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Theorem eldmg 4890
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 4042 . . 3  |-  ( x  =  A  ->  (
x B y  <->  A B
y ) )
21exbidv 1616 . 2  |-  ( x  =  A  ->  ( E. y  x B
y  <->  E. y  A B y ) )
3 df-dm 4715 . 2  |-  dom  B  =  { x  |  E. y  x B y }
42, 3elab2g 2929 1  |-  ( A  e.  V  ->  ( A  e.  dom  B  <->  E. y  A B y ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176   E.wex 1531    = wceq 1632    e. wcel 1696   class class class wbr 4039   dom cdm 4705
This theorem is referenced by:  eldm2g  4891  eldm  4892  breldmg  4900  releldmb  4929  funeu  5294  fneu  5364  ndmfv  5568  erref  6696  ecdmn0  6718  rlimdm  12041  rlimdmo1  12107  iscmet3lem2  18734  dvcnp2  19285  ulmcau  19788  pserulm  19814  mulog2sum  20702  afveu  28121  rlimdmafv  28145
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-br 4040  df-dm 4715
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