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Theorem compss 8002
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 7997 . . 3  |-  `' F  =  F
32imaeq1i 5009 . 2  |-  ( `' F " G )  =  ( F " G )
4 difeq2 3288 . . . . 5  |-  ( x  =  y  ->  ( A  \  x )  =  ( A  \  y
) )
54cbvmptv 4111 . . . 4  |-  ( x  e.  ~P A  |->  ( A  \  x ) )  =  ( y  e.  ~P A  |->  ( A  \  y ) )
61, 5eqtri 2303 . . 3  |-  F  =  ( y  e.  ~P A  |->  ( A  \ 
y ) )
76mptpreima 5166 . 2  |-  ( `' F " G )  =  { y  e. 
~P A  |  ( A  \  y )  e.  G }
83, 7eqtr3i 2305 1  |-  ( F
" G )  =  { y  e.  ~P A  |  ( A  \  y )  e.  G }
Colors of variables: wff set class
Syntax hints:    = wceq 1623    e. wcel 1684   {crab 2547    \ cdif 3149   ~Pcpw 3625    e. cmpt 4077   `'ccnv 4688   "cima 4692
This theorem is referenced by:  isf34lem4  8003
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  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702
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