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Theorem compss 8294
Description: Express image under of the complementation isomorphism. (Contributed by Stefan O'Rear, 5-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
compss  |-  ( F
" G )  =  { y  e.  ~P A  |  ( A  \  y )  e.  G }
Distinct variable groups:    x, y, A    y, F    y, G
Allowed substitution hints:    F( x)    G( x)

Proof of Theorem compss
StepHypRef Expression
1 compss.a . . . 4  |-  F  =  ( x  e.  ~P A  |->  ( A  \  x ) )
21compsscnv 8289 . . 3  |-  `' F  =  F
32imaeq1i 5235 . 2  |-  ( `' F " G )  =  ( F " G )
4 difeq2 3448 . . . . 5  |-  ( x  =  y  ->  ( A  \  x )  =  ( A  \  y
) )
54cbvmptv 4331 . . . 4  |-  ( x  e.  ~P A  |->  ( A  \  x ) )  =  ( y  e.  ~P A  |->  ( A  \  y ) )
61, 5eqtri 2463 . . 3  |-  F  =  ( y  e.  ~P A  |->  ( A  \ 
y ) )
76mptpreima 5398 . 2  |-  ( `' F " G )  =  { y  e. 
~P A  |  ( A  \  y )  e.  G }
83, 7eqtr3i 2465 1  |-  ( F
" G )  =  { y  e.  ~P A  |  ( A  \  y )  e.  G }
Colors of variables: wff set class
Syntax hints:    = wceq 1654    e. wcel 1728   {crab 2716    \ cdif 3306   ~Pcpw 3828    e. cmpt 4297   `'ccnv 4912   "cima 4916
This theorem is referenced by:  isf34lem4  8295
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  df-dm 4923  df-rn 4924  df-res 4925  df-ima 4926
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