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Theorem compsscnv 7997
Description: Complementation on a power set lattice is an involution. (Contributed by Mario Carneiro, 17-May-2015.)
Hypothesis
Ref Expression
compss.a  |-  F  =  ( x  e.  ~P A  |->  ( A  \  x ) )
Assertion
Ref Expression
compsscnv  |-  `' F  =  F
Distinct variable group:    x, A
Allowed substitution hint:    F( x)

Proof of Theorem compsscnv
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 cnvopab 5083 . 2  |-  `' { <. y ,  x >.  |  ( y  e.  ~P A  /\  x  =  ( A  \  y ) ) }  =  { <. x ,  y >.  |  ( y  e. 
~P A  /\  x  =  ( A  \ 
y ) ) }
2 compss.a . . . 4  |-  F  =  ( x  e.  ~P A  |->  ( A  \  x ) )
3 difeq2 3288 . . . . 5  |-  ( x  =  y  ->  ( A  \  x )  =  ( A  \  y
) )
43cbvmptv 4111 . . . 4  |-  ( x  e.  ~P A  |->  ( A  \  x ) )  =  ( y  e.  ~P A  |->  ( A  \  y ) )
5 df-mpt 4079 . . . 4  |-  ( y  e.  ~P A  |->  ( A  \  y ) )  =  { <. y ,  x >.  |  ( y  e.  ~P A  /\  x  =  ( A  \  y ) ) }
62, 4, 53eqtri 2307 . . 3  |-  F  =  { <. y ,  x >.  |  ( y  e. 
~P A  /\  x  =  ( A  \ 
y ) ) }
76cnveqi 4856 . 2  |-  `' F  =  `' { <. y ,  x >.  |  ( y  e. 
~P A  /\  x  =  ( A  \ 
y ) ) }
8 df-mpt 4079 . . 3  |-  ( x  e.  ~P A  |->  ( A  \  x ) )  =  { <. x ,  y >.  |  ( x  e.  ~P A  /\  y  =  ( A  \  x ) ) }
9 compsscnvlem 7996 . . . . 5  |-  ( ( y  e.  ~P A  /\  x  =  ( A  \  y ) )  ->  ( x  e. 
~P A  /\  y  =  ( A  \  x ) ) )
10 compsscnvlem 7996 . . . . 5  |-  ( ( x  e.  ~P A  /\  y  =  ( A  \  x ) )  ->  ( y  e. 
~P A  /\  x  =  ( A  \ 
y ) ) )
119, 10impbii 180 . . . 4  |-  ( ( y  e.  ~P A  /\  x  =  ( A  \  y ) )  <-> 
( x  e.  ~P A  /\  y  =  ( A  \  x ) ) )
1211opabbii 4083 . . 3  |-  { <. x ,  y >.  |  ( y  e.  ~P A  /\  x  =  ( A  \  y ) ) }  =  { <. x ,  y >.  |  ( x  e.  ~P A  /\  y  =  ( A  \  x ) ) }
138, 2, 123eqtr4i 2313 . 2  |-  F  =  { <. x ,  y
>.  |  ( y  e.  ~P A  /\  x  =  ( A  \ 
y ) ) }
141, 7, 133eqtr4i 2313 1  |-  `' F  =  F
Colors of variables: wff set class
Syntax hints:    /\ wa 358    = wceq 1623    e. wcel 1684    \ cdif 3149   ~Pcpw 3625   {copab 4076    e. cmpt 4077   `'ccnv 4688
This theorem is referenced by:  compssiso  8000  isf34lem3  8001  compss  8002  isf34lem5  8004
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-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-br 4024  df-opab 4078  df-mpt 4079  df-xp 4695  df-rel 4696  df-cnv 4697
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