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Theorem elimag 4990
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
StepHypRef Expression
1 breq2 3987 . . 3  |-  ( y  =  A  ->  (
x B y  <->  x B A ) )
21rexbidv 2537 . 2  |-  ( y  =  A  ->  ( E. x  e.  C  x B y  <->  E. x  e.  C  x B A ) )
3 dfima2 4988 . 2  |-  ( B
" C )  =  { y  |  E. x  e.  C  x B y }
42, 3elab2g 2884 1  |-  ( A  e.  V  ->  ( A  e.  ( B " C )  <->  E. x  e.  C  x B A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 6    <-> wb 178    = wceq 1619    e. wcel 1621   E.wrex 2517   class class class wbr 3983   "cima 4650
This theorem is referenced by:  elima  4991  fvelima  5494
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 2237  ax-sep 4101  ax-nul 4109  ax-pr 4172
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 2121  df-mo 2122  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ne 2421  df-ral 2521  df-rex 2522  df-rab 2525  df-v 2759  df-dif 3116  df-un 3118  df-in 3120  df-ss 3127  df-nul 3417  df-if 3526  df-sn 3606  df-pr 3607  df-op 3609  df-br 3984  df-opab 4038  df-xp 4661  df-cnv 4663  df-dm 4665  df-rn 4666  df-res 4667  df-ima 4668
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