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Theorem eldm2 4649
 Description: Membership in a domain. Theorem 4 of [Suppes] p. 59. (Contributed by NM, 1-Aug-1994.)
Hypothesis
Ref Expression
eldm.1 𝐴 ∈ V
Assertion
Ref Expression
eldm2 (𝐴 ∈ dom 𝐵 ↔ ∃𝑦𝐴, 𝑦⟩ ∈ 𝐵)
Distinct variable groups:   𝑦,𝐴   𝑦,𝐵

Proof of Theorem eldm2
StepHypRef Expression
1 eldm.1 . 2 𝐴 ∈ V
2 eldm2g 4647 . 2 (𝐴 ∈ V → (𝐴 ∈ dom 𝐵 ↔ ∃𝑦𝐴, 𝑦⟩ ∈ 𝐵))
31, 2ax-mp 7 1 (𝐴 ∈ dom 𝐵 ↔ ∃𝑦𝐴, 𝑦⟩ ∈ 𝐵)
 Colors of variables: wff set class Syntax hints:   ↔ wb 104  ∃wex 1427   ∈ wcel 1439  Vcvv 2622  ⟨cop 3455  dom cdm 4454 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071 This theorem depends on definitions:  df-bi 116  df-3an 927  df-tru 1293  df-nf 1396  df-sb 1694  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-v 2624  df-un 3006  df-sn 3458  df-pr 3459  df-op 3461  df-br 3854  df-dm 4464 This theorem is referenced by:  dmss  4650  opeldm  4654  dmin  4659  dmiun  4660  dmuni  4661  dm0  4665  reldm0  4669  dmrnssfld  4711  dmcoss  4717  dmcosseq  4719  dmres  4749  iss  4773  dmxpss  4876  dmsnopg  4917  relssdmrn  4966  funssres  5071  fun11iun  5289  tfrlemibxssdm  6108  tfr1onlembxssdm  6124  tfrcllembxssdm  6137
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