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Theorem compssiso 8000
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 4162 . . . . 5  |-  ( A  e.  V  ->  ( A  \  x )  e. 
_V )
21ralrimivw 2627 . . . 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 5370 . . . 4  |-  ( A. x  e.  ~P  A
( A  \  x
)  e.  _V  ->  F  Fn  ~P A )
52, 4syl 15 . . 3  |-  ( A  e.  V  ->  F  Fn  ~P A )
63compsscnv 7997 . . . . 5  |-  `' F  =  F
76fneq1i 5338 . . . 4  |-  ( `' F  Fn  ~P A  <->  F  Fn  ~P A )
85, 7sylibr 203 . . 3  |-  ( A  e.  V  ->  `' F  Fn  ~P A
)
9 dff1o4 5480 . . 3  |-  ( F : ~P A -1-1-onto-> ~P A  <->  ( F  Fn  ~P A  /\  `' F  Fn  ~P A ) )
105, 8, 9sylanbrc 645 . 2  |-  ( A  e.  V  ->  F : ~P A -1-1-onto-> ~P A )
11 elpwi 3633 . . . . . . . . 9  |-  ( b  e.  ~P A  -> 
b  C_  A )
1211ad2antll 709 . . . . . . . 8  |-  ( ( A  e.  V  /\  ( a  e.  ~P A  /\  b  e.  ~P A ) )  -> 
b  C_  A )
133isf34lem1 7998 . . . . . . . 8  |-  ( ( A  e.  V  /\  b  C_  A )  -> 
( F `  b
)  =  ( A 
\  b ) )
1412, 13syldan 456 . . . . . . 7  |-  ( ( A  e.  V  /\  ( a  e.  ~P A  /\  b  e.  ~P A ) )  -> 
( F `  b
)  =  ( A 
\  b ) )
15 elpwi 3633 . . . . . . . . 9  |-  ( a  e.  ~P A  -> 
a  C_  A )
1615ad2antrl 708 . . . . . . . 8  |-  ( ( A  e.  V  /\  ( a  e.  ~P A  /\  b  e.  ~P A ) )  -> 
a  C_  A )
173isf34lem1 7998 . . . . . . . 8  |-  ( ( A  e.  V  /\  a  C_  A )  -> 
( F `  a
)  =  ( A 
\  a ) )
1816, 17syldan 456 . . . . . . 7  |-  ( ( A  e.  V  /\  ( a  e.  ~P A  /\  b  e.  ~P A ) )  -> 
( F `  a
)  =  ( A 
\  a ) )
1914, 18psseq12d 3270 . . . . . 6  |-  ( ( A  e.  V  /\  ( a  e.  ~P A  /\  b  e.  ~P A ) )  -> 
( ( F `  b )  C.  ( F `  a )  <->  ( A  \  b ) 
C.  ( A  \ 
a ) ) )
20 difss 3303 . . . . . . 7  |-  ( A 
\  a )  C_  A
21 pssdifcom1 3539 . . . . . . 7  |-  ( ( b  C_  A  /\  ( A  \  a
)  C_  A )  ->  ( ( A  \ 
b )  C.  ( A  \  a )  <->  ( A  \  ( A  \  a
) )  C.  b
) )
2212, 20, 21sylancl 643 . . . . . 6  |-  ( ( A  e.  V  /\  ( a  e.  ~P A  /\  b  e.  ~P A ) )  -> 
( ( A  \ 
b )  C.  ( A  \  a )  <->  ( A  \  ( A  \  a
) )  C.  b
) )
23 dfss4 3403 . . . . . . . 8  |-  ( a 
C_  A  <->  ( A  \  ( A  \  a
) )  =  a )
2416, 23sylib 188 . . . . . . 7  |-  ( ( A  e.  V  /\  ( a  e.  ~P A  /\  b  e.  ~P A ) )  -> 
( A  \  ( A  \  a ) )  =  a )
2524psseq1d 3268 . . . . . 6  |-  ( ( A  e.  V  /\  ( a  e.  ~P A  /\  b  e.  ~P A ) )  -> 
( ( A  \ 
( A  \  a
) )  C.  b  <->  a 
C.  b ) )
2619, 22, 253bitrrd 271 . . . . 5  |-  ( ( A  e.  V  /\  ( a  e.  ~P A  /\  b  e.  ~P A ) )  -> 
( a  C.  b  <->  ( F `  b ) 
C.  ( F `  a ) ) )
27 vex 2791 . . . . . 6  |-  b  e. 
_V
2827brrpss 6280 . . . . 5  |-  ( a [
C.]  b  <->  a  C.  b )
29 fvex 5539 . . . . . 6  |-  ( F `
 a )  e. 
_V
3029brrpss 6280 . . . . 5  |-  ( ( F `  b ) [
C.]  ( F `  a )  <->  ( F `  b )  C.  ( F `  a )
)
3126, 28, 303bitr4g 279 . . . 4  |-  ( ( A  e.  V  /\  ( a  e.  ~P A  /\  b  e.  ~P A ) )  -> 
( a [ C.]  b  <->  ( F `  b ) [
C.]  ( F `  a ) ) )
32 relrpss 6278 . . . . 5  |-  Rel [ C.]
3332relbrcnv 5054 . . . 4  |-  ( ( F `  a ) `' [ C.]  ( F `  b )  <->  ( F `  b ) [ C.]  ( F `  a )
)
3431, 33syl6bbr 254 . . 3  |-  ( ( A  e.  V  /\  ( a  e.  ~P A  /\  b  e.  ~P A ) )  -> 
( a [ C.]  b  <->  ( F `  a ) `' [ C.]  ( F `  b ) ) )
3534ralrimivva 2635 . 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 5264 . 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 645 1  |-  ( A  e.  V  ->  F  Isom [ C.]  ,  `' [ C.]  ( ~P A ,  ~P A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684   A.wral 2543   _Vcvv 2788    \ cdif 3149    C_ wss 3152    C. wpss 3153   ~Pcpw 3625   class class class wbr 4023    e. cmpt 4077   `'ccnv 4688    Fn wfn 5250   -1-1-onto->wf1o 5254   ` cfv 5255    Isom wiso 5256   [ C.] crpss 6276
This theorem is referenced by:  isf34lem3  8001  isf34lem5  8004  isfin1-4  8013
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-rpss 6277
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