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Theorem elima2 5278
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 5277 . 2  |-  ( A  e.  ( B " C )  <->  E. x  e.  C  x B A )
3 df-rex 2802 . 2  |-  ( E. x  e.  C  x B A  <->  E. x
( x  e.  C  /\  x B A ) )
42, 3bitri 249 1  |-  ( A  e.  ( B " C )  <->  E. x
( x  e.  C  /\  x B A ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 369   E.wex 1587    e. wcel 1758   E.wrex 2797   _Vcvv 3072   class class class wbr 4395   "cima 4946
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1954  ax-ext 2431  ax-sep 4516  ax-nul 4524  ax-pr 4634
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2265  df-mo 2266  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2602  df-ne 2647  df-ral 2801  df-rex 2802  df-rab 2805  df-v 3074  df-dif 3434  df-un 3436  df-in 3438  df-ss 3445  df-nul 3741  df-if 3895  df-sn 3981  df-pr 3983  df-op 3987  df-br 4396  df-opab 4454  df-xp 4949  df-cnv 4951  df-dm 4953  df-rn 4954  df-res 4955  df-ima 4956
This theorem is referenced by:  elima3  5279  dminss  5354  imainss  5355  imadif  5596  metcld2  20944  isch2  24773  dfdm5  27726  dfrn5  27727  brimg  28107
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