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Theorem eldm2 4877
Description: Membership in a domain. Theorem 4 of [Suppes] p. 59. (Contributed by NM, 1-Aug-1994.)
Hypothesis
Ref Expression
eldm.1  |-  A  e. 
_V
Assertion
Ref Expression
eldm2  |-  ( A  e.  dom  B  <->  E. y <. A ,  y >.  e.  B )
Distinct variable groups:    y, A    y, B

Proof of Theorem eldm2
StepHypRef Expression
1 eldm.1 . 2  |-  A  e. 
_V
2 eldm2g 4875 . 2  |-  ( A  e.  _V  ->  ( A  e.  dom  B  <->  E. y <. A ,  y >.  e.  B ) )
31, 2ax-mp 8 1  |-  ( A  e.  dom  B  <->  E. y <. A ,  y >.  e.  B )
Colors of variables: wff set class
Syntax hints:    <-> wb 176   E.wex 1528    e. wcel 1684   _Vcvv 2788   <.cop 3643   dom cdm 4689
This theorem is referenced by:  dmss  4878  opeldm  4882  dmin  4886  dmiun  4887  dmuni  4888  dm0  4892  reldm0  4896  dmrnssfld  4938  dmcoss  4944  dmcosseq  4946  dmres  4976  iss  4998  dmsnopg  5144  relssdmrn  5193  funssres  5294  fun11iun  5493  dmfco  5593  axdc3lem2  8077  gsum2d2  15225  cnlnssadj  22660  eldm3  24119  dfdm5  24132  wfrlem12  24267  frrlem11  24293  tfrqfree  24489  dmrngcmp  25751
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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