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Theorem elima 5099
Description: Membership in an image. Theorem 34 of [Suppes] p. 65. (Contributed by NM, 19-Apr-2004.)
Hypothesis
Ref Expression
elima.1  |-  A  e. 
_V
Assertion
Ref Expression
elima  |-  ( A  e.  ( B " C )  <->  E. x  e.  C  x B A )
Distinct variable groups:    x, A    x, B    x, C

Proof of Theorem elima
StepHypRef Expression
1 elima.1 . 2  |-  A  e. 
_V
2 elimag 5098 . 2  |-  ( A  e.  _V  ->  ( A  e.  ( B " C )  <->  E. x  e.  C  x B A ) )
31, 2ax-mp 8 1  |-  ( A  e.  ( B " C )  <->  E. x  e.  C  x B A )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    e. wcel 1710   E.wrex 2620   _Vcvv 2864   class class class wbr 4104   "cima 4774
This theorem is referenced by:  elima2  5100  rninxp  5199  imaco  5260  isarep1  5413  funimass4  5656  isomin  5921  dfsup2  7285  dfsup2OLD  7286  dfac10b  7855  hausmapdom  17332  pi1blem  18641  adjbd1o  22779  brimage  25023  brimg  25034  dfrdg4  25047  tfrqfree  25048
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-sep 4222  ax-nul 4230  ax-pr 4295
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-ral 2624  df-rex 2625  df-rab 2628  df-v 2866  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-sn 3722  df-pr 3723  df-op 3725  df-br 4105  df-opab 4159  df-xp 4777  df-cnv 4779  df-dm 4781  df-rn 4782  df-res 4783  df-ima 4784
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