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Theorem eldmg 5007
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 4158 . . 3  |-  ( x  =  A  ->  (
x B y  <->  A B
y ) )
21exbidv 1633 . 2  |-  ( x  =  A  ->  ( E. y  x B
y  <->  E. y  A B y ) )
3 df-dm 4830 . 2  |-  dom  B  =  { x  |  E. y  x B y }
42, 3elab2g 3029 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 1547    = wceq 1649    e. wcel 1717   class class class wbr 4155   dom cdm 4820
This theorem is referenced by:  eldm2g  5008  eldm  5009  breldmg  5017  releldmb  5046  funeu  5419  fneu  5491  ndmfv  5697  erref  6863  ecdmn0  6885  rlimdm  12274  rlimdmo1  12340  iscmet3lem2  19118  dvcnp2  19675  ulmcau  20180  pserulm  20207  mulog2sum  21100  afveu  27688  rlimdmafv  27712
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-rab 2660  df-v 2903  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-nul 3574  df-if 3685  df-sn 3765  df-pr 3766  df-op 3768  df-br 4156  df-dm 4830
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