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Theorem fniniseg 5790
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 5789 . 2  |-  ( F  Fn  A  ->  ( C  e.  ( `' F " { B }
)  <->  ( C  e.  A  /\  ( F `
 C )  e. 
{ B } ) ) )
2 fvex 5682 . . . 4  |-  ( F `
 C )  e. 
_V
32elsnc 3780 . . 3  |-  ( ( F `  C )  e.  { B }  <->  ( F `  C )  =  B )
43anbi2i 676 . 2  |-  ( ( C  e.  A  /\  ( F `  C )  e.  { B }
)  <->  ( C  e.  A  /\  ( F `
 C )  =  B ) )
51, 4syl6bb 253 1  |-  ( F  Fn  A  ->  ( C  e.  ( `' F " { B }
)  <->  ( C  e.  A  /\  ( F `
 C )  =  B ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1717   {csn 3757   `'ccnv 4817   "cima 4821    Fn wfn 5389   ` cfv 5394
This theorem is referenced by:  fparlem1  6385  fparlem2  6386  pw2f1olem  7148  recmulnq  8774  dmrecnq  8778  vdwlem1  13276  vdwlem2  13277  vdwlem6  13281  vdwlem8  13283  vdwlem9  13284  vdwlem12  13287  vdwlem13  13288  ramval  13303  ramub1lem1  13321  ghmeqker  14959  efgrelexlemb  15309  efgredeu  15311  qtopeu  17669  itg1addlem1  19451  i1faddlem  19452  i1fmullem  19453  i1fmulclem  19461  i1fres  19464  itg10a  19469  itg1ge0a  19470  itg1climres  19473  mbfi1fseqlem4  19477  ply1remlem  19952  ply1rem  19953  fta1glem1  19955  fta1glem2  19956  fta1g  19957  fta1blem  19958  plyco0  19978  ofmulrt  20066  plyremlem  20088  plyrem  20089  fta1lem  20091  fta1  20092  vieta1lem1  20094  vieta1lem2  20095  vieta1  20096  plyexmo  20097  elaa  20100  aannenlem1  20112  aalioulem2  20117  pilem1  20234  efif1olem3  20313  efif1olem4  20314  efifo  20316  eff1olem  20317  basellem4  20733  lgsqrlem2  20993  lgsqrlem3  20994  rpvmasum2  21073  dirith  21090  qqhre  24182  indpi1  24215  indpreima  24218  cvmliftlem6  24756  cvmliftlem7  24757  cvmliftlem8  24758  cvmliftlem9  24759  itg2addnclem  25957  itg2addnclem2  25958  pw2f1o2val2  26802  dnnumch3  26813  proot1mul  27184  proot1hash  27188  proot1ex  27189
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-sep 4271  ax-nul 4279  ax-pr 4344
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-ral 2654  df-rex 2655  df-rab 2658  df-v 2901  df-sbc 3105  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-if 3683  df-sn 3763  df-pr 3764  df-op 3766  df-uni 3958  df-br 4154  df-opab 4208  df-id 4439  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fn 5397  df-fv 5402
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