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Theorem eldm2 5192
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 5190 . 2  |-  ( A  e.  _V  ->  ( A  e.  dom  B  <->  E. y <. A ,  y >.  e.  B ) )
31, 2ax-mp 5 1  |-  ( A  e.  dom  B  <->  E. y <. A ,  y >.  e.  B )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184   E.wex 1591    e. wcel 1762   _Vcvv 3106   <.cop 4026   dom cdm 4992
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-rab 2816  df-v 3108  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-sn 4021  df-pr 4023  df-op 4027  df-br 4441  df-dm 5002
This theorem is referenced by:  dmss  5193  opeldm  5197  dmin  5201  dmiun  5202  dmuni  5203  dm0  5207  reldm0  5211  dmrnssfld  5252  dmcoss  5253  dmcosseq  5255  dmres  5285  iss  5312  dmsnopg  5470  relssdmrn  5519  funssres  5619  dmfco  5932  fun11iun  6734  axdc3lem2  8820  gsum2d2  16786  cnlnssadj  26661  prsdm  27518  eldm3  28754  dfdm5  28769  wfrlem12  28917  frrlem11  28962  tfrqfree  29164
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