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Theorem elima3 5196
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 5195 . 2  |-  ( A  e.  ( B " C )  <->  E. x
( x  e.  C  /\  x B A ) )
3 df-br 4200 . . . 4  |-  ( x B A  <->  <. x ,  A >.  e.  B
)
43anbi2i 676 . . 3  |-  ( ( x  e.  C  /\  x B A )  <->  ( x  e.  C  /\  <. x ,  A >.  e.  B
) )
54exbii 1592 . 2  |-  ( E. x ( x  e.  C  /\  x B A )  <->  E. x
( x  e.  C  /\  <. x ,  A >.  e.  B ) )
62, 5bitri 241 1  |-  ( A  e.  ( B " C )  <->  E. x
( x  e.  C  /\  <. x ,  A >.  e.  B ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 177    /\ wa 359   E.wex 1550    e. wcel 1725   _Vcvv 2943   <.cop 3804   class class class wbr 4199   "cima 4867
This theorem is referenced by:  cnvresima  5345  imaiun  5978
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2411  ax-sep 4317  ax-nul 4325  ax-pr 4390
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2417  df-cleq 2423  df-clel 2426  df-nfc 2555  df-ne 2595  df-ral 2697  df-rex 2698  df-rab 2701  df-v 2945  df-dif 3310  df-un 3312  df-in 3314  df-ss 3321  df-nul 3616  df-if 3727  df-sn 3807  df-pr 3808  df-op 3810  df-br 4200  df-opab 4254  df-xp 4870  df-cnv 4872  df-dm 4874  df-rn 4875  df-res 4876  df-ima 4877
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