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Theorem eldmg 5058
Description: Domain membership. Theorem 4 of [Suppes] p. 59. (Contributed by Mario Carneiro, 9-Jul-2014.)
Assertion
Ref Expression
eldmg  |-  ( A  e.  V  ->  ( A  e.  dom  B  <->  E. y  A B y ) )
Distinct variable groups:    y, A    y, B
Allowed substitution hint:    V( y)

Proof of Theorem eldmg
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 breq1 4208 . . 3  |-  ( x  =  A  ->  (
x B y  <->  A B
y ) )
21exbidv 1636 . 2  |-  ( x  =  A  ->  ( E. y  x B
y  <->  E. y  A B y ) )
3 df-dm 4881 . 2  |-  dom  B  =  { x  |  E. y  x B y }
42, 3elab2g 3077 1  |-  ( A  e.  V  ->  ( A  e.  dom  B  <->  E. y  A B y ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177   E.wex 1550    = wceq 1652    e. wcel 1725   class class class wbr 4205   dom cdm 4871
This theorem is referenced by:  eldm2g  5059  eldm  5060  breldmg  5068  releldmb  5097  funeu  5470  fneu  5542  ndmfv  5748  erref  6918  ecdmn0  6940  rlimdm  12338  rlimdmo1  12404  iscmet3lem2  19238  dvcnp2  19799  ulmcau  20304  pserulm  20331  mulog2sum  21224  afveu  27985  rlimdmafv  28009
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-rab 2707  df-v 2951  df-dif 3316  df-un 3318  df-in 3320  df-ss 3327  df-nul 3622  df-if 3733  df-sn 3813  df-pr 3814  df-op 3816  df-br 4206  df-dm 4881
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