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Theorem fvelima 5780
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 5209 . . . 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 5767 . . . 4  |-  ( Fun 
F  ->  ( x F A  ->  ( F `
 x )  =  A ) )
43reximdv 2819 . . 3  |-  ( Fun 
F  ->  ( E. x  e.  B  x F A  ->  E. x  e.  B  ( F `  x )  =  A ) )
52, 4syl5 31 . 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 4    /\ wa 360    = wceq 1653    e. wcel 1726   E.wrex 2708   class class class wbr 4214   "cima 4883   Fun wfun 5450   ` cfv 5456
This theorem is referenced by:  ssimaex  5790  isofrlem  6062  tz7.49  6704  rankwflemb  7721  tcrank  7810  zorn2lem5  8382  zorn2lem6  8383  uniimadom  8421  wunr1om  8596  tskr1om  8644  tskr1om2  8645  grur1  8697  iscldtop  17161  kqfvima  17764  fmfnfmlem4  17991  fmfnfm  17992  divstgpopn  18151  c1liplem1  19882  plypf1  20133  htthlem  22422  xrofsup  24128  erdszelem7  24885  erdszelem8  24886  axcontlem9  25913  ftc2nc  26291  ivthALT  26340  heibor1lem  26520  ismrc  26757
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pr 4405
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4215  df-opab 4269  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fv 5464
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