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Theorem compssiso 10411
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 5334 . . . . 5 (𝐴𝑉 → (𝐴𝑥) ∈ V)
21ralrimivw 3147 . . . 4 (𝐴𝑉 → ∀𝑥 ∈ 𝒫 𝐴(𝐴𝑥) ∈ V)
3 compss.a . . . . 5 𝐹 = (𝑥 ∈ 𝒫 𝐴 ↦ (𝐴𝑥))
43fnmpt 6708 . . . 4 (∀𝑥 ∈ 𝒫 𝐴(𝐴𝑥) ∈ V → 𝐹 Fn 𝒫 𝐴)
52, 4syl 17 . . 3 (𝐴𝑉𝐹 Fn 𝒫 𝐴)
63compsscnv 10408 . . . . 5 𝐹 = 𝐹
76fneq1i 6665 . . . 4 (𝐹 Fn 𝒫 𝐴𝐹 Fn 𝒫 𝐴)
85, 7sylibr 234 . . 3 (𝐴𝑉𝐹 Fn 𝒫 𝐴)
9 dff1o4 6856 . . 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 4611 . . . . . . . . 9 (𝑏 ∈ 𝒫 𝐴𝑏𝐴)
1211ad2antll 729 . . . . . . . 8 ((𝐴𝑉 ∧ (𝑎 ∈ 𝒫 𝐴𝑏 ∈ 𝒫 𝐴)) → 𝑏𝐴)
133isf34lem1 10409 . . . . . . . 8 ((𝐴𝑉𝑏𝐴) → (𝐹𝑏) = (𝐴𝑏))
1412, 13syldan 591 . . . . . . 7 ((𝐴𝑉 ∧ (𝑎 ∈ 𝒫 𝐴𝑏 ∈ 𝒫 𝐴)) → (𝐹𝑏) = (𝐴𝑏))
15 elpwi 4611 . . . . . . . . 9 (𝑎 ∈ 𝒫 𝐴𝑎𝐴)
1615ad2antrl 728 . . . . . . . 8 ((𝐴𝑉 ∧ (𝑎 ∈ 𝒫 𝐴𝑏 ∈ 𝒫 𝐴)) → 𝑎𝐴)
173isf34lem1 10409 . . . . . . . 8 ((𝐴𝑉𝑎𝐴) → (𝐹𝑎) = (𝐴𝑎))
1816, 17syldan 591 . . . . . . 7 ((𝐴𝑉 ∧ (𝑎 ∈ 𝒫 𝐴𝑏 ∈ 𝒫 𝐴)) → (𝐹𝑎) = (𝐴𝑎))
1914, 18psseq12d 4106 . . . . . 6 ((𝐴𝑉 ∧ (𝑎 ∈ 𝒫 𝐴𝑏 ∈ 𝒫 𝐴)) → ((𝐹𝑏) ⊊ (𝐹𝑎) ↔ (𝐴𝑏) ⊊ (𝐴𝑎)))
20 difss 4145 . . . . . . 7 (𝐴𝑎) ⊆ 𝐴
21 pssdifcom1 4495 . . . . . . 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 4274 . . . . . . . 8 (𝑎𝐴 ↔ (𝐴 ∖ (𝐴𝑎)) = 𝑎)
2416, 23sylib 218 . . . . . . 7 ((𝐴𝑉 ∧ (𝑎 ∈ 𝒫 𝐴𝑏 ∈ 𝒫 𝐴)) → (𝐴 ∖ (𝐴𝑎)) = 𝑎)
2524psseq1d 4104 . . . . . 6 ((𝐴𝑉 ∧ (𝑎 ∈ 𝒫 𝐴𝑏 ∈ 𝒫 𝐴)) → ((𝐴 ∖ (𝐴𝑎)) ⊊ 𝑏𝑎𝑏))
2619, 22, 253bitrrd 306 . . . . 5 ((𝐴𝑉 ∧ (𝑎 ∈ 𝒫 𝐴𝑏 ∈ 𝒫 𝐴)) → (𝑎𝑏 ↔ (𝐹𝑏) ⊊ (𝐹𝑎)))
27 vex 3481 . . . . . 6 𝑏 ∈ V
2827brrpss 7744 . . . . 5 (𝑎 [] 𝑏𝑎𝑏)
29 fvex 6919 . . . . . 6 (𝐹𝑎) ∈ V
3029brrpss 7744 . . . . 5 ((𝐹𝑏) [] (𝐹𝑎) ↔ (𝐹𝑏) ⊊ (𝐹𝑎))
3126, 28, 303bitr4g 314 . . . 4 ((𝐴𝑉 ∧ (𝑎 ∈ 𝒫 𝐴𝑏 ∈ 𝒫 𝐴)) → (𝑎 [] 𝑏 ↔ (𝐹𝑏) [] (𝐹𝑎)))
32 relrpss 7742 . . . . 5 Rel []
3332relbrcnv 6127 . . . 4 ((𝐹𝑎) [] (𝐹𝑏) ↔ (𝐹𝑏) [] (𝐹𝑎))
3431, 33bitr4di 289 . . 3 ((𝐴𝑉 ∧ (𝑎 ∈ 𝒫 𝐴𝑏 ∈ 𝒫 𝐴)) → (𝑎 [] 𝑏 ↔ (𝐹𝑎) [] (𝐹𝑏)))
3534ralrimivva 3199 . 2 (𝐴𝑉 → ∀𝑎 ∈ 𝒫 𝐴𝑏 ∈ 𝒫 𝐴(𝑎 [] 𝑏 ↔ (𝐹𝑎) [] (𝐹𝑏)))
36 df-isom 6571 . 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 206  wa 395   = wceq 1536  wcel 2105  wral 3058  Vcvv 3477  cdif 3959  wss 3962  wpss 3963  𝒫 cpw 4604   class class class wbr 5147  cmpt 5230  ccnv 5687   Fn wfn 6557  1-1-ontowf1o 6561  cfv 6562   Isom wiso 6563   [] crpss 7740
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pr 5437
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3433  df-v 3479  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-pss 3982  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5582  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-isom 6571  df-rpss 7741
This theorem is referenced by:  isf34lem3  10412  isf34lem5  10415  isfin1-4  10424
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