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Theorem breldm 4567
Description: Membership of first of a binary relation in a domain. (Contributed by NM, 30-Jul-1995.)
Hypotheses
Ref Expression
opeldm.1  |-  A  e. 
_V
opeldm.2  |-  B  e. 
_V
Assertion
Ref Expression
breldm  |-  ( A R B  ->  A  e.  dom  R )

Proof of Theorem breldm
StepHypRef Expression
1 df-br 3794 . 2  |-  ( A R B  <->  <. A ,  B >.  e.  R )
2 opeldm.1 . . 3  |-  A  e. 
_V
3 opeldm.2 . . 3  |-  B  e. 
_V
42, 3opeldm 4566 . 2  |-  ( <. A ,  B >.  e.  R  ->  A  e.  dom  R )
51, 4sylbi 119 1  |-  ( A R B  ->  A  e.  dom  R )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1434   _Vcvv 2602   <.cop 3409   class class class wbr 3793   dom cdm 4371
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-v 2604  df-un 2978  df-sn 3412  df-pr 3413  df-op 3415  df-br 3794  df-dm 4381
This theorem is referenced by:  exse2  4729  funcnv3  4992  dff13  5439  reldmtpos  5902  rntpos  5906  dftpos4  5912  tpostpos  5913  iserd  6198
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