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Theorem breldmg 4900
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 4043 . . . . 5  |-  ( x  =  B  ->  ( A R x  <->  A R B ) )
21spcegv 2882 . . . 4  |-  ( B  e.  D  ->  ( A R B  ->  E. x  A R x ) )
32imp 418 . . 3  |-  ( ( B  e.  D  /\  A R B )  ->  E. x  A R x )
433adant1 973 . 2  |-  ( ( A  e.  C  /\  B  e.  D  /\  A R B )  ->  E. x  A R x )
5 eldmg 4890 . . 3  |-  ( A  e.  C  ->  ( A  e.  dom  R  <->  E. x  A R x ) )
653ad2ant1 976 . 2  |-  ( ( A  e.  C  /\  B  e.  D  /\  A R B )  -> 
( A  e.  dom  R  <->  E. x  A R x ) )
74, 6mpbird 223 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 176    /\ w3a 934   E.wex 1531    e. wcel 1696   class class class wbr 4039   dom cdm 4705
This theorem is referenced by:  brelrng  4924  releldm  4927  sossfld  5136  brtpos  6259  tfrlem9a  6418  lmdvg  23391  domintreflemb  25165  tz6.12-afv  28141  rlimdmafv  28145
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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