢ (𝑥𝐶𝑦 ↔ (𝑥(𝐴 × 𝐴)𝑦 ∧ ¬ 𝑥𝑅𝑦))
7	6	baib 545	. . . . . . . 8 ⊢ (𝑥(𝐴 × 𝐴)𝑦 → (𝑥𝐶𝑦 ↔ ¬ 𝑥𝑅𝑦))
8	2, 7	sylbir 238	. . . . . . 7 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (𝑥𝐶𝑦 ↔ ¬ 𝑥𝑅𝑦))
9	8	adantl 487	. . . . . 6 ⊢ ((𝐻:𝐴–1-1-onto→𝐵 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → (𝑥𝐶𝑦 ↔ ¬ 𝑥𝑅𝑦))
10		f1of 6824	. . . . . . . 8 ⊢ (𝐻:𝐴–1-1-onto→𝐵 → 𝐻:𝐴⟶𝐵)
11		ffvelcdm 7081	. . . . . . . . . 10 ⊢ ((𝐻:𝐴⟶𝐵 ∧ 𝑥 ∈ 𝐴) → (𝐻‘𝑥) ∈ 𝐵)
12		ffvelcdm 7081	. . . . . . . . . 10 ⊢ ((𝐻:𝐴⟶𝐵 ∧ 𝑦 ∈ 𝐴) → (𝐻‘𝑦) ∈ 𝐵)
13	11, 12	anim12dan 631	. . . . . . . . 9 ⊢ ((𝐻:𝐴⟶𝐵 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → ((𝐻‘𝑥) ∈ 𝐵 ∧ (𝐻‘𝑦) ∈ 𝐵))
14		brxp 5700	. . . . . . . . 9 ⊢ ((𝐻‘𝑥)(𝐵 × 𝐵)(𝐻‘𝑦) ↔ ((𝐻‘𝑥) ∈ 𝐵 ∧ (𝐻‘𝑦) ∈ 𝐵))
15	13, 14	sylibr 237	. . . . . . . 8 ⊢ ((𝐻:𝐴⟶𝐵 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → (𝐻‘𝑥)(𝐵 × 𝐵)(𝐻‘𝑦))
16	10, 15	sylan 592	. . . . . . 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 5109	. . . . . . . . 9 ⊢ ((𝐻‘𝑥)𝐷(𝐻‘𝑦) ↔ (𝐻‘𝑥)((𝐵 × 𝐵) ∖ 𝑆)(𝐻‘𝑦))
19		brdif 5158	. . . . . . . . 9 ⊢ ((𝐻‘𝑥)((𝐵 × 𝐵) ∖ 𝑆)(𝐻‘𝑦) ↔ ((𝐻‘𝑥)(𝐵 × 𝐵)(𝐻‘𝑦) ∧ ¬ (𝐻‘𝑥)𝑆(𝐻‘𝑦)))
20	18, 19	bitri 278	. . . . . . . 8 ⊢ ((𝐻‘𝑥)𝐷(𝐻‘𝑦) ↔ ((𝐻‘𝑥)(𝐵 × 𝐵)(𝐻‘𝑦) ∧ ¬ (𝐻‘𝑥)𝑆(𝐻‘𝑦)))
21	20	baib 545	. . . . . . 7 ⊢ ((𝐻‘𝑥)(𝐵 × 𝐵)(𝐻‘𝑦) → ((𝐻‘𝑥)𝐷(𝐻‘𝑦) ↔ ¬ (𝐻‘𝑥)𝑆(𝐻‘𝑦)))
22	16, 21	syl 18	. . . . . 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 348	. . . . 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 293	. . . 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 3225	. . 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 585	. 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 6547	. 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 6547	. 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 306	1 ⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ↔ 𝐻 Isom 𝐶, 𝐷 (𝐴, 𝐵))

This proof depends on syntax axioms:  ¬ wn 3   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2145  ∀wral 3077   ∖ cdif 3896   class class class wbr 5103   × cxp 5649  ⟶wf 6534  –1-1-onto→wf1o 6537  ‘cfv 6538   Isom wiso 6539

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-12 2213  ax-ext 2733  ax-sep 5249  ax-nul 5260  ax-pr 5391

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-ne 2957  df-ral 3078  df-rex 3088  df-rab 3414  df-v 3453  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-f1o 6545  df-fv 6546  df-isom 6547

This theorem is used by:  leiso  14604  gtiso  33294