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Theorem fniniseg 5843
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 5842 . 2  |-  ( F  Fn  A  ->  ( C  e.  ( `' F " { B }
)  <->  ( C  e.  A  /\  ( F `
 C )  e. 
{ B } ) ) )
2 fvex 5734 . . . 4  |-  ( F `
 C )  e. 
_V
32elsnc 3829 . . 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 1652    e. wcel 1725   {csn 3806   `'ccnv 4869   "cima 4873    Fn wfn 5441   ` cfv 5446
This theorem is referenced by:  fparlem1  6438  fparlem2  6439  pw2f1olem  7204  recmulnq  8833  dmrecnq  8837  vdwlem1  13341  vdwlem2  13342  vdwlem6  13346  vdwlem8  13348  vdwlem9  13349  vdwlem12  13352  vdwlem13  13353  ramval  13368  ramub1lem1  13386  ghmeqker  15024  efgrelexlemb  15374  efgredeu  15376  qtopeu  17740  itg1addlem1  19576  i1faddlem  19577  i1fmullem  19578  i1fmulclem  19586  i1fres  19589  itg10a  19594  itg1ge0a  19595  itg1climres  19598  mbfi1fseqlem4  19602  ply1remlem  20077  ply1rem  20078  fta1glem1  20080  fta1glem2  20081  fta1g  20082  fta1blem  20083  plyco0  20103  ofmulrt  20191  plyremlem  20213  plyrem  20214  fta1lem  20216  fta1  20217  vieta1lem1  20219  vieta1lem2  20220  vieta1  20221  plyexmo  20222  elaa  20225  aannenlem1  20237  aalioulem2  20242  pilem1  20359  efif1olem3  20438  efif1olem4  20439  efifo  20441  eff1olem  20442  basellem4  20858  lgsqrlem2  21118  lgsqrlem3  21119  rpvmasum2  21198  dirith  21215  ofpreima  24073  qqhre  24378  indpi1  24411  indpreima  24414  sibfof  24646  cvmliftlem6  24969  cvmliftlem7  24970  cvmliftlem8  24971  cvmliftlem9  24972  itg2addnclem  26246  itg2addnclem2  26247  pw2f1o2val2  27102  dnnumch3  27113  proot1mul  27483  proot1hash  27487  proot1ex  27488
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-fv 5454
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