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Theorem fvelima 5508
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 5004 . . . 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 5495 . . . 4  |-  ( Fun 
F  ->  ( x F A  ->  ( F `
 x )  =  A ) )
43reximdv 2629 . . 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 2519   class class class wbr 3997   "cima 4664   Fun wfun 4667   ` cfv 4673
This theorem is referenced by:  ssimaex  5518  isofrlem  5771  tz7.49  6425  rankwflemb  7433  tcrank  7522  zorn2lem5  8095  zorn2lem6  8096  uniimadom  8134  wunr1om  8309  tskr1om  8357  tskr1om2  8358  grur1  8410  iscldtop  16795  kqfvima  17384  fmfnfmlem4  17615  fmfnfm  17616  divstgpopn  17765  c1liplem1  19306  plypf1  19557  htthlem  21458  erdszelem7  23101  erdszelem8  23102  axcontlem9  23976  ivthALT  25626  heibor1lem  25901  ismrc  26144
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 1927  ax-ext 2239  ax-sep 4115  ax-nul 4123  ax-pr 4186
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 1884  df-eu 2122  df-mo 2123  df-clab 2245  df-cleq 2251  df-clel 2254  df-nfc 2383  df-ne 2423  df-ral 2523  df-rex 2524  df-rab 2527  df-v 2765  df-sbc 2967  df-dif 3130  df-un 3132  df-in 3134  df-ss 3141  df-nul 3431  df-if 3540  df-sn 3620  df-pr 3621  df-op 3623  df-uni 3802  df-br 3998  df-opab 4052  df-id 4281  df-xp 4675  df-rel 4676  df-cnv 4677  df-co 4678  df-dm 4679  df-rn 4680  df-res 4681  df-ima 4682  df-fun 4683  df-fv 4689
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