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Theorem isocnv3 6579
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 5145 . . . . . . . 8 (𝑥(𝐴 × 𝐴)𝑦 ↔ (𝑥𝐴𝑦𝐴))
2 isocnv3.1 . . . . . . . . . . 11 𝐶 = ((𝐴 × 𝐴) ∖ 𝑅)
32breqi 4657 . . . . . . . . . 10 (𝑥𝐶𝑦𝑥((𝐴 × 𝐴) ∖ 𝑅)𝑦)
4 brdif 4703 . . . . . . . . . 10 (𝑥((𝐴 × 𝐴) ∖ 𝑅)𝑦 ↔ (𝑥(𝐴 × 𝐴)𝑦 ∧ ¬ 𝑥𝑅𝑦))
53, 4bitri 264 . . . . . . . . 9 (𝑥𝐶𝑦 ↔ (𝑥(𝐴 × 𝐴)𝑦 ∧ ¬ 𝑥𝑅𝑦))
65baib 944 . . . . . . . 8 (𝑥(𝐴 × 𝐴)𝑦 → (𝑥𝐶𝑦 ↔ ¬ 𝑥𝑅𝑦))
71, 6sylbir 225 . . . . . . 7 ((𝑥𝐴𝑦𝐴) → (𝑥𝐶𝑦 ↔ ¬ 𝑥𝑅𝑦))
87adantl 482 . . . . . 6 ((𝐻:𝐴1-1-onto𝐵 ∧ (𝑥𝐴𝑦𝐴)) → (𝑥𝐶𝑦 ↔ ¬ 𝑥𝑅𝑦))
9 f1of 6135 . . . . . . . 8 (𝐻:𝐴1-1-onto𝐵𝐻:𝐴𝐵)
10 ffvelrn 6355 . . . . . . . . . 10 ((𝐻:𝐴𝐵𝑥𝐴) → (𝐻𝑥) ∈ 𝐵)
11 ffvelrn 6355 . . . . . . . . . 10 ((𝐻:𝐴𝐵𝑦𝐴) → (𝐻𝑦) ∈ 𝐵)
1210, 11anim12dan 882 . . . . . . . . 9 ((𝐻:𝐴𝐵 ∧ (𝑥𝐴𝑦𝐴)) → ((𝐻𝑥) ∈ 𝐵 ∧ (𝐻𝑦) ∈ 𝐵))
13 brxp 5145 . . . . . . . . 9 ((𝐻𝑥)(𝐵 × 𝐵)(𝐻𝑦) ↔ ((𝐻𝑥) ∈ 𝐵 ∧ (𝐻𝑦) ∈ 𝐵))
1412, 13sylibr 224 . . . . . . . 8 ((𝐻:𝐴𝐵 ∧ (𝑥𝐴𝑦𝐴)) → (𝐻𝑥)(𝐵 × 𝐵)(𝐻𝑦))
159, 14sylan 488 . . . . . . 7 ((𝐻:𝐴1-1-onto𝐵 ∧ (𝑥𝐴𝑦𝐴)) → (𝐻𝑥)(𝐵 × 𝐵)(𝐻𝑦))
16 isocnv3.2 . . . . . . . . . 10 𝐷 = ((𝐵 × 𝐵) ∖ 𝑆)
1716breqi 4657 . . . . . . . . 9 ((𝐻𝑥)𝐷(𝐻𝑦) ↔ (𝐻𝑥)((𝐵 × 𝐵) ∖ 𝑆)(𝐻𝑦))
18 brdif 4703 . . . . . . . . 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 944 . . . . . . 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 335 . . . . 5 ((𝐻:𝐴1-1-onto𝐵 ∧ (𝑥𝐴𝑦𝐴)) → ((𝑥𝐶𝑦 ↔ (𝐻𝑥)𝐷(𝐻𝑦)) ↔ (¬ 𝑥𝑅𝑦 ↔ ¬ (𝐻𝑥)𝑆(𝐻𝑦))))
23 notbi 309 . . . . 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 2987 . . 3 (𝐻:𝐴1-1-onto𝐵 → (∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦 ↔ (𝐻𝑥)𝑆(𝐻𝑦)) ↔ ∀𝑥𝐴𝑦𝐴 (𝑥𝐶𝑦 ↔ (𝐻𝑥)𝐷(𝐻𝑦))))
2625pm5.32i 669 . 2 ((𝐻:𝐴1-1-onto𝐵 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦 ↔ (𝐻𝑥)𝑆(𝐻𝑦))) ↔ (𝐻:𝐴1-1-onto𝐵 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝐶𝑦 ↔ (𝐻𝑥)𝐷(𝐻𝑦))))
27 df-isom 5895 . 2 (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ↔ (𝐻:𝐴1-1-onto𝐵 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦 ↔ (𝐻𝑥)𝑆(𝐻𝑦))))
28 df-isom 5895 . 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 384   = wceq 1482  wcel 1989  wral 2911  cdif 3569   class class class wbr 4651   × cxp 5110  wf 5882  1-1-ontowf1o 5885  cfv 5886   Isom wiso 5887
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601  ax-sep 4779  ax-nul 4787  ax-pr 4904
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-ral 2916  df-rex 2917  df-rab 2920  df-v 3200  df-sbc 3434  df-dif 3575  df-un 3577  df-in 3579  df-ss 3586  df-nul 3914  df-if 4085  df-sn 4176  df-pr 4178  df-op 4182  df-uni 4435  df-br 4652  df-opab 4711  df-id 5022  df-xp 5118  df-rel 5119  df-cnv 5120  df-co 5121  df-dm 5122  df-rn 5123  df-iota 5849  df-fun 5888  df-fn 5889  df-f 5890  df-f1 5891  df-f1o 5893  df-fv 5894  df-isom 5895
This theorem is referenced by:  leiso  13238  gtiso  29463
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