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Theorem eldmg 4874
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
Dummy variable  x is distinct from all other variables.
Allowed substitution hint:    V( y)

Proof of Theorem eldmg
StepHypRef Expression
1 breq1 4028 . . 3  |-  ( x  =  A  ->  (
x B y  <->  A B
y ) )
21exbidv 1613 . 2  |-  ( x  =  A  ->  ( E. y  x B
y  <->  E. y  A B y ) )
3 df-dm 4699 . 2  |-  dom  B  =  { x  |  E. y  x B y }
42, 3elab2g 2918 1  |-  ( A  e.  V  ->  ( A  e.  dom  B  <->  E. y  A B y ) )
Colors of variables: wff set class
Syntax hints:    -> wi 6    <-> wb 178   E.wex 1529    = wceq 1624    e. wcel 1685   class class class wbr 4025   dom cdm 4689
This theorem is referenced by:  eldm2g  4875  eldm  4876  breldmg  4884  releldmb  4913  funeu  5245  fneu  5314  ndmfv  5514  erref  6676  ecdmn0  6698  rlimdm  12020  rlimdmo1  12086  iscmet3lem2  18713  dvcnp2  19264  ulmcau  19767  pserulm  19793  mulog2sum  20681
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1868  ax-ext 2266
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 938  df-tru 1312  df-ex 1530  df-nf 1533  df-sb 1632  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-rab 2554  df-v 2792  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-nul 3458  df-if 3568  df-sn 3648  df-pr 3649  df-op 3651  df-br 4026  df-dm 4699
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