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Theorem eldmg 4827
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
StepHypRef Expression
1 breq1 3966 . . 3  |-  ( x  =  A  ->  (
x B y  <->  A B
y ) )
21exbidv 2006 . 2  |-  ( x  =  A  ->  ( E. y  x B
y  <->  E. y  A B y ) )
3 df-dm 4644 . 2  |-  dom  B  =  { x  |  E. y  x B y }
42, 3elab2g 2867 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 1537    = wceq 1619    e. wcel 1621   class class class wbr 3963   dom cdm 4626
This theorem is referenced by:  eldm2g  4828  eldm  4829  breldmg  4837  releldmb  4866  funeu  5182  fneu  5251  ndmfv  5451  erref  6613  ecdmn0  6635  rlimdm  11955  rlimdmo1  12021  iscmet3lem2  18645  dvcnp2  19196  ulmcau  19699  pserulm  19725  mulog2sum  20613
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-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2237
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-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-rab 2523  df-v 2742  df-dif 3097  df-un 3099  df-in 3101  df-ss 3108  df-nul 3398  df-if 3507  df-sn 3587  df-pr 3588  df-op 3590  df-br 3964  df-dm 4644
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