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Theorem isocnv3 6724
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 5287 . . . . . . . 8 (𝑥(𝐴 × 𝐴)𝑦 ↔ (𝑥𝐴𝑦𝐴))
2 isocnv3.1 . . . . . . . . . . 11 𝐶 = ((𝐴 × 𝐴) ∖ 𝑅)
32breqi 4790 . . . . . . . . . 10 (𝑥𝐶𝑦𝑥((𝐴 × 𝐴) ∖ 𝑅)𝑦)
4 brdif 4837 . . . . . . . . . 10 (𝑥((𝐴 × 𝐴) ∖ 𝑅)𝑦 ↔ (𝑥(𝐴 × 𝐴)𝑦 ∧ ¬ 𝑥𝑅𝑦))
53, 4bitri 264 . . . . . . . . 9 (𝑥𝐶𝑦 ↔ (𝑥(𝐴 × 𝐴)𝑦 ∧ ¬ 𝑥𝑅𝑦))
65baib 517 . . . . . . . 8 (𝑥(𝐴 × 𝐴)𝑦 → (𝑥𝐶𝑦 ↔ ¬ 𝑥𝑅𝑦))
71, 6sylbir 225 . . . . . . 7 ((𝑥𝐴𝑦𝐴) → (𝑥𝐶𝑦 ↔ ¬ 𝑥𝑅𝑦))
87adantl 467 . . . . . 6 ((𝐻:𝐴1-1-onto𝐵 ∧ (𝑥𝐴𝑦𝐴)) → (𝑥𝐶𝑦 ↔ ¬ 𝑥𝑅𝑦))
9 f1of 6278 . . . . . . . 8 (𝐻:𝐴1-1-onto𝐵𝐻:𝐴𝐵)
10 ffvelrn 6500 . . . . . . . . . 10 ((𝐻:𝐴𝐵𝑥𝐴) → (𝐻𝑥) ∈ 𝐵)
11 ffvelrn 6500 . . . . . . . . . 10 ((𝐻:𝐴𝐵𝑦𝐴) → (𝐻𝑦) ∈ 𝐵)
1210, 11anim12dan 597 . . . . . . . . 9 ((𝐻:𝐴𝐵 ∧ (𝑥𝐴𝑦𝐴)) → ((𝐻𝑥) ∈ 𝐵 ∧ (𝐻𝑦) ∈ 𝐵))
13 brxp 5287 . . . . . . . . 9 ((𝐻𝑥)(𝐵 × 𝐵)(𝐻𝑦) ↔ ((𝐻𝑥) ∈ 𝐵 ∧ (𝐻𝑦) ∈ 𝐵))
1412, 13sylibr 224 . . . . . . . 8 ((𝐻:𝐴𝐵 ∧ (𝑥𝐴𝑦𝐴)) → (𝐻𝑥)(𝐵 × 𝐵)(𝐻𝑦))
159, 14sylan 561 . . . . . . 7 ((𝐻:𝐴1-1-onto𝐵 ∧ (𝑥𝐴𝑦𝐴)) → (𝐻𝑥)(𝐵 × 𝐵)(𝐻𝑦))
16 isocnv3.2 . . . . . . . . . 10 𝐷 = ((𝐵 × 𝐵) ∖ 𝑆)
1716breqi 4790 . . . . . . . . 9 ((𝐻𝑥)𝐷(𝐻𝑦) ↔ (𝐻𝑥)((𝐵 × 𝐵) ∖ 𝑆)(𝐻𝑦))
18 brdif 4837 . . . . . . . . 9 ((𝐻𝑥)((𝐵 × 𝐵) ∖ 𝑆)(𝐻𝑦) ↔ ((𝐻𝑥)(𝐵 × 𝐵)(𝐻𝑦) ∧ ¬ (𝐻𝑥)𝑆(𝐻𝑦)))
1917, 18bitri 264 . . . . . . . 8 ((𝐻𝑥)𝐷(𝐻𝑦) ↔ ((𝐻𝑥)(𝐵 × 𝐵)(𝐻𝑦) ∧ ¬ (𝐻𝑥)𝑆(𝐻𝑦)))
2019baib 517 . . . . . . 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 279 . . . 4 ((𝐻:𝐴1-1-onto𝐵 ∧ (𝑥𝐴𝑦𝐴)) → ((𝑥𝑅𝑦 ↔ (𝐻𝑥)𝑆(𝐻𝑦)) ↔ (𝑥𝐶𝑦 ↔ (𝐻𝑥)𝐷(𝐻𝑦))))
25242ralbidva 3136 . . 3 (𝐻:𝐴1-1-onto𝐵 → (∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦 ↔ (𝐻𝑥)𝑆(𝐻𝑦)) ↔ ∀𝑥𝐴𝑦𝐴 (𝑥𝐶𝑦 ↔ (𝐻𝑥)𝐷(𝐻𝑦))))
2625pm5.32i 556 . 2 ((𝐻:𝐴1-1-onto𝐵 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦 ↔ (𝐻𝑥)𝑆(𝐻𝑦))) ↔ (𝐻:𝐴1-1-onto𝐵 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝐶𝑦 ↔ (𝐻𝑥)𝐷(𝐻𝑦))))
27 df-isom 6040 . 2 (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ↔ (𝐻:𝐴1-1-onto𝐵 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦 ↔ (𝐻𝑥)𝑆(𝐻𝑦))))
28 df-isom 6040 . 2 (𝐻 Isom 𝐶, 𝐷 (𝐴, 𝐵) ↔ (𝐻:𝐴1-1-onto𝐵 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝐶𝑦 ↔ (𝐻𝑥)𝐷(𝐻𝑦))))
2926, 27, 283bitr4i 292 1 (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ↔ 𝐻 Isom 𝐶, 𝐷 (𝐴, 𝐵))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 196  wa 382   = wceq 1630  wcel 2144  wral 3060  cdif 3718   class class class wbr 4784   × cxp 5247  wf 6027  1-1-ontowf1o 6030  cfv 6031   Isom wiso 6032
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-sep 4912  ax-nul 4920  ax-pr 5034
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ral 3065  df-rex 3066  df-rab 3069  df-v 3351  df-sbc 3586  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-nul 4062  df-if 4224  df-sn 4315  df-pr 4317  df-op 4321  df-uni 4573  df-br 4785  df-opab 4845  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-f1o 6038  df-fv 6039  df-isom 6040
This theorem is referenced by:  leiso  13444  gtiso  29812
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