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Theorem fniniseg 5435
Description: Membership in the preimage of a singleton, under a function. (Contributed by Mario Carneiro, 12-May-2014.) (Proof shortened by Mario Carneiro, 28-Apr-2015.)
Assertion
Ref Expression
fniniseg  |-  ( F  Fn  A  ->  ( C  e.  ( `' F " { B }
)  <->  ( C  e.  A  /\  ( F `
 C )  =  B ) ) )

Proof of Theorem fniniseg
StepHypRef Expression
1 elpreima 5434 . 2  |-  ( F  Fn  A  ->  ( C  e.  ( `' F " { B }
)  <->  ( C  e.  A  /\  ( F `
 C )  e. 
{ B } ) ) )
2 funfvex 5337 . . . . 5  |-  ( ( Fun  F  /\  C  e.  dom  F )  -> 
( F `  C
)  e.  _V )
3 elsng 3467 . . . . 5  |-  ( ( F `  C )  e.  _V  ->  (
( F `  C
)  e.  { B } 
<->  ( F `  C
)  =  B ) )
42, 3syl 14 . . . 4  |-  ( ( Fun  F  /\  C  e.  dom  F )  -> 
( ( F `  C )  e.  { B }  <->  ( F `  C )  =  B ) )
54funfni 5129 . . 3  |-  ( ( F  Fn  A  /\  C  e.  A )  ->  ( ( F `  C )  e.  { B }  <->  ( F `  C )  =  B ) )
65pm5.32da 441 . 2  |-  ( F  Fn  A  ->  (
( C  e.  A  /\  ( F `  C
)  e.  { B } )  <->  ( C  e.  A  /\  ( F `  C )  =  B ) ) )
71, 6bitrd 187 1  |-  ( F  Fn  A  ->  ( C  e.  ( `' F " { B }
)  <->  ( C  e.  A  /\  ( F `
 C )  =  B ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    = wceq 1290    e. wcel 1439   _Vcvv 2622   {csn 3452   `'ccnv 4453   dom cdm 4454   "cima 4457   Fun wfun 5024    Fn wfn 5025   ` cfv 5030
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-14 1451  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071  ax-sep 3965  ax-pow 4017  ax-pr 4047
This theorem depends on definitions:  df-bi 116  df-3an 927  df-tru 1293  df-nf 1396  df-sb 1694  df-eu 1952  df-mo 1953  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ral 2365  df-rex 2366  df-v 2624  df-sbc 2844  df-un 3006  df-in 3008  df-ss 3015  df-pw 3437  df-sn 3458  df-pr 3459  df-op 3461  df-uni 3662  df-br 3854  df-opab 3908  df-id 4131  df-xp 4460  df-rel 4461  df-cnv 4462  df-co 4463  df-dm 4464  df-rn 4465  df-res 4466  df-ima 4467  df-iota 4995  df-fun 5032  df-fn 5033  df-fv 5038
This theorem is referenced by: (None)
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