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Theorem compsscnv 8289
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 5309 . 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 3448 . . . . 5  |-  ( x  =  y  ->  ( A  \  x )  =  ( A  \  y
) )
43cbvmptv 4331 . . . 4  |-  ( x  e.  ~P A  |->  ( A  \  x ) )  =  ( y  e.  ~P A  |->  ( A  \  y ) )
5 df-mpt 4299 . . . 4  |-  ( y  e.  ~P A  |->  ( A  \  y ) )  =  { <. y ,  x >.  |  ( y  e.  ~P A  /\  x  =  ( A  \  y ) ) }
62, 4, 53eqtri 2467 . . 3  |-  F  =  { <. y ,  x >.  |  ( y  e. 
~P A  /\  x  =  ( A  \ 
y ) ) }
76cnveqi 5082 . 2  |-  `' F  =  `' { <. y ,  x >.  |  ( y  e. 
~P A  /\  x  =  ( A  \ 
y ) ) }
8 df-mpt 4299 . . 3  |-  ( x  e.  ~P A  |->  ( A  \  x ) )  =  { <. x ,  y >.  |  ( x  e.  ~P A  /\  y  =  ( A  \  x ) ) }
9 compsscnvlem 8288 . . . . 5  |-  ( ( y  e.  ~P A  /\  x  =  ( A  \  y ) )  ->  ( x  e. 
~P A  /\  y  =  ( A  \  x ) ) )
10 compsscnvlem 8288 . . . . 5  |-  ( ( x  e.  ~P A  /\  y  =  ( A  \  x ) )  ->  ( y  e. 
~P A  /\  x  =  ( A  \ 
y ) ) )
119, 10impbii 182 . . . 4  |-  ( ( y  e.  ~P A  /\  x  =  ( A  \  y ) )  <-> 
( x  e.  ~P A  /\  y  =  ( A  \  x ) ) )
1211opabbii 4303 . . 3  |-  { <. x ,  y >.  |  ( y  e.  ~P A  /\  x  =  ( A  \  y ) ) }  =  { <. x ,  y >.  |  ( x  e.  ~P A  /\  y  =  ( A  \  x ) ) }
138, 2, 123eqtr4i 2473 . 2  |-  F  =  { <. x ,  y
>.  |  ( y  e.  ~P A  /\  x  =  ( A  \ 
y ) ) }
141, 7, 133eqtr4i 2473 1  |-  `' F  =  F
Colors of variables: wff set class
Syntax hints:    /\ wa 360    = wceq 1654    e. wcel 1728    \ cdif 3306   ~Pcpw 3828   {copab 4296    e. cmpt 4297   `'ccnv 4912
This theorem is referenced by:  compssiso  8292  isf34lem3  8293  compss  8294  isf34lem5  8296
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1628  ax-9 1669  ax-8 1690  ax-14 1732  ax-6 1747  ax-7 1752  ax-11 1764  ax-12 1954  ax-ext 2424  ax-sep 4361  ax-nul 4369  ax-pr 4438
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1661  df-eu 2292  df-mo 2293  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-ral 2717  df-rex 2718  df-rab 2721  df-v 2967  df-dif 3312  df-un 3314  df-in 3316  df-ss 3323  df-nul 3617  df-if 3768  df-pw 3830  df-sn 3849  df-pr 3850  df-op 3852  df-br 4244  df-opab 4298  df-mpt 4299  df-xp 4919  df-rel 4920  df-cnv 4921
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