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Theorem eldm 4892
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 4890 . 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 1531    e. wcel 1696   _Vcvv 2801   class class class wbr 4039   dom cdm 4705
This theorem is referenced by:  dmi  4909  dmcoss  4960  dmcosseq  4962  dminss  5111  dmsnn0  5154  dffun7  5296  dffun8  5297  fnres  5376  fndmdif  5645  dff3  5689  frxp  6241  reldmtpos  6258  dmtpos  6262  opabiota  6309  aceq3lem  7763  axdc2lem  8090  axdclem2  8163  fpwwe2lem12  8279  nqerf  8570  shftdm  11582  xpsfrnel2  13483  bcthlem4  18765  dchrisumlem3  20656  eupath  23920  fundmpss  24193  elfix  24514  fnsingle  24529  fnimage  24539  funpartlem  24552  dfrdg4  24560  dmhmph  25636  prtlem16  26840
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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