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Theorem elimag 4986
Description: Membership in an image. Theorem 34 of [Suppes] p. 65. (Contributed by NM, 20-Jan-2007.)
Assertion
Ref Expression
elimag  |-  ( A  e.  V  ->  ( A  e.  ( B " C )  <->  E. x  e.  C  x B A ) )
Distinct variable groups:    x, A    x, B    x, C
Allowed substitution hint:    V( x)

Proof of Theorem elimag
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 breq2 4019 . . 3  |-  ( y  =  A  ->  (
x B y  <->  x B A ) )
21rexbidv 2488 . 2  |-  ( y  =  A  ->  ( E. x  e.  C  x B y  <->  E. x  e.  C  x B A ) )
3 dfima2 4984 . 2  |-  ( B
" C )  =  { y  |  E. x  e.  C  x B y }
42, 3elab2g 2896 1  |-  ( A  e.  V  ->  ( A  e.  ( B " C )  <->  E. x  e.  C  x B A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 105    = wceq 1363    e. wcel 2158   E.wrex 2466   class class class wbr 4015   "cima 4641
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-14 2161  ax-ext 2169  ax-sep 4133  ax-pow 4186  ax-pr 4221
This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ral 2470  df-rex 2471  df-v 2751  df-un 3145  df-in 3147  df-ss 3154  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-br 4016  df-opab 4077  df-xp 4644  df-cnv 4646  df-dm 4648  df-rn 4649  df-res 4650  df-ima 4651
This theorem is referenced by:  elima  4987  elrelimasn  5006  fvelima  5580  ecexr  6554
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