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Theorem fniniseg 5547
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 5546 . 2  |-  ( F  Fn  A  ->  ( C  e.  ( `' F " { B }
)  <->  ( C  e.  A  /\  ( F `
 C )  e. 
{ B } ) ) )
2 funfvex 5445 . . . . 5  |-  ( ( Fun  F  /\  C  e.  dom  F )  -> 
( F `  C
)  e.  _V )
3 elsng 3546 . . . . 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 5230 . . 3  |-  ( ( F  Fn  A  /\  C  e.  A )  ->  ( ( F `  C )  e.  { B }  <->  ( F `  C )  =  B ) )
65pm5.32da 448 . 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 1332    e. wcel 1481   _Vcvv 2689   {csn 3531   `'ccnv 4545   dom cdm 4546   "cima 4549   Fun wfun 5124    Fn wfn 5125   ` cfv 5130
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4053  ax-pow 4105  ax-pr 4138
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-v 2691  df-sbc 2913  df-un 3079  df-in 3081  df-ss 3088  df-pw 3516  df-sn 3537  df-pr 3538  df-op 3540  df-uni 3744  df-br 3937  df-opab 3997  df-id 4222  df-xp 4552  df-rel 4553  df-cnv 4554  df-co 4555  df-dm 4556  df-rn 4557  df-res 4558  df-ima 4559  df-iota 5095  df-fun 5132  df-fn 5133  df-fv 5138
This theorem is referenced by:  pilem1  12906  taupi  13428
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