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Theorem eldmg 4862
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 4000 . . 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 4679 . 2  |-  dom  B  =  { x  |  E. y  x B y }
42, 3elab2g 2891 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 3997   dom cdm 4661
This theorem is referenced by:  eldm2g  4863  eldm  4864  breldmg  4872  releldmb  4901  funeu  5217  fneu  5286  ndmfv  5486  erref  6648  ecdmn0  6670  rlimdm  11990  rlimdmo1  12056  iscmet3lem2  18680  dvcnp2  19231  ulmcau  19734  pserulm  19760  mulog2sum  20648
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 2239
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 2245  df-cleq 2251  df-clel 2254  df-nfc 2383  df-rab 2527  df-v 2765  df-dif 3130  df-un 3132  df-in 3134  df-ss 3141  df-nul 3431  df-if 3540  df-sn 3620  df-pr 3621  df-op 3623  df-br 3998  df-dm 4679
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