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Theorem elimag 5140
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 4150 . . 3  |-  ( y  =  A  ->  (
x B y  <->  x B A ) )
21rexbidv 2663 . 2  |-  ( y  =  A  ->  ( E. x  e.  C  x B y  <->  E. x  e.  C  x B A ) )
3 dfima2 5138 . 2  |-  ( B
" C )  =  { y  |  E. x  e.  C  x B y }
42, 3elab2g 3020 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 177    = wceq 1649    e. wcel 1717   E.wrex 2643   class class class wbr 4146   "cima 4814
This theorem is referenced by:  elima  5141  fvelima  5710  afvelima  27693
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-sep 4264  ax-nul 4272  ax-pr 4337
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-ral 2647  df-rex 2648  df-rab 2651  df-v 2894  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-nul 3565  df-if 3676  df-sn 3756  df-pr 3757  df-op 3759  df-br 4147  df-opab 4201  df-xp 4817  df-cnv 4819  df-dm 4821  df-rn 4822  df-res 4823  df-ima 4824
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