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Theorem eldm 4876
Description: Membership in a domain. Theorem 4 of [Suppes] p. 59. (Contributed by NM, 2-Apr-2004.)
Hypothesis
Ref Expression
eldm.1  |-  A  e. 
_V
Assertion
Ref Expression
eldm  |-  ( A  e.  dom  B  <->  E. y  A B y )
Distinct variable groups:    y, A    y, B

Proof of Theorem eldm
StepHypRef Expression
1 eldm.1 . 2  |-  A  e. 
_V
2 eldmg 4874 . 2  |-  ( A  e.  _V  ->  ( A  e.  dom  B  <->  E. y  A B y ) )
31, 2ax-mp 8 1  |-  ( A  e.  dom  B  <->  E. y  A B y )
Colors of variables: wff set class
Syntax hints:    <-> wb 176   E.wex 1528    e. wcel 1684   _Vcvv 2788   class class class wbr 4023   dom cdm 4689
This theorem is referenced by:  dmi  4893  dmcoss  4944  dmcosseq  4946  dminss  5095  dmsnn0  5138  dffun7  5280  dffun8  5281  fnres  5360  fndmdif  5629  dff3  5673  frxp  6225  reldmtpos  6242  dmtpos  6246  opabiota  6293  aceq3lem  7747  axdc2lem  8074  axdclem2  8147  fpwwe2lem12  8263  nqerf  8554  shftdm  11566  xpsfrnel2  13467  bcthlem4  18749  dchrisumlem3  20640  eupath  23905  fundmpss  24122  elfix  24443  fnsingle  24458  fnimage  24468  funpartfun  24481  funpartfv  24483  dfrdg4  24488  dmhmph  25533  prtlem16  26737
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-br 4024  df-dm 4699
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