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Theorem fncnvima2 5637
Description: Inverse images under functions expressed as abstractions. (Contributed by Stefan O'Rear, 1-Feb-2015.)
Assertion
Ref Expression
fncnvima2  |-  ( F  Fn  A  ->  ( `' F " B )  =  { x  e.  A  |  ( F `
 x )  e.  B } )
Distinct variable groups:    x, A    x, F    x, B

Proof of Theorem fncnvima2
StepHypRef Expression
1 elpreima 5635 . . 3  |-  ( F  Fn  A  ->  (
x  e.  ( `' F " B )  <-> 
( x  e.  A  /\  ( F `  x
)  e.  B ) ) )
21abbi2dv 2296 . 2  |-  ( F  Fn  A  ->  ( `' F " B )  =  { x  |  ( x  e.  A  /\  ( F `  x
)  e.  B ) } )
3 df-rab 2464 . 2  |-  { x  e.  A  |  ( F `  x )  e.  B }  =  {
x  |  ( x  e.  A  /\  ( F `  x )  e.  B ) }
42, 3eqtr4di 2228 1  |-  ( F  Fn  A  ->  ( `' F " B )  =  { x  e.  A  |  ( F `
 x )  e.  B } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    = wceq 1353    e. wcel 2148   {cab 2163   {crab 2459   `'ccnv 4625   "cima 4629    Fn wfn 5211   ` cfv 5216
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4121  ax-pow 4174  ax-pr 4209
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2739  df-sbc 2963  df-un 3133  df-in 3135  df-ss 3142  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-br 4004  df-opab 4065  df-id 4293  df-xp 4632  df-rel 4633  df-cnv 4634  df-co 4635  df-dm 4636  df-rn 4637  df-res 4638  df-ima 4639  df-iota 5178  df-fun 5218  df-fn 5219  df-fv 5224
This theorem is referenced by:  fniniseg2  5638  fnniniseg2  5639
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