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Theorem eldmg 5204
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 4456 . . 3  |-  ( x  =  A  ->  (
x B y  <->  A B
y ) )
21exbidv 1690 . 2  |-  ( x  =  A  ->  ( E. y  x B
y  <->  E. y  A B y ) )
3 df-dm 5015 . 2  |-  dom  B  =  { x  |  E. y  x B y }
42, 3elab2g 3257 1  |-  ( A  e.  V  ->  ( A  e.  dom  B  <->  E. y  A B y ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    = wceq 1379   E.wex 1596    e. wcel 1767   class class class wbr 4453   dom cdm 5005
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-rab 2826  df-v 3120  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-sn 4034  df-pr 4036  df-op 4040  df-br 4454  df-dm 5015
This theorem is referenced by:  eldm2g  5205  eldm  5206  breldmg  5214  releldmb  5243  funeu  5618  fneu  5691  ndmfv  5896  suppvalbr  6917  erref  7343  ecdmn0  7366  rlimdm  13354  rlimdmo1  13420  iscmet3lem2  21599  dvcnp2  22191  ulmcau  22657  pserulm  22684  mulog2sum  23588  fperdvper  31571  afveu  32028  rlimdmafv  32052
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