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Theorem eldm2 5191
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 5189 . 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 1599    e. wcel 1804   _Vcvv 3095   <.cop 4020   dom cdm 4989
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-rab 2802  df-v 3097  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3771  df-if 3927  df-sn 4015  df-pr 4017  df-op 4021  df-br 4438  df-dm 4999
This theorem is referenced by:  dmss  5192  opeldm  5196  dmin  5200  dmiun  5201  dmuni  5202  dm0  5206  reldm0  5210  dmrnssfld  5251  dmcoss  5252  dmcosseq  5254  dmres  5284  iss  5311  dmsnopg  5469  relssdmrn  5518  funssres  5618  dmfco  5932  fun11iun  6745  axdc3lem2  8834  gsum2d2  16876  cnlnssadj  26871  prsdm  27769  eldm3  29166  dfdm5  29181  wfrlem12  29329  frrlem11  29374  tfrqfree  29576
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