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Theorem fniniseg 5367
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 5366 . 2  |-  ( F  Fn  A  ->  ( C  e.  ( `' F " { B }
)  <->  ( C  e.  A  /\  ( F `
 C )  e. 
{ B } ) ) )
2 funfvex 5270 . . . . 5  |-  ( ( Fun  F  /\  C  e.  dom  F )  -> 
( F `  C
)  e.  _V )
3 elsng 3440 . . . . 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 5070 . . 3  |-  ( ( F  Fn  A  /\  C  e.  A )  ->  ( ( F `  C )  e.  { B }  <->  ( F `  C )  =  B ) )
65pm5.32da 440 . 2  |-  ( F  Fn  A  ->  (
( C  e.  A  /\  ( F `  C
)  e.  { B } )  <->  ( C  e.  A  /\  ( F `  C )  =  B ) ) )
71, 6bitrd 186 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 102    <-> wb 103    = wceq 1287    e. wcel 1436   _Vcvv 2614   {csn 3425   `'ccnv 4403   dom cdm 4404   "cima 4407   Fun wfun 4966    Fn wfn 4967   ` cfv 4972
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-14 1448  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067  ax-sep 3925  ax-pow 3977  ax-pr 4003
This theorem depends on definitions:  df-bi 115  df-3an 924  df-tru 1290  df-nf 1393  df-sb 1690  df-eu 1948  df-mo 1949  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-ral 2360  df-rex 2361  df-v 2616  df-sbc 2829  df-un 2990  df-in 2992  df-ss 2999  df-pw 3411  df-sn 3431  df-pr 3432  df-op 3434  df-uni 3631  df-br 3815  df-opab 3869  df-id 4087  df-xp 4410  df-rel 4411  df-cnv 4412  df-co 4413  df-dm 4414  df-rn 4415  df-res 4416  df-ima 4417  df-iota 4937  df-fun 4974  df-fn 4975  df-fv 4980
This theorem is referenced by: (None)
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