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Theorem compssiso 9788
Description: Complementation is an antiautomorphism on power set lattices. (Contributed by Stefan O'Rear, 4-Nov-2014.) (Proof shortened by Mario Carneiro, 17-May-2015.)
Hypothesis
Ref Expression
compss.a 𝐹 = (𝑥 ∈ 𝒫 𝐴 ↦ (𝐴𝑥))
Assertion
Ref Expression
compssiso (𝐴𝑉𝐹 Isom [] , [] (𝒫 𝐴, 𝒫 𝐴))
Distinct variable groups:   𝑥,𝐴   𝑥,𝑉
Allowed substitution hint:   𝐹(𝑥)

Proof of Theorem compssiso
Dummy variables 𝑎 𝑏 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 difexg 5227 . . . . 5 (𝐴𝑉 → (𝐴𝑥) ∈ V)
21ralrimivw 3187 . . . 4 (𝐴𝑉 → ∀𝑥 ∈ 𝒫 𝐴(𝐴𝑥) ∈ V)
3 compss.a . . . . 5 𝐹 = (𝑥 ∈ 𝒫 𝐴 ↦ (𝐴𝑥))
43fnmpt 6484 . . . 4 (∀𝑥 ∈ 𝒫 𝐴(𝐴𝑥) ∈ V → 𝐹 Fn 𝒫 𝐴)
52, 4syl 17 . . 3 (𝐴𝑉𝐹 Fn 𝒫 𝐴)
63compsscnv 9785 . . . . 5 𝐹 = 𝐹
76fneq1i 6446 . . . 4 (𝐹 Fn 𝒫 𝐴𝐹 Fn 𝒫 𝐴)
85, 7sylibr 235 . . 3 (𝐴𝑉𝐹 Fn 𝒫 𝐴)
9 dff1o4 6619 . . 3 (𝐹:𝒫 𝐴1-1-onto→𝒫 𝐴 ↔ (𝐹 Fn 𝒫 𝐴𝐹 Fn 𝒫 𝐴))
105, 8, 9sylanbrc 583 . 2 (𝐴𝑉𝐹:𝒫 𝐴1-1-onto→𝒫 𝐴)
11 elpwi 4553 . . . . . . . . 9 (𝑏 ∈ 𝒫 𝐴𝑏𝐴)
1211ad2antll 725 . . . . . . . 8 ((𝐴𝑉 ∧ (𝑎 ∈ 𝒫 𝐴𝑏 ∈ 𝒫 𝐴)) → 𝑏𝐴)
133isf34lem1 9786 . . . . . . . 8 ((𝐴𝑉𝑏𝐴) → (𝐹𝑏) = (𝐴𝑏))
1412, 13syldan 591 . . . . . . 7 ((𝐴𝑉 ∧ (𝑎 ∈ 𝒫 𝐴𝑏 ∈ 𝒫 𝐴)) → (𝐹𝑏) = (𝐴𝑏))
15 elpwi 4553 . . . . . . . . 9 (𝑎 ∈ 𝒫 𝐴𝑎𝐴)
1615ad2antrl 724 . . . . . . . 8 ((𝐴𝑉 ∧ (𝑎 ∈ 𝒫 𝐴𝑏 ∈ 𝒫 𝐴)) → 𝑎𝐴)
173isf34lem1 9786 . . . . . . . 8 ((𝐴𝑉𝑎𝐴) → (𝐹𝑎) = (𝐴𝑎))
1816, 17syldan 591 . . . . . . 7 ((𝐴𝑉 ∧ (𝑎 ∈ 𝒫 𝐴𝑏 ∈ 𝒫 𝐴)) → (𝐹𝑎) = (𝐴𝑎))
1914, 18psseq12d 4074 . . . . . 6 ((𝐴𝑉 ∧ (𝑎 ∈ 𝒫 𝐴𝑏 ∈ 𝒫 𝐴)) → ((𝐹𝑏) ⊊ (𝐹𝑎) ↔ (𝐴𝑏) ⊊ (𝐴𝑎)))
20 difss 4111 . . . . . . 7 (𝐴𝑎) ⊆ 𝐴
21 pssdifcom1 4437 . . . . . . 7 ((𝑏𝐴 ∧ (𝐴𝑎) ⊆ 𝐴) → ((𝐴𝑏) ⊊ (𝐴𝑎) ↔ (𝐴 ∖ (𝐴𝑎)) ⊊ 𝑏))
2212, 20, 21sylancl 586 . . . . . 6 ((𝐴𝑉 ∧ (𝑎 ∈ 𝒫 𝐴𝑏 ∈ 𝒫 𝐴)) → ((𝐴𝑏) ⊊ (𝐴𝑎) ↔ (𝐴 ∖ (𝐴𝑎)) ⊊ 𝑏))
23 dfss4 4238 . . . . . . . 8 (𝑎𝐴 ↔ (𝐴 ∖ (𝐴𝑎)) = 𝑎)
2416, 23sylib 219 . . . . . . 7 ((𝐴𝑉 ∧ (𝑎 ∈ 𝒫 𝐴𝑏 ∈ 𝒫 𝐴)) → (𝐴 ∖ (𝐴𝑎)) = 𝑎)
2524psseq1d 4072 . . . . . 6 ((𝐴𝑉 ∧ (𝑎 ∈ 𝒫 𝐴𝑏 ∈ 𝒫 𝐴)) → ((𝐴 ∖ (𝐴𝑎)) ⊊ 𝑏𝑎𝑏))
2619, 22, 253bitrrd 307 . . . . 5 ((𝐴𝑉 ∧ (𝑎 ∈ 𝒫 𝐴𝑏 ∈ 𝒫 𝐴)) → (𝑎𝑏 ↔ (𝐹𝑏) ⊊ (𝐹𝑎)))
27 vex 3502 . . . . . 6 𝑏 ∈ V
2827brrpss 7445 . . . . 5 (𝑎 [] 𝑏𝑎𝑏)
29 fvex 6679 . . . . . 6 (𝐹𝑎) ∈ V
3029brrpss 7445 . . . . 5 ((𝐹𝑏) [] (𝐹𝑎) ↔ (𝐹𝑏) ⊊ (𝐹𝑎))
3126, 28, 303bitr4g 315 . . . 4 ((𝐴𝑉 ∧ (𝑎 ∈ 𝒫 𝐴𝑏 ∈ 𝒫 𝐴)) → (𝑎 [] 𝑏 ↔ (𝐹𝑏) [] (𝐹𝑎)))
32 relrpss 7443 . . . . 5 Rel []
3332relbrcnv 5967 . . . 4 ((𝐹𝑎) [] (𝐹𝑏) ↔ (𝐹𝑏) [] (𝐹𝑎))
3431, 33syl6bbr 290 . . 3 ((𝐴𝑉 ∧ (𝑎 ∈ 𝒫 𝐴𝑏 ∈ 𝒫 𝐴)) → (𝑎 [] 𝑏 ↔ (𝐹𝑎) [] (𝐹𝑏)))
3534ralrimivva 3195 . 2 (𝐴𝑉 → ∀𝑎 ∈ 𝒫 𝐴𝑏 ∈ 𝒫 𝐴(𝑎 [] 𝑏 ↔ (𝐹𝑎) [] (𝐹𝑏)))
36 df-isom 6360 . 2 (𝐹 Isom [] , [] (𝒫 𝐴, 𝒫 𝐴) ↔ (𝐹:𝒫 𝐴1-1-onto→𝒫 𝐴 ∧ ∀𝑎 ∈ 𝒫 𝐴𝑏 ∈ 𝒫 𝐴(𝑎 [] 𝑏 ↔ (𝐹𝑎) [] (𝐹𝑏))))
3710, 35, 36sylanbrc 583 1 (𝐴𝑉𝐹 Isom [] , [] (𝒫 𝐴, 𝒫 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396   = wceq 1530  wcel 2107  wral 3142  Vcvv 3499  cdif 3936  wss 3939  wpss 3940  𝒫 cpw 4541   class class class wbr 5062  cmpt 5142  ccnv 5552   Fn wfn 6346  1-1-ontowf1o 6350  cfv 6351   Isom wiso 6352   [] crpss 7441
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2797  ax-sep 5199  ax-nul 5206  ax-pr 5325
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2619  df-eu 2651  df-clab 2804  df-cleq 2818  df-clel 2897  df-nfc 2967  df-ne 3021  df-ral 3147  df-rex 3148  df-rab 3151  df-v 3501  df-sbc 3776  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-pss 3957  df-nul 4295  df-if 4470  df-pw 4543  df-sn 4564  df-pr 4566  df-op 4570  df-uni 4837  df-br 5063  df-opab 5125  df-mpt 5143  df-id 5458  df-xp 5559  df-rel 5560  df-cnv 5561  df-co 5562  df-dm 5563  df-rn 5564  df-iota 6311  df-fun 6353  df-fn 6354  df-f 6355  df-f1 6356  df-fo 6357  df-f1o 6358  df-fv 6359  df-isom 6360  df-rpss 7442
This theorem is referenced by:  isf34lem3  9789  isf34lem5  9792  isfin1-4  9801
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