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Theorem eldm2 5900
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 5898 . 2 (𝐴 ∈ V → (𝐴 ∈ dom 𝐵 ↔ ∃𝑦𝐴, 𝑦⟩ ∈ 𝐵))
31, 2ax-mp 5 1 (𝐴 ∈ dom 𝐵 ↔ ∃𝑦𝐴, 𝑦⟩ ∈ 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wb 205  wex 1774  wcel 2099  Vcvv 3462  cop 4629  dom cdm 5674
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 2697
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2704  df-cleq 2718  df-clel 2803  df-rab 3420  df-v 3464  df-dif 3949  df-un 3951  df-ss 3963  df-nul 4323  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-br 5146  df-dm 5684
This theorem is referenced by:  dmss  5901  opeldm  5906  dmin  5910  dmiun  5912  dmuni  5913  dm0  5919  reldm0  5926  dmrnssfld  5969  dmcoss  5970  dmcosseq  5972  dmres  6013  iss  6036  dmsnopg  6216  relssdmrnOLD  6272  funssres  6595  dmfco  6990  fiun  7948  f1iun  7949  frrlem8  8300  frrlem10  8302  wfrlem12OLD  8342  axdc3lem2  10485  fnpr2ob  17568  gsum2d2  19968  cnlnssadj  32010  prsdm  33742  eldm3  35596  dfdm5  35609  iss2  38055
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