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Theorem breldmg 5077
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 4218 . . . . 5  |-  ( x  =  B  ->  ( A R x  <->  A R B ) )
21spcegv 3039 . . . 4  |-  ( B  e.  D  ->  ( A R B  ->  E. x  A R x ) )
32imp 420 . . 3  |-  ( ( B  e.  D  /\  A R B )  ->  E. x  A R x )
433adant1 976 . 2  |-  ( ( A  e.  C  /\  B  e.  D  /\  A R B )  ->  E. x  A R x )
5 eldmg 5067 . . 3  |-  ( A  e.  C  ->  ( A  e.  dom  R  <->  E. x  A R x ) )
653ad2ant1 979 . 2  |-  ( ( A  e.  C  /\  B  e.  D  /\  A R B )  -> 
( A  e.  dom  R  <->  E. x  A R x ) )
74, 6mpbird 225 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 178    /\ w3a 937   E.wex 1551    e. wcel 1726   class class class wbr 4214   dom cdm 4880
This theorem is referenced by:  brelrng  5101  releldm  5104  sossfld  5319  brtpos  6490  tfrlem9a  6649  lmdvg  24340  tz6.12-afv  28015  rlimdmafv  28019
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-rab 2716  df-v 2960  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-br 4215  df-dm 4890
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