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Theorem elima3 5007
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 5006 . 2  |-  ( A  e.  ( B " C )  <->  E. x
( x  e.  C  /\  x B A ) )
3 df-br 3998 . . . 4  |-  ( x B A  <->  <. x ,  A >.  e.  B
)
43anbi2i 678 . . 3  |-  ( ( x  e.  C  /\  x B A )  <->  ( x  e.  C  /\  <. x ,  A >.  e.  B
) )
54exbii 1580 . 2  |-  ( E. x ( x  e.  C  /\  x B A )  <->  E. x
( x  e.  C  /\  <. x ,  A >.  e.  B ) )
62, 5bitri 242 1  |-  ( A  e.  ( B " C )  <->  E. x
( x  e.  C  /\  <. x ,  A >.  e.  B ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 178    /\ wa 360   E.wex 1537    e. wcel 1621   _Vcvv 2763   <.cop 3617   class class class wbr 3997   "cima 4664
This theorem is referenced by:  cnvresima  5149  imaiun  5705  cnvresimaOLD  25593
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2239  ax-sep 4115  ax-nul 4123  ax-pr 4186
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2122  df-mo 2123  df-clab 2245  df-cleq 2251  df-clel 2254  df-nfc 2383  df-ne 2423  df-ral 2523  df-rex 2524  df-rab 2527  df-v 2765  df-dif 3130  df-un 3132  df-in 3134  df-ss 3141  df-nul 3431  df-if 3540  df-sn 3620  df-pr 3621  df-op 3623  df-br 3998  df-opab 4052  df-xp 4675  df-cnv 4677  df-dm 4679  df-rn 4680  df-res 4681  df-ima 4682
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