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Theorem compssiso 9193
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 4806 . . . . 5 (𝐴𝑉 → (𝐴𝑥) ∈ V)
21ralrimivw 2966 . . . 4 (𝐴𝑉 → ∀𝑥 ∈ 𝒫 𝐴(𝐴𝑥) ∈ V)
3 compss.a . . . . 5 𝐹 = (𝑥 ∈ 𝒫 𝐴 ↦ (𝐴𝑥))
43fnmpt 6018 . . . 4 (∀𝑥 ∈ 𝒫 𝐴(𝐴𝑥) ∈ V → 𝐹 Fn 𝒫 𝐴)
52, 4syl 17 . . 3 (𝐴𝑉𝐹 Fn 𝒫 𝐴)
63compsscnv 9190 . . . . 5 𝐹 = 𝐹
76fneq1i 5983 . . . 4 (𝐹 Fn 𝒫 𝐴𝐹 Fn 𝒫 𝐴)
85, 7sylibr 224 . . 3 (𝐴𝑉𝐹 Fn 𝒫 𝐴)
9 dff1o4 6143 . . 3 (𝐹:𝒫 𝐴1-1-onto→𝒫 𝐴 ↔ (𝐹 Fn 𝒫 𝐴𝐹 Fn 𝒫 𝐴))
105, 8, 9sylanbrc 698 . 2 (𝐴𝑉𝐹:𝒫 𝐴1-1-onto→𝒫 𝐴)
11 elpwi 4166 . . . . . . . . 9 (𝑏 ∈ 𝒫 𝐴𝑏𝐴)
1211ad2antll 765 . . . . . . . 8 ((𝐴𝑉 ∧ (𝑎 ∈ 𝒫 𝐴𝑏 ∈ 𝒫 𝐴)) → 𝑏𝐴)
133isf34lem1 9191 . . . . . . . 8 ((𝐴𝑉𝑏𝐴) → (𝐹𝑏) = (𝐴𝑏))
1412, 13syldan 487 . . . . . . 7 ((𝐴𝑉 ∧ (𝑎 ∈ 𝒫 𝐴𝑏 ∈ 𝒫 𝐴)) → (𝐹𝑏) = (𝐴𝑏))
15 elpwi 4166 . . . . . . . . 9 (𝑎 ∈ 𝒫 𝐴𝑎𝐴)
1615ad2antrl 764 . . . . . . . 8 ((𝐴𝑉 ∧ (𝑎 ∈ 𝒫 𝐴𝑏 ∈ 𝒫 𝐴)) → 𝑎𝐴)
173isf34lem1 9191 . . . . . . . 8 ((𝐴𝑉𝑎𝐴) → (𝐹𝑎) = (𝐴𝑎))
1816, 17syldan 487 . . . . . . 7 ((𝐴𝑉 ∧ (𝑎 ∈ 𝒫 𝐴𝑏 ∈ 𝒫 𝐴)) → (𝐹𝑎) = (𝐴𝑎))
1914, 18psseq12d 3699 . . . . . 6 ((𝐴𝑉 ∧ (𝑎 ∈ 𝒫 𝐴𝑏 ∈ 𝒫 𝐴)) → ((𝐹𝑏) ⊊ (𝐹𝑎) ↔ (𝐴𝑏) ⊊ (𝐴𝑎)))
20 difss 3735 . . . . . . 7 (𝐴𝑎) ⊆ 𝐴
21 pssdifcom1 4052 . . . . . . 7 ((𝑏𝐴 ∧ (𝐴𝑎) ⊆ 𝐴) → ((𝐴𝑏) ⊊ (𝐴𝑎) ↔ (𝐴 ∖ (𝐴𝑎)) ⊊ 𝑏))
2212, 20, 21sylancl 694 . . . . . 6 ((𝐴𝑉 ∧ (𝑎 ∈ 𝒫 𝐴𝑏 ∈ 𝒫 𝐴)) → ((𝐴𝑏) ⊊ (𝐴𝑎) ↔ (𝐴 ∖ (𝐴𝑎)) ⊊ 𝑏))
23 dfss4 3856 . . . . . . . 8 (𝑎𝐴 ↔ (𝐴 ∖ (𝐴𝑎)) = 𝑎)
2416, 23sylib 208 . . . . . . 7 ((𝐴𝑉 ∧ (𝑎 ∈ 𝒫 𝐴𝑏 ∈ 𝒫 𝐴)) → (𝐴 ∖ (𝐴𝑎)) = 𝑎)
2524psseq1d 3697 . . . . . 6 ((𝐴𝑉 ∧ (𝑎 ∈ 𝒫 𝐴𝑏 ∈ 𝒫 𝐴)) → ((𝐴 ∖ (𝐴𝑎)) ⊊ 𝑏𝑎𝑏))
2619, 22, 253bitrrd 295 . . . . 5 ((𝐴𝑉 ∧ (𝑎 ∈ 𝒫 𝐴𝑏 ∈ 𝒫 𝐴)) → (𝑎𝑏 ↔ (𝐹𝑏) ⊊ (𝐹𝑎)))
27 vex 3201 . . . . . 6 𝑏 ∈ V
2827brrpss 6937 . . . . 5 (𝑎 [] 𝑏𝑎𝑏)
29 fvex 6199 . . . . . 6 (𝐹𝑎) ∈ V
3029brrpss 6937 . . . . 5 ((𝐹𝑏) [] (𝐹𝑎) ↔ (𝐹𝑏) ⊊ (𝐹𝑎))
3126, 28, 303bitr4g 303 . . . 4 ((𝐴𝑉 ∧ (𝑎 ∈ 𝒫 𝐴𝑏 ∈ 𝒫 𝐴)) → (𝑎 [] 𝑏 ↔ (𝐹𝑏) [] (𝐹𝑎)))
32 relrpss 6935 . . . . 5 Rel []
3332relbrcnv 5504 . . . 4 ((𝐹𝑎) [] (𝐹𝑏) ↔ (𝐹𝑏) [] (𝐹𝑎))
3431, 33syl6bbr 278 . . 3 ((𝐴𝑉 ∧ (𝑎 ∈ 𝒫 𝐴𝑏 ∈ 𝒫 𝐴)) → (𝑎 [] 𝑏 ↔ (𝐹𝑎) [] (𝐹𝑏)))
3534ralrimivva 2970 . 2 (𝐴𝑉 → ∀𝑎 ∈ 𝒫 𝐴𝑏 ∈ 𝒫 𝐴(𝑎 [] 𝑏 ↔ (𝐹𝑎) [] (𝐹𝑏)))
36 df-isom 5895 . 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 698 1 (𝐴𝑉𝐹 Isom [] , [] (𝒫 𝐴, 𝒫 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1482  wcel 1989  wral 2911  Vcvv 3198  cdif 3569  wss 3572  wpss 3573  𝒫 cpw 4156   class class class wbr 4651  cmpt 4727  ccnv 5111   Fn wfn 5881  1-1-ontowf1o 5885  cfv 5886   Isom wiso 5887   [] crpss 6933
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-ne 2794  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-pss 3588  df-nul 3914  df-if 4085  df-pw 4158  df-sn 4176  df-pr 4178  df-op 4182  df-uni 4435  df-br 4652  df-opab 4711  df-mpt 4728  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-fo 5892  df-f1o 5893  df-fv 5894  df-isom 5895  df-rpss 6934
This theorem is referenced by:  isf34lem3  9194  isf34lem5  9197  isfin1-4  9206
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