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Theorem breldm 4899
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 4040 . 2  |-  ( A R B  <->  <. A ,  B >.  e.  R )
2 opeldm.1 . . 3  |-  A  e. 
_V
3 opeldm.2 . . 3  |-  B  e. 
_V
42, 3opeldm 4898 . 2  |-  ( <. A ,  B >.  e.  R  ->  A  e.  dom  R )
51, 4sylbi 187 1  |-  ( A R B  ->  A  e.  dom  R )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1696   _Vcvv 2801   <.cop 3656   class class class wbr 4039   dom cdm 4705
This theorem is referenced by:  exse2  5063  funcnv3  5327  dffv2  5608  dff13  5799  reldmtpos  6258  rntpos  6263  dftpos4  6269  tpostpos  6270  opabiota  6309  iserd  6702  dcomex  8089  axdc2lem  8090  axdclem2  8163  dmrecnq  8608  shftfval  11581  geolim2  12343  geomulcvg  12348  geoisum1c  12352  cvgrat  12355  eftlub  12405  eflegeo  12417  rpnnen2lem5  12513  imasleval  13459  psdmrn  14332  psssdm2  14340  ovoliunnul  18882  vitalilem5  18983  dvcj  19315  dvrec  19320  dvef  19343  ftc1cn  19406  aaliou3lem3  19740  ulmdv  19796  dvradcnv  19813  abelthlem7  19830  abelthlem9  19832  logtayllem  20022  leibpi  20254  log2tlbnd  20257  hhcms  21798  hhsscms  21872  occl  21899  zetacvg  23704  wfrlem5  24331  frrlem5  24356  imageval  24540  ftc1cnnc  25025  domfldrefc  25160  domrngref  25163  domintrefb  25166  tolat  25389  cntrset  25705  geomcau  26578
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-br 4040  df-dm 4715
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