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Theorem breldm 5207
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 4448 . 2  |-  ( A R B  <->  <. A ,  B >.  e.  R )
2 opeldm.1 . . 3  |-  A  e. 
_V
3 opeldm.2 . . 3  |-  B  e. 
_V
42, 3opeldm 5206 . 2  |-  ( <. A ,  B >.  e.  R  ->  A  e.  dom  R )
51, 4sylbi 195 1  |-  ( A R B  ->  A  e.  dom  R )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    e. wcel 1767   _Vcvv 3113   <.cop 4033   class class class wbr 4447   dom cdm 4999
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-rab 2823  df-v 3115  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-br 4448  df-dm 5009
This theorem is referenced by:  funcnv3  5649  opabiota  5930  dffv2  5940  dff13  6154  exse2  6723  reldmtpos  6963  rntpos  6968  dftpos4  6974  tpostpos  6975  iserd  7337  dcomex  8827  axdc2lem  8828  axdclem2  8900  dmrecnq  9346  shftfval  12866  geolim2  13643  geomulcvg  13648  geoisum1c  13652  cvgrat  13655  eftlub  13705  eflegeo  13717  rpnnen2lem5  13813  imasleval  14796  psdmrn  15694  psssdm2  15702  ovoliunnul  21681  vitalilem5  21784  dvcj  22116  dvrec  22121  dvef  22144  ftc1cn  22207  aaliou3lem3  22502  ulmdv  22560  dvradcnv  22578  abelthlem7  22595  abelthlem9  22597  logtayllem  22796  leibpi  23029  log2tlbnd  23032  hhcms  25824  hhsscms  25899  occl  25926  gsummpt2co  27462  zetacvg  28225  ntrivcvg  28636  iprodgam  28730  wfrlem5  28952  frrlem5  28996  imageval  29185  ftc1cnnc  29694  geomcau  29883
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