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Theorem eldm 5243
 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 5241 . 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 195  ∃wex 1695   ∈ wcel 1977  Vcvv 3173   class class class wbr 4583  dom cdm 5038 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-br 4584  df-dm 5048 This theorem is referenced by:  dmi  5261  dmcoss  5306  dmcosseq  5308  dminss  5466  dmsnn0  5518  dffun7  5830  dffun8  5831  fnres  5921  opabiota  6171  fndmdif  6229  dff3  6280  frxp  7174  suppvalbr  7186  reldmtpos  7247  dmtpos  7251  aceq3lem  8826  axdc2lem  9153  axdclem2  9225  fpwwe2lem12  9342  nqerf  9631  shftdm  13659  xpsfrnel2  16048  bcthlem4  22932  dchrisumlem3  24980  eupath  26508  fundmpss  30910  elfix  31180  fnsingle  31196  fnimage  31206  funpartlem  31219  dfrecs2  31227  dfrdg4  31228  knoppcnlem9  31661  prtlem16  33172  undmrnresiss  36929  eulerpath  41409
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