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Theorem fniniseg 5810
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 5809 . 2  |-  ( F  Fn  A  ->  ( C  e.  ( `' F " { B }
)  <->  ( C  e.  A  /\  ( F `
 C )  e. 
{ B } ) ) )
2 fvex 5701 . . . 4  |-  ( F `
 C )  e. 
_V
32elsnc 3797 . . 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 1721   {csn 3774   `'ccnv 4836   "cima 4840    Fn wfn 5408   ` cfv 5413
This theorem is referenced by:  fparlem1  6405  fparlem2  6406  pw2f1olem  7171  recmulnq  8797  dmrecnq  8801  vdwlem1  13304  vdwlem2  13305  vdwlem6  13309  vdwlem8  13311  vdwlem9  13312  vdwlem12  13315  vdwlem13  13316  ramval  13331  ramub1lem1  13349  ghmeqker  14987  efgrelexlemb  15337  efgredeu  15339  qtopeu  17701  itg1addlem1  19537  i1faddlem  19538  i1fmullem  19539  i1fmulclem  19547  i1fres  19550  itg10a  19555  itg1ge0a  19556  itg1climres  19559  mbfi1fseqlem4  19563  ply1remlem  20038  ply1rem  20039  fta1glem1  20041  fta1glem2  20042  fta1g  20043  fta1blem  20044  plyco0  20064  ofmulrt  20152  plyremlem  20174  plyrem  20175  fta1lem  20177  fta1  20178  vieta1lem1  20180  vieta1lem2  20181  vieta1  20182  plyexmo  20183  elaa  20186  aannenlem1  20198  aalioulem2  20203  pilem1  20320  efif1olem3  20399  efif1olem4  20400  efifo  20402  eff1olem  20403  basellem4  20819  lgsqrlem2  21079  lgsqrlem3  21080  rpvmasum2  21159  dirith  21176  ofpreima  24034  qqhre  24339  indpi1  24372  indpreima  24375  sibfof  24607  cvmliftlem6  24930  cvmliftlem7  24931  cvmliftlem8  24932  cvmliftlem9  24933  itg2addnclem  26155  itg2addnclem2  26156  pw2f1o2val2  27001  dnnumch3  27012  proot1mul  27383  proot1hash  27387  proot1ex  27388
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 1662  ax-8 1683  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pr 4363
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 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-sbc 3122  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-br 4173  df-opab 4227  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-fv 5421
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