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Theorem fvelima 5925
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 5351 . . . 4  |-  ( A  e.  ( F " B )  ->  ( A  e.  ( F " B )  <->  E. x  e.  B  x F A ) )
21ibi 241 . . 3  |-  ( A  e.  ( F " B )  ->  E. x  e.  B  x F A )
3 funbrfv 5911 . . . 4  |-  ( Fun 
F  ->  ( x F A  ->  ( F `
 x )  =  A ) )
43reximdv 2931 . . 3  |-  ( Fun 
F  ->  ( E. x  e.  B  x F A  ->  E. x  e.  B  ( F `  x )  =  A ) )
52, 4syl5 32 . 2  |-  ( Fun 
F  ->  ( A  e.  ( F " B
)  ->  E. x  e.  B  ( F `  x )  =  A ) )
65imp 429 1  |-  ( ( Fun  F  /\  A  e.  ( F " B
) )  ->  E. x  e.  B  ( F `  x )  =  A )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1395    e. wcel 1819   E.wrex 2808   class class class wbr 4456   "cima 5011   Fun wfun 5588   ` cfv 5594
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pr 4695
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-br 4457  df-opab 4516  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fv 5602
This theorem is referenced by:  ssimaex  5938  isofrlem  6237  tz7.49  7128  rankwflemb  8228  tcrank  8319  zorn2lem5  8897  zorn2lem6  8898  uniimadom  8936  wunr1om  9114  tskr1om  9162  tskr1om2  9163  grur1  9215  iscldtop  19723  kqfvima  20357  fmfnfmlem4  20584  fmfnfm  20585  qustgpopn  20744  c1liplem1  22523  plypf1  22735  ltgseg  24108  axcontlem9  24402  htthlem  25961  xrofsup  27742  fimaproj  27997  txomap  27998  qtophaus  28000  erdszelem7  28838  erdszelem8  28839  mrsub0  29073  mrsubccat  29075  mrsubcn  29076  msubrn  29086  mthmblem  29137  ftc2nc  30304  ivthALT  30358  heibor1lem  30510  ismrc  30838  icccncfext  31893  dirkercncflem2  32089
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