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Theorem eldm 5756
Description: Membership in a domain. Theorem 4 of [Suppes] p. 59. (Contributed by NM, 2-Apr-2004.)
Hypothesis
Ref Expression
eldm.1 𝐴 ∈ V
Assertion
Ref Expression
eldm (𝐴 ∈ dom 𝐵 ↔ ∃𝑦 𝐴𝐵𝑦)
Distinct variable groups:   𝑦,𝐴   𝑦,𝐵

Proof of Theorem eldm
StepHypRef Expression
1 eldm.1 . 2 𝐴 ∈ V
2 eldmg 5754 . 2 (𝐴 ∈ V → (𝐴 ∈ dom 𝐵 ↔ ∃𝑦 𝐴𝐵𝑦))
31, 2ax-mp 5 1 (𝐴 ∈ dom 𝐵 ↔ ∃𝑦 𝐴𝐵𝑦)
Colors of variables: wff setvar class
Syntax hints:  wb 209  wex 1781  wcel 2115  Vcvv 3480   class class class wbr 5052  dom cdm 5542
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-v 3482  df-un 3924  df-sn 4551  df-pr 4553  df-op 4557  df-br 5053  df-dm 5552
This theorem is referenced by:  dmi  5778  dmep  5780  dmcoss  5829  dmcosseq  5831  dminss  5997  dmsnn0  6051  dffun7  6370  dffun8  6371  fnres  6463  opabiota  6737  fndmdif  6803  dff3  6857  frxp  7816  suppvalbr  7830  reldmtpos  7896  dmtpos  7900  aceq3lem  9544  axdc2lem  9868  axdclem2  9940  fpwwe2lem12  10061  nqerf  10350  shftdm  14430  bcthlem4  23934  dchrisumlem3  26078  eulerpath  28029  fundmpss  33066  elfix  33421  fnsingle  33437  fnimage  33447  funpartlem  33460  dfrecs2  33468  dfrdg4  33469  knoppcnlem9  33897  prtlem16  36110  undmrnresiss  40220
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