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Theorem compssiso 8754
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 4595 . . . . 5  |-  ( A  e.  V  ->  ( A  \  x )  e. 
_V )
21ralrimivw 2879 . . . 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 5707 . . . 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 8751 . . . . 5  |-  `' F  =  F
76fneq1i 5675 . . . 4  |-  ( `' F  Fn  ~P A  <->  F  Fn  ~P A )
85, 7sylibr 212 . . 3  |-  ( A  e.  V  ->  `' F  Fn  ~P A
)
9 dff1o4 5824 . . 3  |-  ( F : ~P A -1-1-onto-> ~P A  <->  ( F  Fn  ~P A  /\  `' F  Fn  ~P A ) )
105, 8, 9sylanbrc 664 . 2  |-  ( A  e.  V  ->  F : ~P A -1-1-onto-> ~P A )
11 elpwi 4019 . . . . . . . . 9  |-  ( b  e.  ~P A  -> 
b  C_  A )
1211ad2antll 728 . . . . . . . 8  |-  ( ( A  e.  V  /\  ( a  e.  ~P A  /\  b  e.  ~P A ) )  -> 
b  C_  A )
133isf34lem1 8752 . . . . . . . 8  |-  ( ( A  e.  V  /\  b  C_  A )  -> 
( F `  b
)  =  ( A 
\  b ) )
1412, 13syldan 470 . . . . . . 7  |-  ( ( A  e.  V  /\  ( a  e.  ~P A  /\  b  e.  ~P A ) )  -> 
( F `  b
)  =  ( A 
\  b ) )
15 elpwi 4019 . . . . . . . . 9  |-  ( a  e.  ~P A  -> 
a  C_  A )
1615ad2antrl 727 . . . . . . . 8  |-  ( ( A  e.  V  /\  ( a  e.  ~P A  /\  b  e.  ~P A ) )  -> 
a  C_  A )
173isf34lem1 8752 . . . . . . . 8  |-  ( ( A  e.  V  /\  a  C_  A )  -> 
( F `  a
)  =  ( A 
\  a ) )
1816, 17syldan 470 . . . . . . 7  |-  ( ( A  e.  V  /\  ( a  e.  ~P A  /\  b  e.  ~P A ) )  -> 
( F `  a
)  =  ( A 
\  a ) )
1914, 18psseq12d 3598 . . . . . 6  |-  ( ( A  e.  V  /\  ( a  e.  ~P A  /\  b  e.  ~P A ) )  -> 
( ( F `  b )  C.  ( F `  a )  <->  ( A  \  b ) 
C.  ( A  \ 
a ) ) )
20 difss 3631 . . . . . . 7  |-  ( A 
\  a )  C_  A
21 pssdifcom1 3912 . . . . . . 7  |-  ( ( b  C_  A  /\  ( A  \  a
)  C_  A )  ->  ( ( A  \ 
b )  C.  ( A  \  a )  <->  ( A  \  ( A  \  a
) )  C.  b
) )
2212, 20, 21sylancl 662 . . . . . 6  |-  ( ( A  e.  V  /\  ( a  e.  ~P A  /\  b  e.  ~P A ) )  -> 
( ( A  \ 
b )  C.  ( A  \  a )  <->  ( A  \  ( A  \  a
) )  C.  b
) )
23 dfss4 3732 . . . . . . . 8  |-  ( a 
C_  A  <->  ( A  \  ( A  \  a
) )  =  a )
2416, 23sylib 196 . . . . . . 7  |-  ( ( A  e.  V  /\  ( a  e.  ~P A  /\  b  e.  ~P A ) )  -> 
( A  \  ( A  \  a ) )  =  a )
2524psseq1d 3596 . . . . . 6  |-  ( ( A  e.  V  /\  ( a  e.  ~P A  /\  b  e.  ~P A ) )  -> 
( ( A  \ 
( A  \  a
) )  C.  b  <->  a 
C.  b ) )
2619, 22, 253bitrrd 280 . . . . 5  |-  ( ( A  e.  V  /\  ( a  e.  ~P A  /\  b  e.  ~P A ) )  -> 
( a  C.  b  <->  ( F `  b ) 
C.  ( F `  a ) ) )
27 vex 3116 . . . . . 6  |-  b  e. 
_V
2827brrpss 6567 . . . . 5  |-  ( a [
C.]  b  <->  a  C.  b
)
29 fvex 5876 . . . . . 6  |-  ( F `
 a )  e. 
_V
3029brrpss 6567 . . . . 5  |-  ( ( F `  b ) [
C.]  ( F `  a )  <->  ( F `  b )  C.  ( F `  a )
)
3126, 28, 303bitr4g 288 . . . 4  |-  ( ( A  e.  V  /\  ( a  e.  ~P A  /\  b  e.  ~P A ) )  -> 
( a [ C.]  b  <->  ( F `  b ) [
C.]  ( F `  a ) ) )
32 relrpss 6565 . . . . 5  |-  Rel [ C.]
3332relbrcnv 5377 . . . 4  |-  ( ( F `  a ) `' [ C.]  ( F `  b )  <->  ( F `  b ) [ C.]  ( F `  a )
)
3431, 33syl6bbr 263 . . 3  |-  ( ( A  e.  V  /\  ( a  e.  ~P A  /\  b  e.  ~P A ) )  -> 
( a [ C.]  b  <->  ( F `  a ) `' [ C.]  ( F `  b ) ) )
3534ralrimivva 2885 . 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 5597 . 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 664 1  |-  ( A  e.  V  ->  F  Isom [ C.]  ,  `' [ C.]  ( ~P A ,  ~P A ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1379    e. wcel 1767   A.wral 2814   _Vcvv 3113    \ cdif 3473    C_ wss 3476    C. wpss 3477   ~Pcpw 4010   class class class wbr 4447    |-> cmpt 4505   `'ccnv 4998    Fn wfn 5583   -1-1-onto->wf1o 5587   ` cfv 5588    Isom wiso 5589   [ C.] crpss 6563
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pr 4686
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-isom 5597  df-rpss 6564
This theorem is referenced by:  isf34lem3  8755  isf34lem5  8758  isfin1-4  8767
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