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Theorem breldm 4539
Description: Membership of first of a binary relation in a domain. (Contributed by NM, 30-Jul-1995.)
Hypotheses
Ref Expression
opeldm.1 𝐴 ∈ V
opeldm.2 𝐵 ∈ V
Assertion
Ref Expression
breldm (𝐴𝑅𝐵𝐴 ∈ dom 𝑅)

Proof of Theorem breldm
StepHypRef Expression
1 df-br 3765 . 2 (𝐴𝑅𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ 𝑅)
2 opeldm.1 . . 3 𝐴 ∈ V
3 opeldm.2 . . 3 𝐵 ∈ V
42, 3opeldm 4538 . 2 (⟨𝐴, 𝐵⟩ ∈ 𝑅𝐴 ∈ dom 𝑅)
51, 4sylbi 114 1 (𝐴𝑅𝐵𝐴 ∈ dom 𝑅)
Colors of variables: wff set class
Syntax hints:  wi 4  wcel 1393  Vcvv 2557  cop 3378   class class class wbr 3764  dom cdm 4345
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022
This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-un 2922  df-sn 3381  df-pr 3382  df-op 3384  df-br 3765  df-dm 4355
This theorem is referenced by:  exse2  4699  funcnv3  4961  dff13  5407  reldmtpos  5868  rntpos  5872  dftpos4  5878  tpostpos  5879  iserd  6132
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