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Theorem breldm 4455
 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 3729 . 2 (A𝑅B ↔ ⟨A, B 𝑅)
2 opeldm.1 . . 3 A V
3 opeldm.2 . . 3 B V
42, 3opeldm 4454 . 2 (⟨A, B 𝑅A dom 𝑅)
51, 4sylbi 114 1 (A𝑅BA dom 𝑅)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∈ wcel 1367  Vcvv 2527  ⟨cop 3343   class class class wbr 3728  dom cdm 4261 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 614  ax-5 1310  ax-7 1311  ax-gen 1312  ax-ie1 1356  ax-ie2 1357  ax-8 1369  ax-10 1370  ax-11 1371  ax-i12 1372  ax-bnd 1373  ax-4 1374  ax-17 1393  ax-i9 1397  ax-ial 1401  ax-i5r 1402  ax-ext 1996 This theorem depends on definitions:  df-bi 110  df-3an 869  df-tru 1227  df-nf 1324  df-sb 1620  df-clab 2001  df-cleq 2007  df-clel 2010  df-nfc 2141  df-v 2529  df-un 2891  df-sn 3346  df-pr 3347  df-op 3349  df-br 3729  df-dm 4271 This theorem is referenced by:  exse2  4615  funcnv3  4876  dff13  5321  reldmtpos  5779  rntpos  5783  dftpos4  5789  tpostpos  5790  iserd  6032
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