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Theorem compss 8804
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 8799 . . 3  |-  `' F  =  F
32imaeq1i 5185 . 2  |-  ( `' F " G )  =  ( F " G )
4 difeq2 3583 . . . . 5  |-  ( x  =  y  ->  ( A  \  x )  =  ( A  \  y
) )
54cbvmptv 4518 . . . 4  |-  ( x  e.  ~P A  |->  ( A  \  x ) )  =  ( y  e.  ~P A  |->  ( A  \  y ) )
61, 5eqtri 2458 . . 3  |-  F  =  ( y  e.  ~P A  |->  ( A  \ 
y ) )
76mptpreima 5348 . 2  |-  ( `' F " G )  =  { y  e. 
~P A  |  ( A  \  y )  e.  G }
83, 7eqtr3i 2460 1  |-  ( F
" G )  =  { y  e.  ~P A  |  ( A  \  y )  e.  G }
Colors of variables: wff setvar class
Syntax hints:    = wceq 1437    e. wcel 1870   {crab 2786    \ cdif 3439   ~Pcpw 3985    |-> cmpt 4484   `'ccnv 4853   "cima 4857
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-sep 4548  ax-nul 4556  ax-pr 4661
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-rab 2791  df-v 3089  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-op 4009  df-br 4427  df-opab 4485  df-mpt 4486  df-xp 4860  df-rel 4861  df-cnv 4862  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867
This theorem is referenced by:  isf34lem4  8805
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