MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  compssiso Structured version   Visualization version   GIF version

Theorem compssiso 9784
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 5222 . . . . 5 (𝐴𝑉 → (𝐴𝑥) ∈ V)
21ralrimivw 3180 . . . 4 (𝐴𝑉 → ∀𝑥 ∈ 𝒫 𝐴(𝐴𝑥) ∈ V)
3 compss.a . . . . 5 𝐹 = (𝑥 ∈ 𝒫 𝐴 ↦ (𝐴𝑥))
43fnmpt 6481 . . . 4 (∀𝑥 ∈ 𝒫 𝐴(𝐴𝑥) ∈ V → 𝐹 Fn 𝒫 𝐴)
52, 4syl 17 . . 3 (𝐴𝑉𝐹 Fn 𝒫 𝐴)
63compsscnv 9781 . . . . 5 𝐹 = 𝐹
76fneq1i 6443 . . . 4 (𝐹 Fn 𝒫 𝐴𝐹 Fn 𝒫 𝐴)
85, 7sylibr 235 . . 3 (𝐴𝑉𝐹 Fn 𝒫 𝐴)
9 dff1o4 6616 . . 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 4547 . . . . . . . . 9 (𝑏 ∈ 𝒫 𝐴𝑏𝐴)
1211ad2antll 725 . . . . . . . 8 ((𝐴𝑉 ∧ (𝑎 ∈ 𝒫 𝐴𝑏 ∈ 𝒫 𝐴)) → 𝑏𝐴)
133isf34lem1 9782 . . . . . . . 8 ((𝐴𝑉𝑏𝐴) → (𝐹𝑏) = (𝐴𝑏))
1412, 13syldan 591 . . . . . . 7 ((𝐴𝑉 ∧ (𝑎 ∈ 𝒫 𝐴𝑏 ∈ 𝒫 𝐴)) → (𝐹𝑏) = (𝐴𝑏))
15 elpwi 4547 . . . . . . . . 9 (𝑎 ∈ 𝒫 𝐴𝑎𝐴)
1615ad2antrl 724 . . . . . . . 8 ((𝐴𝑉 ∧ (𝑎 ∈ 𝒫 𝐴𝑏 ∈ 𝒫 𝐴)) → 𝑎𝐴)
173isf34lem1 9782 . . . . . . . 8 ((𝐴𝑉𝑎𝐴) → (𝐹𝑎) = (𝐴𝑎))
1816, 17syldan 591 . . . . . . 7 ((𝐴𝑉 ∧ (𝑎 ∈ 𝒫 𝐴𝑏 ∈ 𝒫 𝐴)) → (𝐹𝑎) = (𝐴𝑎))
1914, 18psseq12d 4068 . . . . . 6 ((𝐴𝑉 ∧ (𝑎 ∈ 𝒫 𝐴𝑏 ∈ 𝒫 𝐴)) → ((𝐹𝑏) ⊊ (𝐹𝑎) ↔ (𝐴𝑏) ⊊ (𝐴𝑎)))
20 difss 4105 . . . . . . 7 (𝐴𝑎) ⊆ 𝐴
21 pssdifcom1 4431 . . . . . . 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 4232 . . . . . . . 8 (𝑎𝐴 ↔ (𝐴 ∖ (𝐴𝑎)) = 𝑎)
2416, 23sylib 219 . . . . . . 7 ((𝐴𝑉 ∧ (𝑎 ∈ 𝒫 𝐴𝑏 ∈ 𝒫 𝐴)) → (𝐴 ∖ (𝐴𝑎)) = 𝑎)
2524psseq1d 4066 . . . . . 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 3495 . . . . . 6 𝑏 ∈ V
2827brrpss 7441 . . . . 5 (𝑎 [] 𝑏𝑎𝑏)
29 fvex 6676 . . . . . 6 (𝐹𝑎) ∈ V
3029brrpss 7441 . . . . 5 ((𝐹𝑏) [] (𝐹𝑎) ↔ (𝐹𝑏) ⊊ (𝐹𝑎))
3126, 28, 303bitr4g 315 . . . 4 ((𝐴𝑉 ∧ (𝑎 ∈ 𝒫 𝐴𝑏 ∈ 𝒫 𝐴)) → (𝑎 [] 𝑏 ↔ (𝐹𝑏) [] (𝐹𝑎)))
32 relrpss 7439 . . . . 5 Rel []
3332relbrcnv 5963 . . . 4 ((𝐹𝑎) [] (𝐹𝑏) ↔ (𝐹𝑏) [] (𝐹𝑎))
3431, 33syl6bbr 290 . . 3 ((𝐴𝑉 ∧ (𝑎 ∈ 𝒫 𝐴𝑏 ∈ 𝒫 𝐴)) → (𝑎 [] 𝑏 ↔ (𝐹𝑎) [] (𝐹𝑏)))
3534ralrimivva 3188 . 2 (𝐴𝑉 → ∀𝑎 ∈ 𝒫 𝐴𝑏 ∈ 𝒫 𝐴(𝑎 [] 𝑏 ↔ (𝐹𝑎) [] (𝐹𝑏)))
36 df-isom 6357 . 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 1528  wcel 2105  wral 3135  Vcvv 3492  cdif 3930  wss 3933  wpss 3934  𝒫 cpw 4535   class class class wbr 5057  cmpt 5137  ccnv 5547   Fn wfn 6343  1-1-ontowf1o 6347  cfv 6348   Isom wiso 6349   [] crpss 7437
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-pss 3951  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-isom 6357  df-rpss 7438
This theorem is referenced by:  isf34lem3  9785  isf34lem5  9788  isfin1-4  9797
  Copyright terms: Public domain W3C validator