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Theorem compsscnv 8215
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 5241 . 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 3427 . . . . 5  |-  ( x  =  y  ->  ( A  \  x )  =  ( A  \  y
) )
43cbvmptv 4268 . . . 4  |-  ( x  e.  ~P A  |->  ( A  \  x ) )  =  ( y  e.  ~P A  |->  ( A  \  y ) )
5 df-mpt 4236 . . . 4  |-  ( y  e.  ~P A  |->  ( A  \  y ) )  =  { <. y ,  x >.  |  ( y  e.  ~P A  /\  x  =  ( A  \  y ) ) }
62, 4, 53eqtri 2436 . . 3  |-  F  =  { <. y ,  x >.  |  ( y  e. 
~P A  /\  x  =  ( A  \ 
y ) ) }
76cnveqi 5014 . 2  |-  `' F  =  `' { <. y ,  x >.  |  ( y  e. 
~P A  /\  x  =  ( A  \ 
y ) ) }
8 df-mpt 4236 . . 3  |-  ( x  e.  ~P A  |->  ( A  \  x ) )  =  { <. x ,  y >.  |  ( x  e.  ~P A  /\  y  =  ( A  \  x ) ) }
9 compsscnvlem 8214 . . . . 5  |-  ( ( y  e.  ~P A  /\  x  =  ( A  \  y ) )  ->  ( x  e. 
~P A  /\  y  =  ( A  \  x ) ) )
10 compsscnvlem 8214 . . . . 5  |-  ( ( x  e.  ~P A  /\  y  =  ( A  \  x ) )  ->  ( y  e. 
~P A  /\  x  =  ( A  \ 
y ) ) )
119, 10impbii 181 . . . 4  |-  ( ( y  e.  ~P A  /\  x  =  ( A  \  y ) )  <-> 
( x  e.  ~P A  /\  y  =  ( A  \  x ) ) )
1211opabbii 4240 . . 3  |-  { <. x ,  y >.  |  ( y  e.  ~P A  /\  x  =  ( A  \  y ) ) }  =  { <. x ,  y >.  |  ( x  e.  ~P A  /\  y  =  ( A  \  x ) ) }
138, 2, 123eqtr4i 2442 . 2  |-  F  =  { <. x ,  y
>.  |  ( y  e.  ~P A  /\  x  =  ( A  \ 
y ) ) }
141, 7, 133eqtr4i 2442 1  |-  `' F  =  F
Colors of variables: wff set class
Syntax hints:    /\ wa 359    = wceq 1649    e. wcel 1721    \ cdif 3285   ~Pcpw 3767   {copab 4233    e. cmpt 4234   `'ccnv 4844
This theorem is referenced by:  compssiso  8218  isf34lem3  8219  compss  8220  isf34lem5  8222
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-sep 4298  ax-nul 4306  ax-pr 4371
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-ral 2679  df-rex 2680  df-rab 2683  df-v 2926  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-op 3791  df-br 4181  df-opab 4235  df-mpt 4236  df-xp 4851  df-rel 4852  df-cnv 4853
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