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Theorem breldmg 4835
Description: Membership of first of a binary relation in a domain. (Contributed by NM, 21-Mar-2007.)
Assertion
Ref Expression
breldmg  |-  ( ( A  e.  C  /\  B  e.  D  /\  A R B )  ->  A  e.  dom  R )

Proof of Theorem breldmg
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 breq2 4009 . . . . 5  |-  ( x  =  B  ->  ( A R x  <->  A R B ) )
21spcegv 2827 . . . 4  |-  ( B  e.  D  ->  ( A R B  ->  E. x  A R x ) )
32imp 124 . . 3  |-  ( ( B  e.  D  /\  A R B )  ->  E. x  A R x )
433adant1 1015 . 2  |-  ( ( A  e.  C  /\  B  e.  D  /\  A R B )  ->  E. x  A R x )
5 eldmg 4824 . . 3  |-  ( A  e.  C  ->  ( A  e.  dom  R  <->  E. x  A R x ) )
653ad2ant1 1018 . 2  |-  ( ( A  e.  C  /\  B  e.  D  /\  A R B )  -> 
( A  e.  dom  R  <->  E. x  A R x ) )
74, 6mpbird 167 1  |-  ( ( A  e.  C  /\  B  e.  D  /\  A R B )  ->  A  e.  dom  R )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 105    /\ w3a 978   E.wex 1492    e. wcel 2148   class class class wbr 4005   dom cdm 4628
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2741  df-un 3135  df-sn 3600  df-pr 3601  df-op 3603  df-br 4006  df-dm 4638
This theorem is referenced by:  brelrng  4860  releldm  4864  brtposg  6257  shftfvalg  10829  shftfval  10832  geolim2  11522  geoisum1c  11530  ntrivcvgap  11558  eftlub  11700  eflegeo  11711  dvcj  14212  dvrecap  14216  dvef  14227  trilpolemisumle  14825  trilpolemeq1  14827
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