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Theorem compssiso 8804
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 4551 . . . . 5  |-  ( A  e.  V  ->  ( A  \  x )  e. 
_V )
21ralrimivw 2803 . . . 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 5704 . . . 4  |-  ( A. x  e.  ~P  A
( A  \  x
)  e.  _V  ->  F  Fn  ~P A )
52, 4syl 17 . . 3  |-  ( A  e.  V  ->  F  Fn  ~P A )
63compsscnv 8801 . . . . 5  |-  `' F  =  F
76fneq1i 5670 . . . 4  |-  ( `' F  Fn  ~P A  <->  F  Fn  ~P A )
85, 7sylibr 216 . . 3  |-  ( A  e.  V  ->  `' F  Fn  ~P A
)
9 dff1o4 5822 . . 3  |-  ( F : ~P A -1-1-onto-> ~P A  <->  ( F  Fn  ~P A  /\  `' F  Fn  ~P A ) )
105, 8, 9sylanbrc 670 . 2  |-  ( A  e.  V  ->  F : ~P A -1-1-onto-> ~P A )
11 elpwi 3960 . . . . . . . . 9  |-  ( b  e.  ~P A  -> 
b  C_  A )
1211ad2antll 735 . . . . . . . 8  |-  ( ( A  e.  V  /\  ( a  e.  ~P A  /\  b  e.  ~P A ) )  -> 
b  C_  A )
133isf34lem1 8802 . . . . . . . 8  |-  ( ( A  e.  V  /\  b  C_  A )  -> 
( F `  b
)  =  ( A 
\  b ) )
1412, 13syldan 473 . . . . . . 7  |-  ( ( A  e.  V  /\  ( a  e.  ~P A  /\  b  e.  ~P A ) )  -> 
( F `  b
)  =  ( A 
\  b ) )
15 elpwi 3960 . . . . . . . . 9  |-  ( a  e.  ~P A  -> 
a  C_  A )
1615ad2antrl 734 . . . . . . . 8  |-  ( ( A  e.  V  /\  ( a  e.  ~P A  /\  b  e.  ~P A ) )  -> 
a  C_  A )
173isf34lem1 8802 . . . . . . . 8  |-  ( ( A  e.  V  /\  a  C_  A )  -> 
( F `  a
)  =  ( A 
\  a ) )
1816, 17syldan 473 . . . . . . 7  |-  ( ( A  e.  V  /\  ( a  e.  ~P A  /\  b  e.  ~P A ) )  -> 
( F `  a
)  =  ( A 
\  a ) )
1914, 18psseq12d 3527 . . . . . 6  |-  ( ( A  e.  V  /\  ( a  e.  ~P A  /\  b  e.  ~P A ) )  -> 
( ( F `  b )  C.  ( F `  a )  <->  ( A  \  b ) 
C.  ( A  \ 
a ) ) )
20 difss 3560 . . . . . . 7  |-  ( A 
\  a )  C_  A
21 pssdifcom1 3853 . . . . . . 7  |-  ( ( b  C_  A  /\  ( A  \  a
)  C_  A )  ->  ( ( A  \ 
b )  C.  ( A  \  a )  <->  ( A  \  ( A  \  a
) )  C.  b
) )
2212, 20, 21sylancl 668 . . . . . 6  |-  ( ( A  e.  V  /\  ( a  e.  ~P A  /\  b  e.  ~P A ) )  -> 
( ( A  \ 
b )  C.  ( A  \  a )  <->  ( A  \  ( A  \  a
) )  C.  b
) )
23 dfss4 3677 . . . . . . . 8  |-  ( a 
C_  A  <->  ( A  \  ( A  \  a
) )  =  a )
2416, 23sylib 200 . . . . . . 7  |-  ( ( A  e.  V  /\  ( a  e.  ~P A  /\  b  e.  ~P A ) )  -> 
( A  \  ( A  \  a ) )  =  a )
2524psseq1d 3525 . . . . . 6  |-  ( ( A  e.  V  /\  ( a  e.  ~P A  /\  b  e.  ~P A ) )  -> 
( ( A  \ 
( A  \  a
) )  C.  b  <->  a 
C.  b ) )
2619, 22, 253bitrrd 284 . . . . 5  |-  ( ( A  e.  V  /\  ( a  e.  ~P A  /\  b  e.  ~P A ) )  -> 
( a  C.  b  <->  ( F `  b ) 
C.  ( F `  a ) ) )
27 vex 3048 . . . . . 6  |-  b  e. 
_V
2827brrpss 6574 . . . . 5  |-  ( a [ C.]  b  <->  a  C.  b
)
29 fvex 5875 . . . . . 6  |-  ( F `
 a )  e. 
_V
3029brrpss 6574 . . . . 5  |-  ( ( F `  b ) [ C.]  ( F `  a
)  <->  ( F `  b )  C.  ( F `  a )
)
3126, 28, 303bitr4g 292 . . . 4  |-  ( ( A  e.  V  /\  ( a  e.  ~P A  /\  b  e.  ~P A ) )  -> 
( a [ C.]  b  <->  ( F `  b ) [ C.]  ( F `  a
) ) )
32 relrpss 6572 . . . . 5  |-  Rel [ C.]
3332relbrcnv 5210 . . . 4  |-  ( ( F `  a ) `' [ C.]  ( F `  b )  <->  ( F `  b ) [ C.]  ( F `  a )
)
3431, 33syl6bbr 267 . . 3  |-  ( ( A  e.  V  /\  ( a  e.  ~P A  /\  b  e.  ~P A ) )  -> 
( a [ C.]  b  <->  ( F `  a ) `' [ C.]  ( F `  b ) ) )
3534ralrimivva 2809 . 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 5591 . 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 670 1  |-  ( A  e.  V  ->  F  Isom [
C.]  ,  `' [ C.]  ( ~P A ,  ~P A
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 188    /\ wa 371    = wceq 1444    e. wcel 1887   A.wral 2737   _Vcvv 3045    \ cdif 3401    C_ wss 3404    C. wpss 3405   ~Pcpw 3951   class class class wbr 4402    |-> cmpt 4461   `'ccnv 4833    Fn wfn 5577   -1-1-onto->wf1o 5581   ` cfv 5582    Isom wiso 5583   [ C.] crpss 6570
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-sep 4525  ax-nul 4534  ax-pr 4639
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-ral 2742  df-rex 2743  df-rab 2746  df-v 3047  df-sbc 3268  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-pss 3420  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-op 3975  df-uni 4199  df-br 4403  df-opab 4462  df-mpt 4463  df-id 4749  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-isom 5591  df-rpss 6571
This theorem is referenced by:  isf34lem3  8805  isf34lem5  8808  isfin1-4  8817
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