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Theorem elima 5167
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 5166 . 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 177    e. wcel 1721   E.wrex 2667   _Vcvv 2916   class class class wbr 4172   "cima 4840
This theorem is referenced by:  elima2  5168  rninxp  5269  imaco  5334  isarep1  5491  funimass4  5736  isomin  6016  dfsup2  7405  dfsup2OLD  7406  dfac10b  7975  hausmapdom  17516  pi1blem  19017  adjbd1o  23541  brimage  25679  brimg  25690  dfrdg4  25703  tfrqfree  25704
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pr 4363
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-sn 3780  df-pr 3781  df-op 3783  df-br 4173  df-opab 4227  df-xp 4843  df-cnv 4845  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850
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