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Theorem elima2 4925
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 4924 . 2  |-  ( A  e.  ( B " C )  <->  E. x  e.  C  x B A )
3 df-rex 2514 . 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 1537    e. wcel 1621   E.wrex 2510   _Vcvv 2727   class class class wbr 3920   "cima 4583
This theorem is referenced by:  elima3  4926  dminss  5002  imainss  5003  imadif  5184  metcld2  18564  isch2  21633  dfdm5  23300  dfrn5  23301
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 1926  ax-ext 2234  ax-sep 4038  ax-nul 4046  ax-pr 4108
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 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-ral 2513  df-rex 2514  df-rab 2516  df-v 2729  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-nul 3363  df-if 3471  df-sn 3550  df-pr 3551  df-op 3553  df-br 3921  df-opab 3975  df-xp 4594  df-cnv 4596  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601
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