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Theorem breldm 4703
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 3896 . 2  |-  ( A R B  <->  <. A ,  B >.  e.  R )
2 opeldm.1 . . 3  |-  A  e. 
_V
3 opeldm.2 . . 3  |-  B  e. 
_V
42, 3opeldm 4702 . 2  |-  ( <. A ,  B >.  e.  R  ->  A  e.  dom  R )
51, 4sylbi 120 1  |-  ( A R B  ->  A  e.  dom  R )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1463   _Vcvv 2657   <.cop 3496   class class class wbr 3895   dom cdm 4499
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2244  df-v 2659  df-un 3041  df-sn 3499  df-pr 3500  df-op 3502  df-br 3896  df-dm 4509
This theorem is referenced by:  exse2  4871  funcnv3  5143  dff13  5623  reldmtpos  6104  rntpos  6108  dftpos4  6114  tpostpos  6115  iserd  6409
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