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Theorem eldmg 5146
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 4406 . . 3  |-  ( x  =  A  ->  (
x B y  <->  A B
y ) )
21exbidv 1681 . 2  |-  ( x  =  A  ->  ( E. y  x B
y  <->  E. y  A B y ) )
3 df-dm 4961 . 2  |-  dom  B  =  { x  |  E. y  x B y }
42, 3elab2g 3215 1  |-  ( A  e.  V  ->  ( A  e.  dom  B  <->  E. y  A B y ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    = wceq 1370   E.wex 1587    e. wcel 1758   class class class wbr 4403   dom cdm 4951
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-rab 2808  df-v 3080  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-sn 3989  df-pr 3991  df-op 3995  df-br 4404  df-dm 4961
This theorem is referenced by:  eldm2g  5147  eldm  5148  breldmg  5156  releldmb  5185  funeu  5553  fneu  5626  ndmfv  5826  suppvalbr  6807  erref  7234  ecdmn0  7256  rlimdm  13151  rlimdmo1  13217  iscmet3lem2  20945  dvcnp2  21537  ulmcau  22003  pserulm  22030  mulog2sum  22929  afveu  30230  rlimdmafv  30254
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