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Theorem elimag 4855
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 3903 . . 3  |-  ( y  =  A  ->  (
x B y  <->  x B A ) )
21rexbidv 2415 . 2  |-  ( y  =  A  ->  ( E. x  e.  C  x B y  <->  E. x  e.  C  x B A ) )
3 dfima2 4853 . 2  |-  ( B
" C )  =  { y  |  E. x  e.  C  x B y }
42, 3elab2g 2804 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 104    = wceq 1316    e. wcel 1465   E.wrex 2394   class class class wbr 3899   "cima 4512
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 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-v 2662  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-br 3900  df-opab 3960  df-xp 4515  df-cnv 4517  df-dm 4519  df-rn 4520  df-res 4521  df-ima 4522
This theorem is referenced by:  elima  4856  fvelima  5441  ecexr  6402
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