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Theorem compss 8756
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 8751 . . 3  |-  `' F  =  F
32imaeq1i 5334 . 2  |-  ( `' F " G )  =  ( F " G )
4 difeq2 3616 . . . . 5  |-  ( x  =  y  ->  ( A  \  x )  =  ( A  \  y
) )
54cbvmptv 4538 . . . 4  |-  ( x  e.  ~P A  |->  ( A  \  x ) )  =  ( y  e.  ~P A  |->  ( A  \  y ) )
61, 5eqtri 2496 . . 3  |-  F  =  ( y  e.  ~P A  |->  ( A  \ 
y ) )
76mptpreima 5500 . 2  |-  ( `' F " G )  =  { y  e. 
~P A  |  ( A  \  y )  e.  G }
83, 7eqtr3i 2498 1  |-  ( F
" G )  =  { y  e.  ~P A  |  ( A  \  y )  e.  G }
Colors of variables: wff setvar class
Syntax hints:    = wceq 1379    e. wcel 1767   {crab 2818    \ cdif 3473   ~Pcpw 4010    |-> cmpt 4505   `'ccnv 4998   "cima 5002
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pr 4686
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-br 4448  df-opab 4506  df-mpt 4507  df-xp 5005  df-rel 5006  df-cnv 5007  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012
This theorem is referenced by:  isf34lem4  8757
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