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

Theorem compssiso 8016
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 4178 . . . . 5  |-  ( A  e.  V  ->  ( A  \  x )  e. 
_V )
21ralrimivw 2640 . . . 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 5386 . . . 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 8013 . . . . 5  |-  `' F  =  F
76fneq1i 5354 . . . 4  |-  ( `' F  Fn  ~P A  <->  F  Fn  ~P A )
85, 7sylibr 203 . . 3  |-  ( A  e.  V  ->  `' F  Fn  ~P A
)
9 dff1o4 5496 . . 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 3646 . . . . . . . . 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 8014 . . . . . . . 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 3646 . . . . . . . . 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 8014 . . . . . . . 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 3283 . . . . . 6  |-  ( ( A  e.  V  /\  ( a  e.  ~P A  /\  b  e.  ~P A ) )  -> 
( ( F `  b )  C.  ( F `  a )  <->  ( A  \  b ) 
C.  ( A  \ 
a ) ) )
20 difss 3316 . . . . . . 7  |-  ( A 
\  a )  C_  A
21 pssdifcom1 3552 . . . . . . 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 3416 . . . . . . . 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 3281 . . . . . 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 2804 . . . . . 6  |-  b  e. 
_V
2827brrpss 6296 . . . . 5  |-  ( a [
C.]  b  <->  a  C.  b )
29 fvex 5555 . . . . . 6  |-  ( F `
 a )  e. 
_V
3029brrpss 6296 . . . . 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 6294 . . . . 5  |-  Rel [ C.]
3332relbrcnv 5070 . . . 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 2648 . 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 5280 . 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 1632    e. wcel 1696   A.wral 2556   _Vcvv 2801    \ cdif 3162    C_ wss 3165    C. wpss 3166   ~Pcpw 3638   class class class wbr 4039    e. cmpt 4093   `'ccnv 4704    Fn wfn 5266   -1-1-onto->wf1o 5270   ` cfv 5271    Isom wiso 5272   [ C.] crpss 6292
This theorem is referenced by:  isf34lem3  8017  isf34lem5  8020  isfin1-4  8029
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-isom 5280  df-rpss 6293
  Copyright terms: Public domain W3C validator