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Theorem eldmg 4619
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 3840 . . 3  |-  ( x  =  A  ->  (
x B y  <->  A B
y ) )
21exbidv 1753 . 2  |-  ( x  =  A  ->  ( E. y  x B
y  <->  E. y  A B y ) )
3 df-dm 4438 . 2  |-  dom  B  =  { x  |  E. y  x B y }
42, 3elab2g 2760 1  |-  ( A  e.  V  ->  ( A  e.  dom  B  <->  E. y  A B y ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 103    = wceq 1289   E.wex 1426    e. wcel 1438   class class class wbr 3837   dom cdm 4428
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-v 2621  df-un 3001  df-sn 3447  df-pr 3448  df-op 3450  df-br 3838  df-dm 4438
This theorem is referenced by:  eldm2g  4620  eldm  4621  breldmg  4630  releldmb  4660  funeu  5026  fneu  5104  ndmfvg  5319  erref  6292  ecdmn0m  6314  shftdm  10221
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