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Theorem fvelima 5576
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 5018 . . . 4  |-  ( A  e.  ( F " B )  ->  ( A  e.  ( F " B )  <->  E. x  e.  B  x F A ) )
21ibi 232 . . 3  |-  ( A  e.  ( F " B )  ->  E. x  e.  B  x F A )
3 funbrfv 5563 . . . 4  |-  ( Fun 
F  ->  ( x F A  ->  ( F `
 x )  =  A ) )
43reximdv 2656 . . 3  |-  ( Fun 
F  ->  ( E. x  e.  B  x F A  ->  E. x  e.  B  ( F `  x )  =  A ) )
52, 4syl5 28 . 2  |-  ( Fun 
F  ->  ( A  e.  ( F " B
)  ->  E. x  e.  B  ( F `  x )  =  A ) )
65imp 418 1  |-  ( ( Fun  F  /\  A  e.  ( F " B
) )  ->  E. x  e.  B  ( F `  x )  =  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1625    e. wcel 1686   E.wrex 2546   class class class wbr 4025   "cima 4694   Fun wfun 5251   ` cfv 5257
This theorem is referenced by:  ssimaex  5586  isofrlem  5839  tz7.49  6459  rankwflemb  7467  tcrank  7556  zorn2lem5  8129  zorn2lem6  8130  uniimadom  8168  wunr1om  8343  tskr1om  8391  tskr1om2  8392  grur1  8444  iscldtop  16834  kqfvima  17423  fmfnfmlem4  17654  fmfnfm  17655  divstgpopn  17804  c1liplem1  19345  plypf1  19596  htthlem  21499  xrofsup  23257  erdszelem7  23730  erdszelem8  23731  axcontlem9  24602  ivthALT  26269  heibor1lem  26544  ismrc  26787
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1637  ax-8 1645  ax-14 1690  ax-6 1705  ax-7 1710  ax-11 1717  ax-12 1868  ax-ext 2266  ax-sep 4143  ax-nul 4151  ax-pr 4216
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1531  df-nf 1534  df-sb 1632  df-eu 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-ral 2550  df-rex 2551  df-rab 2554  df-v 2792  df-sbc 2994  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-nul 3458  df-if 3568  df-sn 3648  df-pr 3649  df-op 3651  df-uni 3830  df-br 4026  df-opab 4080  df-id 4311  df-xp 4697  df-rel 4698  df-cnv 4699  df-co 4700  df-dm 4701  df-rn 4702  df-res 4703  df-ima 4704  df-iota 5221  df-fun 5259  df-fv 5265
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