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Theorem elimag 5170
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 4180 . . 3  |-  ( y  =  A  ->  (
x B y  <->  x B A ) )
21rexbidv 2691 . 2  |-  ( y  =  A  ->  ( E. x  e.  C  x B y  <->  E. x  e.  C  x B A ) )
3 dfima2 5168 . 2  |-  ( B
" C )  =  { y  |  E. x  e.  C  x B y }
42, 3elab2g 3048 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 1721   E.wrex 2671   class class class wbr 4176   "cima 4844
This theorem is referenced by:  elima  5171  fvelima  5741  afvelima  27902
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 1662  ax-8 1683  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389  ax-sep 4294  ax-nul 4302  ax-pr 4367
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 2262  df-mo 2263  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ne 2573  df-ral 2675  df-rex 2676  df-rab 2679  df-v 2922  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-nul 3593  df-if 3704  df-sn 3784  df-pr 3785  df-op 3787  df-br 4177  df-opab 4231  df-xp 4847  df-cnv 4849  df-dm 4851  df-rn 4852  df-res 4853  df-ima 4854
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