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Theorem compss 8216
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 8211 . . 3  |-  `' F  =  F
32imaeq1i 5163 . 2  |-  ( `' F " G )  =  ( F " G )
4 difeq2 3423 . . . . 5  |-  ( x  =  y  ->  ( A  \  x )  =  ( A  \  y
) )
54cbvmptv 4264 . . . 4  |-  ( x  e.  ~P A  |->  ( A  \  x ) )  =  ( y  e.  ~P A  |->  ( A  \  y ) )
61, 5eqtri 2428 . . 3  |-  F  =  ( y  e.  ~P A  |->  ( A  \ 
y ) )
76mptpreima 5326 . 2  |-  ( `' F " G )  =  { y  e. 
~P A  |  ( A  \  y )  e.  G }
83, 7eqtr3i 2430 1  |-  ( F
" G )  =  { y  e.  ~P A  |  ( A  \  y )  e.  G }
Colors of variables: wff set class
Syntax hints:    = wceq 1649    e. wcel 1721   {crab 2674    \ cdif 3281   ~Pcpw 3763    e. cmpt 4230   `'ccnv 4840   "cima 4844
This theorem is referenced by:  isf34lem4  8217
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389  ax-sep 4294  ax-nul 4302  ax-pr 4367
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2262  df-mo 2263  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ne 2573  df-ral 2675  df-rex 2676  df-rab 2679  df-v 2922  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-nul 3593  df-if 3704  df-pw 3765  df-sn 3784  df-pr 3785  df-op 3787  df-br 4177  df-opab 4231  df-mpt 4232  df-xp 4847  df-rel 4848  df-cnv 4849  df-dm 4851  df-rn 4852  df-res 4853  df-ima 4854
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