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Theorem elima3 4675
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 4674 . 2  |-  ( A  e.  ( B " C )  <->  E. x
( x  e.  C  /\  x B A ) )
3 df-br 3765 . . . 4  |-  ( x B A  <->  <. x ,  A >.  e.  B
)
43anbi2i 430 . . 3  |-  ( ( x  e.  C  /\  x B A )  <->  ( x  e.  C  /\  <. x ,  A >.  e.  B
) )
54exbii 1496 . 2  |-  ( E. x ( x  e.  C  /\  x B A )  <->  E. x
( x  e.  C  /\  <. x ,  A >.  e.  B ) )
62, 5bitri 173 1  |-  ( A  e.  ( B " C )  <->  E. x
( x  e.  C  /\  <. x ,  A >.  e.  B ) )
Colors of variables: wff set class
Syntax hints:    /\ wa 97    <-> wb 98   E.wex 1381    e. wcel 1393   _Vcvv 2557   <.cop 3378   class class class wbr 3764   "cima 4348
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944
This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-br 3765  df-opab 3819  df-xp 4351  df-cnv 4353  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358
This theorem is referenced by:  cnvresima  4810  imaiun  5399
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