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Theorem fniniseg2 5636
Description: Inverse point images under functions expressed as abstractions. (Contributed by Stefan O'Rear, 1-Feb-2015.)
Assertion
Ref Expression
fniniseg2  |-  ( F  Fn  A  ->  ( `' F " { B } )  =  {
x  e.  A  | 
( F `  x
)  =  B }
)
Distinct variable groups:    x, A    x, F    x, B

Proof of Theorem fniniseg2
StepHypRef Expression
1 fncnvima2 5635 . 2  |-  ( F  Fn  A  ->  ( `' F " { B } )  =  {
x  e.  A  | 
( F `  x
)  e.  { B } } )
2 funfvex 5530 . . . . 5  |-  ( ( Fun  F  /\  x  e.  dom  F )  -> 
( F `  x
)  e.  _V )
3 elsng 3607 . . . . 5  |-  ( ( F `  x )  e.  _V  ->  (
( F `  x
)  e.  { B } 
<->  ( F `  x
)  =  B ) )
42, 3syl 14 . . . 4  |-  ( ( Fun  F  /\  x  e.  dom  F )  -> 
( ( F `  x )  e.  { B }  <->  ( F `  x )  =  B ) )
54funfni 5314 . . 3  |-  ( ( F  Fn  A  /\  x  e.  A )  ->  ( ( F `  x )  e.  { B }  <->  ( F `  x )  =  B ) )
65rabbidva 2725 . 2  |-  ( F  Fn  A  ->  { x  e.  A  |  ( F `  x )  e.  { B } }  =  { x  e.  A  |  ( F `  x )  =  B } )
71, 6eqtrd 2210 1  |-  ( F  Fn  A  ->  ( `' F " { B } )  =  {
x  e.  A  | 
( F `  x
)  =  B }
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    <-> wb 105    = wceq 1353    e. wcel 2148   {crab 2459   _Vcvv 2737   {csn 3592   `'ccnv 4624   dom cdm 4625   "cima 4628   Fun wfun 5208    Fn wfn 5209   ` cfv 5214
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 4120  ax-pow 4173  ax-pr 4208
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 4003  df-opab 4064  df-id 4292  df-xp 4631  df-rel 4632  df-cnv 4633  df-co 4634  df-dm 4635  df-rn 4636  df-res 4637  df-ima 4638  df-iota 5176  df-fun 5216  df-fn 5217  df-fv 5222
This theorem is referenced by: (None)
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