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Theorem fvelima 5426
Description: Function value in an image. Part of Theorem 4.4(iii) of [Monk1] p. 42. (Contributed by NM, 29-Apr-2004.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)
Assertion
Ref Expression
fvelima  |-  ( ( Fun  F  /\  A  e.  ( F " B
) )  ->  E. x  e.  B  ( F `  x )  =  A )
Distinct variable groups:    x, A    x, B    x, F

Proof of Theorem fvelima
StepHypRef Expression
1 elimag 4923 . . . 4  |-  ( A  e.  ( F " B )  ->  ( A  e.  ( F " B )  <->  E. x  e.  B  x F A ) )
21ibi 234 . . 3  |-  ( A  e.  ( F " B )  ->  E. x  e.  B  x F A )
3 funbrfv 5413 . . . 4  |-  ( Fun 
F  ->  ( x F A  ->  ( F `
 x )  =  A ) )
43reximdv 2616 . . 3  |-  ( Fun 
F  ->  ( E. x  e.  B  x F A  ->  E. x  e.  B  ( F `  x )  =  A ) )
52, 4syl5 30 . 2  |-  ( Fun 
F  ->  ( A  e.  ( F " B
)  ->  E. x  e.  B  ( F `  x )  =  A ) )
65imp 420 1  |-  ( ( Fun  F  /\  A  e.  ( F " B
) )  ->  E. x  e.  B  ( F `  x )  =  A )
Colors of variables: wff set class
Syntax hints:    -> wi 6    /\ wa 360    = wceq 1619    e. wcel 1621   E.wrex 2510   class class class wbr 3920   "cima 4583   Fun wfun 4586   ` cfv 4592
This theorem is referenced by:  ssimaex  5436  isofrlem  5689  tz7.49  6343  rankwflemb  7349  tcrank  7438  zorn2lem5  8011  zorn2lem6  8012  uniimadom  8050  wunr1om  8221  tskr1om  8269  tskr1om2  8270  grur1  8322  iscldtop  16664  kqfvima  17253  fmfnfmlem4  17484  fmfnfm  17485  divstgpopn  17634  c1liplem1  19175  plypf1  19426  htthlem  21327  erdszelem7  22899  erdszelem8  22900  axcontlem9  23774  ivthALT  25424  heibor1lem  25699  ismrc  25942
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1926  ax-ext 2234  ax-sep 4038  ax-nul 4046  ax-pr 4108
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-ral 2513  df-rex 2514  df-rab 2516  df-v 2729  df-sbc 2922  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-nul 3363  df-if 3471  df-sn 3550  df-pr 3551  df-op 3553  df-uni 3728  df-br 3921  df-opab 3975  df-id 4202  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-fun 4602  df-fv 4608
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