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Theorem elima3 4892
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 4891 . 2  |-  ( A  e.  ( B " C )  <->  E. x
( x  e.  C  /\  x B A ) )
3 df-br 3934 . . . 4  |-  ( x B A  <->  <. x ,  A >.  e.  B
)
43anbi2i 453 . . 3  |-  ( ( x  e.  C  /\  x B A )  <->  ( x  e.  C  /\  <. x ,  A >.  e.  B
) )
54exbii 1585 . 2  |-  ( E. x ( x  e.  C  /\  x B A )  <->  E. x
( x  e.  C  /\  <. x ,  A >.  e.  B ) )
62, 5bitri 183 1  |-  ( A  e.  ( B " C )  <->  E. x
( x  e.  C  /\  <. x ,  A >.  e.  B ) )
Colors of variables: wff set class
Syntax hints:    /\ wa 103    <-> wb 104   E.wex 1469    e. wcel 1481   _Vcvv 2687   <.cop 3531   class class class wbr 3933   "cima 4546
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4050  ax-pow 4102  ax-pr 4135
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-v 2689  df-un 3076  df-in 3078  df-ss 3085  df-pw 3513  df-sn 3534  df-pr 3535  df-op 3537  df-br 3934  df-opab 3994  df-xp 4549  df-cnv 4551  df-dm 4553  df-rn 4554  df-res 4555  df-ima 4556
This theorem is referenced by:  cnvresima  5032  imaiun  5665
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