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Theorem elima 5005
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 5004 . 2  |-  ( A  e.  _V  ->  ( A  e.  ( B " C )  <->  E. x  e.  C  x B A ) )
31, 2ax-mp 10 1  |-  ( A  e.  ( B " C )  <->  E. x  e.  C  x B A )
Colors of variables: wff set class
Syntax hints:    <-> wb 178    e. wcel 1621   E.wrex 2519   _Vcvv 2763   class class class wbr 3997   "cima 4664
This theorem is referenced by:  elima2  5006  rninxp  5105  imaco  5165  isarep1  5269  funimass4  5507  isomin  5768  dfsup2  7163  dfsup2OLD  7164  dfac10b  7733  hausmapdom  17189  pi1blem  18500  adjbd1o  22626  brimage  23841  brimg  23852  dfrdg4  23864  tfrqfree  23865  prj1b  24446  prj3  24447
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2239  ax-sep 4115  ax-nul 4123  ax-pr 4186
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2122  df-mo 2123  df-clab 2245  df-cleq 2251  df-clel 2254  df-nfc 2383  df-ne 2423  df-ral 2523  df-rex 2524  df-rab 2527  df-v 2765  df-dif 3130  df-un 3132  df-in 3134  df-ss 3141  df-nul 3431  df-if 3540  df-sn 3620  df-pr 3621  df-op 3623  df-br 3998  df-opab 4052  df-xp 4675  df-cnv 4677  df-dm 4679  df-rn 4680  df-res 4681  df-ima 4682
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