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Theorem compsscnv 8087
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 5165 . 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 3364 . . . . 5  |-  ( x  =  y  ->  ( A  \  x )  =  ( A  \  y
) )
43cbvmptv 4192 . . . 4  |-  ( x  e.  ~P A  |->  ( A  \  x ) )  =  ( y  e.  ~P A  |->  ( A  \  y ) )
5 df-mpt 4160 . . . 4  |-  ( y  e.  ~P A  |->  ( A  \  y ) )  =  { <. y ,  x >.  |  ( y  e.  ~P A  /\  x  =  ( A  \  y ) ) }
62, 4, 53eqtri 2382 . . 3  |-  F  =  { <. y ,  x >.  |  ( y  e. 
~P A  /\  x  =  ( A  \ 
y ) ) }
76cnveqi 4938 . 2  |-  `' F  =  `' { <. y ,  x >.  |  ( y  e. 
~P A  /\  x  =  ( A  \ 
y ) ) }
8 df-mpt 4160 . . 3  |-  ( x  e.  ~P A  |->  ( A  \  x ) )  =  { <. x ,  y >.  |  ( x  e.  ~P A  /\  y  =  ( A  \  x ) ) }
9 compsscnvlem 8086 . . . . 5  |-  ( ( y  e.  ~P A  /\  x  =  ( A  \  y ) )  ->  ( x  e. 
~P A  /\  y  =  ( A  \  x ) ) )
10 compsscnvlem 8086 . . . . 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 4164 . . 3  |-  { <. x ,  y >.  |  ( y  e.  ~P A  /\  x  =  ( A  \  y ) ) }  =  { <. x ,  y >.  |  ( x  e.  ~P A  /\  y  =  ( A  \  x ) ) }
138, 2, 123eqtr4i 2388 . 2  |-  F  =  { <. x ,  y
>.  |  ( y  e.  ~P A  /\  x  =  ( A  \ 
y ) ) }
141, 7, 133eqtr4i 2388 1  |-  `' F  =  F
Colors of variables: wff set class
Syntax hints:    /\ wa 358    = wceq 1642    e. wcel 1710    \ cdif 3225   ~Pcpw 3701   {copab 4157    e. cmpt 4158   `'ccnv 4770
This theorem is referenced by:  compssiso  8090  isf34lem3  8091  compss  8092  isf34lem5  8094
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-sep 4222  ax-nul 4230  ax-pr 4295
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-ral 2624  df-rex 2625  df-rab 2628  df-v 2866  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-op 3725  df-br 4105  df-opab 4159  df-mpt 4160  df-xp 4777  df-rel 4778  df-cnv 4779
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