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Theorem fniniseg 5646
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 5645 . 2  |-  ( F  Fn  A  ->  ( C  e.  ( `' F " { B }
)  <->  ( C  e.  A  /\  ( F `
 C )  e. 
{ B } ) ) )
2 fvex 5539 . . . 4  |-  ( F `
 C )  e. 
_V
32elsnc 3663 . . 3  |-  ( ( F `  C )  e.  { B }  <->  ( F `  C )  =  B )
43anbi2i 675 . 2  |-  ( ( C  e.  A  /\  ( F `  C )  e.  { B }
)  <->  ( C  e.  A  /\  ( F `
 C )  =  B ) )
51, 4syl6bb 252 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 176    /\ wa 358    = wceq 1623    e. wcel 1684   {csn 3640   `'ccnv 4688   "cima 4692    Fn wfn 5250   ` cfv 5255
This theorem is referenced by:  fparlem1  6218  fparlem2  6219  pw2f1olem  6966  recmulnq  8588  dmrecnq  8592  vdwlem1  13028  vdwlem2  13029  vdwlem6  13033  vdwlem8  13035  vdwlem9  13036  vdwlem12  13039  vdwlem13  13040  ramval  13055  ramub1lem1  13073  ghmeqker  14709  efgrelexlemb  15059  efgredeu  15061  qtopeu  17407  itg1addlem1  19047  i1faddlem  19048  i1fmullem  19049  i1fmulclem  19057  i1fres  19060  itg10a  19065  itg1ge0a  19066  itg1climres  19069  mbfi1fseqlem4  19073  ply1remlem  19548  ply1rem  19549  fta1glem1  19551  fta1glem2  19552  fta1g  19553  fta1blem  19554  plyco0  19574  ofmulrt  19662  plyremlem  19684  plyrem  19685  fta1lem  19687  fta1  19688  vieta1lem1  19690  vieta1lem2  19691  vieta1  19692  plyexmo  19693  elaa  19696  aannenlem1  19708  aalioulem2  19713  pilem1  19827  efif1olem3  19906  efif1olem4  19907  efifo  19909  eff1olem  19910  basellem4  20321  lgsqrlem2  20581  lgsqrlem3  20582  rpvmasum2  20661  dirith  20678  indpi1  23605  indpreima  23608  cvmliftlem6  23821  cvmliftlem7  23822  cvmliftlem8  23823  cvmliftlem9  23824  pw2f1o2val2  27133  dnnumch3  27144  proot1mul  27515  proot1hash  27519  proot1ex  27520
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-fv 5263
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