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Theorem compssiso 9079
 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 4735 . . . . 5 (𝐴𝑉 → (𝐴𝑥) ∈ V)
21ralrimivw 2950 . . . 4 (𝐴𝑉 → ∀𝑥 ∈ 𝒫 𝐴(𝐴𝑥) ∈ V)
3 compss.a . . . . 5 𝐹 = (𝑥 ∈ 𝒫 𝐴 ↦ (𝐴𝑥))
43fnmpt 5933 . . . 4 (∀𝑥 ∈ 𝒫 𝐴(𝐴𝑥) ∈ V → 𝐹 Fn 𝒫 𝐴)
52, 4syl 17 . . 3 (𝐴𝑉𝐹 Fn 𝒫 𝐴)
63compsscnv 9076 . . . . 5 𝐹 = 𝐹
76fneq1i 5899 . . . 4 (𝐹 Fn 𝒫 𝐴𝐹 Fn 𝒫 𝐴)
85, 7sylibr 223 . . 3 (𝐴𝑉𝐹 Fn 𝒫 𝐴)
9 dff1o4 6058 . . 3 (𝐹:𝒫 𝐴1-1-onto→𝒫 𝐴 ↔ (𝐹 Fn 𝒫 𝐴𝐹 Fn 𝒫 𝐴))
105, 8, 9sylanbrc 695 . 2 (𝐴𝑉𝐹:𝒫 𝐴1-1-onto→𝒫 𝐴)
11 elpwi 4117 . . . . . . . . 9 (𝑏 ∈ 𝒫 𝐴𝑏𝐴)
1211ad2antll 761 . . . . . . . 8 ((𝐴𝑉 ∧ (𝑎 ∈ 𝒫 𝐴𝑏 ∈ 𝒫 𝐴)) → 𝑏𝐴)
133isf34lem1 9077 . . . . . . . 8 ((𝐴𝑉𝑏𝐴) → (𝐹𝑏) = (𝐴𝑏))
1412, 13syldan 486 . . . . . . 7 ((𝐴𝑉 ∧ (𝑎 ∈ 𝒫 𝐴𝑏 ∈ 𝒫 𝐴)) → (𝐹𝑏) = (𝐴𝑏))
15 elpwi 4117 . . . . . . . . 9 (𝑎 ∈ 𝒫 𝐴𝑎𝐴)
1615ad2antrl 760 . . . . . . . 8 ((𝐴𝑉 ∧ (𝑎 ∈ 𝒫 𝐴𝑏 ∈ 𝒫 𝐴)) → 𝑎𝐴)
173isf34lem1 9077 . . . . . . . 8 ((𝐴𝑉𝑎𝐴) → (𝐹𝑎) = (𝐴𝑎))
1816, 17syldan 486 . . . . . . 7 ((𝐴𝑉 ∧ (𝑎 ∈ 𝒫 𝐴𝑏 ∈ 𝒫 𝐴)) → (𝐹𝑎) = (𝐴𝑎))
1914, 18psseq12d 3663 . . . . . 6 ((𝐴𝑉 ∧ (𝑎 ∈ 𝒫 𝐴𝑏 ∈ 𝒫 𝐴)) → ((𝐹𝑏) ⊊ (𝐹𝑎) ↔ (𝐴𝑏) ⊊ (𝐴𝑎)))
20 difss 3699 . . . . . . 7 (𝐴𝑎) ⊆ 𝐴
21 pssdifcom1 4006 . . . . . . 7 ((𝑏𝐴 ∧ (𝐴𝑎) ⊆ 𝐴) → ((𝐴𝑏) ⊊ (𝐴𝑎) ↔ (𝐴 ∖ (𝐴𝑎)) ⊊ 𝑏))
2212, 20, 21sylancl 693 . . . . . 6 ((𝐴𝑉 ∧ (𝑎 ∈ 𝒫 𝐴𝑏 ∈ 𝒫 𝐴)) → ((𝐴𝑏) ⊊ (𝐴𝑎) ↔ (𝐴 ∖ (𝐴𝑎)) ⊊ 𝑏))
23 dfss4 3820 . . . . . . . 8 (𝑎𝐴 ↔ (𝐴 ∖ (𝐴𝑎)) = 𝑎)
2416, 23sylib 207 . . . . . . 7 ((𝐴𝑉 ∧ (𝑎 ∈ 𝒫 𝐴𝑏 ∈ 𝒫 𝐴)) → (𝐴 ∖ (𝐴𝑎)) = 𝑎)
2524psseq1d 3661 . . . . . 6 ((𝐴𝑉 ∧ (𝑎 ∈ 𝒫 𝐴𝑏 ∈ 𝒫 𝐴)) → ((𝐴 ∖ (𝐴𝑎)) ⊊ 𝑏𝑎𝑏))
2619, 22, 253bitrrd 294 . . . . 5 ((𝐴𝑉 ∧ (𝑎 ∈ 𝒫 𝐴𝑏 ∈ 𝒫 𝐴)) → (𝑎𝑏 ↔ (𝐹𝑏) ⊊ (𝐹𝑎)))
27 vex 3176 . . . . . 6 𝑏 ∈ V
2827brrpss 6838 . . . . 5 (𝑎 [] 𝑏𝑎𝑏)
29 fvex 6113 . . . . . 6 (𝐹𝑎) ∈ V
3029brrpss 6838 . . . . 5 ((𝐹𝑏) [] (𝐹𝑎) ↔ (𝐹𝑏) ⊊ (𝐹𝑎))
3126, 28, 303bitr4g 302 . . . 4 ((𝐴𝑉 ∧ (𝑎 ∈ 𝒫 𝐴𝑏 ∈ 𝒫 𝐴)) → (𝑎 [] 𝑏 ↔ (𝐹𝑏) [] (𝐹𝑎)))
32 relrpss 6836 . . . . 5 Rel []
3332relbrcnv 5425 . . . 4 ((𝐹𝑎) [] (𝐹𝑏) ↔ (𝐹𝑏) [] (𝐹𝑎))
3431, 33syl6bbr 277 . . 3 ((𝐴𝑉 ∧ (𝑎 ∈ 𝒫 𝐴𝑏 ∈ 𝒫 𝐴)) → (𝑎 [] 𝑏 ↔ (𝐹𝑎) [] (𝐹𝑏)))
3534ralrimivva 2954 . 2 (𝐴𝑉 → ∀𝑎 ∈ 𝒫 𝐴𝑏 ∈ 𝒫 𝐴(𝑎 [] 𝑏 ↔ (𝐹𝑎) [] (𝐹𝑏)))
36 df-isom 5813 . 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 695 1 (𝐴𝑉𝐹 Isom [] , [] (𝒫 𝐴, 𝒫 𝐴))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∀wral 2896  Vcvv 3173   ∖ cdif 3537   ⊆ wss 3540   ⊊ wpss 3541  𝒫 cpw 4108   class class class wbr 4583   ↦ cmpt 4643  ◡ccnv 5037   Fn wfn 5799  –1-1-onto→wf1o 5803  ‘cfv 5804   Isom wiso 5805   [⊊] crpss 6834 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-ne 2782  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-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  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-fo 5810  df-f1o 5811  df-fv 5812  df-isom 5813  df-rpss 6835 This theorem is referenced by:  isf34lem3  9080  isf34lem5  9083  isfin1-4  9092
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