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Theorem elima2 5017
Description: Membership in an image. Theorem 34 of [Suppes] p. 65. (Contributed by NM, 11-Aug-2004.)
Hypothesis
Ref Expression
elima.1  |-  A  e. 
_V
Assertion
Ref Expression
elima2  |-  ( A  e.  ( B " C )  <->  E. x
( x  e.  C  /\  x B A ) )
Distinct variable groups:    x, A    x, B    x, C

Proof of Theorem elima2
StepHypRef Expression
1 elima.1 . . 3  |-  A  e. 
_V
21elima 5016 . 2  |-  ( A  e.  ( B " C )  <->  E. x  e.  C  x B A )
3 df-rex 2550 . 2  |-  ( E. x  e.  C  x B A  <->  E. x
( x  e.  C  /\  x B A ) )
42, 3bitri 242 1  |-  ( A  e.  ( B " C )  <->  E. x
( x  e.  C  /\  x B A ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 178    /\ wa 360   E.wex 1529    e. wcel 1685   E.wrex 2545   _Vcvv 2789   class class class wbr 4024   "cima 4691
This theorem is referenced by:  elima3  5018  dminss  5094  imainss  5095  imadif  5292  metcld2  18726  isch2  21795  dfdm5  23533  dfrn5  23534
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1867  ax-ext 2265  ax-sep 4142  ax-nul 4150  ax-pr 4213
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 938  df-tru 1312  df-ex 1530  df-nf 1533  df-sb 1632  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-ral 2549  df-rex 2550  df-rab 2553  df-v 2791  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3457  df-if 3567  df-sn 3647  df-pr 3648  df-op 3650  df-br 4025  df-opab 4079  df-xp 4694  df-cnv 4696  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701
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