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Theorem eldm 4633
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 4631 . 2  |-  ( A  e.  _V  ->  ( A  e.  dom  B  <->  E. y  A B y ) )
31, 2ax-mp 7 1  |-  ( A  e.  dom  B  <->  E. y  A B y )
Colors of variables: wff set class
Syntax hints:    <-> wb 103   E.wex 1426    e. wcel 1438   _Vcvv 2619   class class class wbr 3845   dom cdm 4438
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 3003  df-sn 3452  df-pr 3453  df-op 3455  df-br 3846  df-dm 4448
This theorem is referenced by:  dmi  4651  dmcoss  4702  dmcosseq  4704  dminss  4846  dmsnm  4896  dffun7  5042  dffun8  5043  fnres  5130  fndmdif  5404  reldmtpos  6018  dmtpos  6021  tfrexlem  6099
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