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Theorem eldm 5897
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 5895 . 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 205  wex 1774  wcel 2099  Vcvv 3470   class class class wbr 5142  dom cdm 5672
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2699
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2706  df-cleq 2720  df-clel 2806  df-rab 3429  df-v 3472  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-br 5143  df-dm 5682
This theorem is referenced by:  dmi  5918  dmep  5920  dmcoss  5968  dmcosseq  5970  dminss  6151  dmsnn0  6205  dffun7  6574  dffun8  6575  fnres  6676  opabiota  6975  fndmdif  7045  dff3  7104  frxp  8125  suppvalbr  8163  reldmtpos  8233  dmtpos  8237  aceq3lem  10137  axdc2lem  10465  axdclem2  10537  fpwwe2lem11  10658  nqerf  10947  shftdm  15044  bcthlem4  25248  dchrisumlem3  27417  eulerpath  30044  fundmpss  35356  elfix  35493  fnsingle  35509  fnimage  35519  funpartlem  35532  dfrecs2  35540  dfrdg4  35541  knoppcnlem9  35970  prtlem16  38335  undmrnresiss  43028
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