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Theorem elima3 4725
Description: Membership in an image. Theorem 34 of [Suppes] p. 65. (Contributed by NM, 14-Aug-1994.)
Hypothesis
Ref Expression
elima.1  |-  A  e. 
_V
Assertion
Ref Expression
elima3  |-  ( A  e.  ( B " C )  <->  E. x
( x  e.  C  /\  <. x ,  A >.  e.  B ) )
Distinct variable groups:    x, A    x, B    x, C

Proof of Theorem elima3
StepHypRef Expression
1 elima.1 . . 3  |-  A  e. 
_V
21elima2 4724 . 2  |-  ( A  e.  ( B " C )  <->  E. x
( x  e.  C  /\  x B A ) )
3 df-br 3806 . . . 4  |-  ( x B A  <->  <. x ,  A >.  e.  B
)
43anbi2i 445 . . 3  |-  ( ( x  e.  C  /\  x B A )  <->  ( x  e.  C  /\  <. x ,  A >.  e.  B
) )
54exbii 1537 . 2  |-  ( E. x ( x  e.  C  /\  x B A )  <->  E. x
( x  e.  C  /\  <. x ,  A >.  e.  B ) )
62, 5bitri 182 1  |-  ( A  e.  ( B " C )  <->  E. x
( x  e.  C  /\  <. x ,  A >.  e.  B ) )
Colors of variables: wff set class
Syntax hints:    /\ wa 102    <-> wb 103   E.wex 1422    e. wcel 1434   _Vcvv 2610   <.cop 3419   class class class wbr 3805   "cima 4394
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3916  ax-pow 3968  ax-pr 3992
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-rex 2359  df-v 2612  df-un 2986  df-in 2988  df-ss 2995  df-pw 3402  df-sn 3422  df-pr 3423  df-op 3425  df-br 3806  df-opab 3860  df-xp 4397  df-cnv 4399  df-dm 4401  df-rn 4402  df-res 4403  df-ima 4404
This theorem is referenced by:  cnvresima  4860  imaiun  5451
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