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Theorem eldm2 4837
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 4835 . 2 (𝐴 ∈ V → (𝐴 ∈ dom 𝐵 ↔ ∃𝑦𝐴, 𝑦⟩ ∈ 𝐵))
31, 2ax-mp 5 1 (𝐴 ∈ dom 𝐵 ↔ ∃𝑦𝐴, 𝑦⟩ ∈ 𝐵)
Colors of variables: wff set class
Syntax hints:  wb 105  wex 1502  wcel 2158  Vcvv 2749  cop 3607  dom cdm 4638
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-ext 2169
This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-nf 1471  df-sb 1773  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-v 2751  df-un 3145  df-sn 3610  df-pr 3611  df-op 3613  df-br 4016  df-dm 4648
This theorem is referenced by:  dmss  4838  opeldm  4842  dmin  4847  dmiun  4848  dmuni  4849  dm0  4853  reldm0  4857  dmrnssfld  4902  dmcoss  4908  dmcosseq  4910  dmres  4940  iss  4965  dmxpss  5071  dmsnopg  5112  relssdmrn  5161  funssres  5270  fun11iun  5494  tfrlemibxssdm  6341  tfr1onlembxssdm  6357  tfrcllembxssdm  6370  fnpr2ob  12777
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