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Theorem isocnv3 6482
 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.
StepHypRef Expression
1 brxp 5071 . . . . . . . 8 (𝑥(𝐴 × 𝐴)𝑦 ↔ (𝑥𝐴𝑦𝐴))
2 isocnv3.1 . . . . . . . . . . 11 𝐶 = ((𝐴 × 𝐴) ∖ 𝑅)
32breqi 4589 . . . . . . . . . 10 (𝑥𝐶𝑦𝑥((𝐴 × 𝐴) ∖ 𝑅)𝑦)
4 brdif 4635 . . . . . . . . . 10 (𝑥((𝐴 × 𝐴) ∖ 𝑅)𝑦 ↔ (𝑥(𝐴 × 𝐴)𝑦 ∧ ¬ 𝑥𝑅𝑦))
53, 4bitri 263 . . . . . . . . 9 (𝑥𝐶𝑦 ↔ (𝑥(𝐴 × 𝐴)𝑦 ∧ ¬ 𝑥𝑅𝑦))
65baib 942 . . . . . . . 8 (𝑥(𝐴 × 𝐴)𝑦 → (𝑥𝐶𝑦 ↔ ¬ 𝑥𝑅𝑦))
71, 6sylbir 224 . . . . . . 7 ((𝑥𝐴𝑦𝐴) → (𝑥𝐶𝑦 ↔ ¬ 𝑥𝑅𝑦))
87adantl 481 . . . . . 6 ((𝐻:𝐴1-1-onto𝐵 ∧ (𝑥𝐴𝑦𝐴)) → (𝑥𝐶𝑦 ↔ ¬ 𝑥𝑅𝑦))
9 f1of 6050 . . . . . . . 8 (𝐻:𝐴1-1-onto𝐵𝐻:𝐴𝐵)
10 ffvelrn 6265 . . . . . . . . . 10 ((𝐻:𝐴𝐵𝑥𝐴) → (𝐻𝑥) ∈ 𝐵)
11 ffvelrn 6265 . . . . . . . . . 10 ((𝐻:𝐴𝐵𝑦𝐴) → (𝐻𝑦) ∈ 𝐵)
1210, 11anim12dan 878 . . . . . . . . 9 ((𝐻:𝐴𝐵 ∧ (𝑥𝐴𝑦𝐴)) → ((𝐻𝑥) ∈ 𝐵 ∧ (𝐻𝑦) ∈ 𝐵))
13 brxp 5071 . . . . . . . . 9 ((𝐻𝑥)(𝐵 × 𝐵)(𝐻𝑦) ↔ ((𝐻𝑥) ∈ 𝐵 ∧ (𝐻𝑦) ∈ 𝐵))
1412, 13sylibr 223 . . . . . . . 8 ((𝐻:𝐴𝐵 ∧ (𝑥𝐴𝑦𝐴)) → (𝐻𝑥)(𝐵 × 𝐵)(𝐻𝑦))
159, 14sylan 487 . . . . . . 7 ((𝐻:𝐴1-1-onto𝐵 ∧ (𝑥𝐴𝑦𝐴)) → (𝐻𝑥)(𝐵 × 𝐵)(𝐻𝑦))
16 isocnv3.2 . . . . . . . . . 10 𝐷 = ((𝐵 × 𝐵) ∖ 𝑆)
1716breqi 4589 . . . . . . . . 9 ((𝐻𝑥)𝐷(𝐻𝑦) ↔ (𝐻𝑥)((𝐵 × 𝐵) ∖ 𝑆)(𝐻𝑦))
18 brdif 4635 . . . . . . . . 9 ((𝐻𝑥)((𝐵 × 𝐵) ∖ 𝑆)(𝐻𝑦) ↔ ((𝐻𝑥)(𝐵 × 𝐵)(𝐻𝑦) ∧ ¬ (𝐻𝑥)𝑆(𝐻𝑦)))
1917, 18bitri 263 . . . . . . . 8 ((𝐻𝑥)𝐷(𝐻𝑦) ↔ ((𝐻𝑥)(𝐵 × 𝐵)(𝐻𝑦) ∧ ¬ (𝐻𝑥)𝑆(𝐻𝑦)))
2019baib 942 . . . . . . 7 ((𝐻𝑥)(𝐵 × 𝐵)(𝐻𝑦) → ((𝐻𝑥)𝐷(𝐻𝑦) ↔ ¬ (𝐻𝑥)𝑆(𝐻𝑦)))
2115, 20syl 17 . . . . . 6 ((𝐻:𝐴1-1-onto𝐵 ∧ (𝑥𝐴𝑦𝐴)) → ((𝐻𝑥)𝐷(𝐻𝑦) ↔ ¬ (𝐻𝑥)𝑆(𝐻𝑦)))
228, 21bibi12d 334 . . . . 5 ((𝐻:𝐴1-1-onto𝐵 ∧ (𝑥𝐴𝑦𝐴)) → ((𝑥𝐶𝑦 ↔ (𝐻𝑥)𝐷(𝐻𝑦)) ↔ (¬ 𝑥𝑅𝑦 ↔ ¬ (𝐻𝑥)𝑆(𝐻𝑦))))
23 notbi 308 . . . . 5 ((𝑥𝑅𝑦 ↔ (𝐻𝑥)𝑆(𝐻𝑦)) ↔ (¬ 𝑥𝑅𝑦 ↔ ¬ (𝐻𝑥)𝑆(𝐻𝑦)))
2422, 23syl6rbbr 278 . . . 4 ((𝐻:𝐴1-1-onto𝐵 ∧ (𝑥𝐴𝑦𝐴)) → ((𝑥𝑅𝑦 ↔ (𝐻𝑥)𝑆(𝐻𝑦)) ↔ (𝑥𝐶𝑦 ↔ (𝐻𝑥)𝐷(𝐻𝑦))))
25242ralbidva 2971 . . 3 (𝐻:𝐴1-1-onto𝐵 → (∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦 ↔ (𝐻𝑥)𝑆(𝐻𝑦)) ↔ ∀𝑥𝐴𝑦𝐴 (𝑥𝐶𝑦 ↔ (𝐻𝑥)𝐷(𝐻𝑦))))
2625pm5.32i 667 . 2 ((𝐻:𝐴1-1-onto𝐵 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦 ↔ (𝐻𝑥)𝑆(𝐻𝑦))) ↔ (𝐻:𝐴1-1-onto𝐵 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝐶𝑦 ↔ (𝐻𝑥)𝐷(𝐻𝑦))))
27 df-isom 5813 . 2 (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ↔ (𝐻:𝐴1-1-onto𝐵 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦 ↔ (𝐻𝑥)𝑆(𝐻𝑦))))
28 df-isom 5813 . 2 (𝐻 Isom 𝐶, 𝐷 (𝐴, 𝐵) ↔ (𝐻:𝐴1-1-onto𝐵 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝐶𝑦 ↔ (𝐻𝑥)𝐷(𝐻𝑦))))
2926, 27, 283bitr4i 291 1 (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ↔ 𝐻 Isom 𝐶, 𝐷 (𝐴, 𝐵))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∀wral 2896   ∖ cdif 3537   class class class wbr 4583   × cxp 5036  ⟶wf 5800  –1-1-onto→wf1o 5803  ‘cfv 5804   Isom wiso 5805 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-f1o 5811  df-fv 5812  df-isom 5813 This theorem is referenced by:  leiso  13100  gtiso  28861
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