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Theorem eldmg 5028
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 4179 . . 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 4851 . 2  |-  dom  B  =  { x  |  E. y  x B y }
42, 3elab2g 3048 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 1721   class class class wbr 4176   dom cdm 4841
This theorem is referenced by:  eldm2g  5029  eldm  5030  breldmg  5038  releldmb  5067  funeu  5440  fneu  5512  ndmfv  5718  erref  6888  ecdmn0  6910  rlimdm  12304  rlimdmo1  12370  iscmet3lem2  19202  dvcnp2  19763  ulmcau  20268  pserulm  20295  mulog2sum  21188  afveu  27888  rlimdmafv  27912
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 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389
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 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-rab 2679  df-v 2922  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-nul 3593  df-if 3704  df-sn 3784  df-pr 3785  df-op 3787  df-br 4177  df-dm 4851
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