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Theorem compssiso 8254
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  e.  ~P A  |->  ( A  \  x ) )
Assertion
Ref Expression
compssiso  |-  ( A  e.  V  ->  F  Isom [ C.]  ,  `' [ C.]  ( ~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 4351 . . . . 5  |-  ( A  e.  V  ->  ( A  \  x )  e. 
_V )
21ralrimivw 2790 . . . 4  |-  ( A  e.  V  ->  A. x  e.  ~P  A ( A 
\  x )  e. 
_V )
3 compss.a . . . . 5  |-  F  =  ( x  e.  ~P A  |->  ( A  \  x ) )
43fnmpt 5571 . . . 4  |-  ( A. x  e.  ~P  A
( A  \  x
)  e.  _V  ->  F  Fn  ~P A )
52, 4syl 16 . . 3  |-  ( A  e.  V  ->  F  Fn  ~P A )
63compsscnv 8251 . . . . 5  |-  `' F  =  F
76fneq1i 5539 . . . 4  |-  ( `' F  Fn  ~P A  <->  F  Fn  ~P A )
85, 7sylibr 204 . . 3  |-  ( A  e.  V  ->  `' F  Fn  ~P A
)
9 dff1o4 5682 . . 3  |-  ( F : ~P A -1-1-onto-> ~P A  <->  ( F  Fn  ~P A  /\  `' F  Fn  ~P A ) )
105, 8, 9sylanbrc 646 . 2  |-  ( A  e.  V  ->  F : ~P A -1-1-onto-> ~P A )
11 elpwi 3807 . . . . . . . . 9  |-  ( b  e.  ~P A  -> 
b  C_  A )
1211ad2antll 710 . . . . . . . 8  |-  ( ( A  e.  V  /\  ( a  e.  ~P A  /\  b  e.  ~P A ) )  -> 
b  C_  A )
133isf34lem1 8252 . . . . . . . 8  |-  ( ( A  e.  V  /\  b  C_  A )  -> 
( F `  b
)  =  ( A 
\  b ) )
1412, 13syldan 457 . . . . . . 7  |-  ( ( A  e.  V  /\  ( a  e.  ~P A  /\  b  e.  ~P A ) )  -> 
( F `  b
)  =  ( A 
\  b ) )
15 elpwi 3807 . . . . . . . . 9  |-  ( a  e.  ~P A  -> 
a  C_  A )
1615ad2antrl 709 . . . . . . . 8  |-  ( ( A  e.  V  /\  ( a  e.  ~P A  /\  b  e.  ~P A ) )  -> 
a  C_  A )
173isf34lem1 8252 . . . . . . . 8  |-  ( ( A  e.  V  /\  a  C_  A )  -> 
( F `  a
)  =  ( A 
\  a ) )
1816, 17syldan 457 . . . . . . 7  |-  ( ( A  e.  V  /\  ( a  e.  ~P A  /\  b  e.  ~P A ) )  -> 
( F `  a
)  =  ( A 
\  a ) )
1914, 18psseq12d 3441 . . . . . 6  |-  ( ( A  e.  V  /\  ( a  e.  ~P A  /\  b  e.  ~P A ) )  -> 
( ( F `  b )  C.  ( F `  a )  <->  ( A  \  b ) 
C.  ( A  \ 
a ) ) )
20 difss 3474 . . . . . . 7  |-  ( A 
\  a )  C_  A
21 pssdifcom1 3713 . . . . . . 7  |-  ( ( b  C_  A  /\  ( A  \  a
)  C_  A )  ->  ( ( A  \ 
b )  C.  ( A  \  a )  <->  ( A  \  ( A  \  a
) )  C.  b
) )
2212, 20, 21sylancl 644 . . . . . 6  |-  ( ( A  e.  V  /\  ( a  e.  ~P A  /\  b  e.  ~P A ) )  -> 
( ( A  \ 
b )  C.  ( A  \  a )  <->  ( A  \  ( A  \  a
) )  C.  b
) )
23 dfss4 3575 . . . . . . . 8  |-  ( a 
C_  A  <->  ( A  \  ( A  \  a
) )  =  a )
2416, 23sylib 189 . . . . . . 7  |-  ( ( A  e.  V  /\  ( a  e.  ~P A  /\  b  e.  ~P A ) )  -> 
( A  \  ( A  \  a ) )  =  a )
2524psseq1d 3439 . . . . . 6  |-  ( ( A  e.  V  /\  ( a  e.  ~P A  /\  b  e.  ~P A ) )  -> 
( ( A  \ 
( A  \  a
) )  C.  b  <->  a 
C.  b ) )
2619, 22, 253bitrrd 272 . . . . 5  |-  ( ( A  e.  V  /\  ( a  e.  ~P A  /\  b  e.  ~P A ) )  -> 
( a  C.  b  <->  ( F `  b ) 
C.  ( F `  a ) ) )
27 vex 2959 . . . . . 6  |-  b  e. 
_V
2827brrpss 6525 . . . . 5  |-  ( a [
C.]  b  <->  a  C.  b )
29 fvex 5742 . . . . . 6  |-  ( F `
 a )  e. 
_V
3029brrpss 6525 . . . . 5  |-  ( ( F `  b ) [
C.]  ( F `  a )  <->  ( F `  b )  C.  ( F `  a )
)
3126, 28, 303bitr4g 280 . . . 4  |-  ( ( A  e.  V  /\  ( a  e.  ~P A  /\  b  e.  ~P A ) )  -> 
( a [ C.]  b  <->  ( F `  b ) [
C.]  ( F `  a ) ) )
32 relrpss 6523 . . . . 5  |-  Rel [ C.]
3332relbrcnv 5245 . . . 4  |-  ( ( F `  a ) `' [ C.]  ( F `  b )  <->  ( F `  b ) [ C.]  ( F `  a )
)
3431, 33syl6bbr 255 . . 3  |-  ( ( A  e.  V  /\  ( a  e.  ~P A  /\  b  e.  ~P A ) )  -> 
( a [ C.]  b  <->  ( F `  a ) `' [ C.]  ( F `  b ) ) )
3534ralrimivva 2798 . 2  |-  ( A  e.  V  ->  A. a  e.  ~P  A A. b  e.  ~P  A ( a [
C.]  b  <->  ( F `  a ) `' [ C.]  ( F `  b ) ) )
36 df-isom 5463 . 2  |-  ( F 
Isom [ C.]  ,  `' [ C.]  ( ~P A ,  ~P A )  <->  ( F : ~P A -1-1-onto-> ~P A  /\  A. a  e.  ~P  A A. b  e.  ~P  A ( a [ C.]  b 
<->  ( F `  a
) `' [ C.]  ( F `  b )
) ) )
3710, 35, 36sylanbrc 646 1  |-  ( A  e.  V  ->  F  Isom [ C.]  ,  `' [ C.]  ( ~P A ,  ~P A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1652    e. wcel 1725   A.wral 2705   _Vcvv 2956    \ cdif 3317    C_ wss 3320    C. wpss 3321   ~Pcpw 3799   class class class wbr 4212    e. cmpt 4266   `'ccnv 4877    Fn wfn 5449   -1-1-onto->wf1o 5453   ` cfv 5454    Isom wiso 5455   [ C.] crpss 6521
This theorem is referenced by:  isf34lem3  8255  isf34lem5  8258  isfin1-4  8267
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pr 4403
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-isom 5463  df-rpss 6522
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