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Theorem breldmg 5214
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 4457 . . . . 5  |-  ( x  =  B  ->  ( A R x  <->  A R B ) )
21spcegv 3204 . . . 4  |-  ( B  e.  D  ->  ( A R B  ->  E. x  A R x ) )
32imp 429 . . 3  |-  ( ( B  e.  D  /\  A R B )  ->  E. x  A R x )
433adant1 1014 . 2  |-  ( ( A  e.  C  /\  B  e.  D  /\  A R B )  ->  E. x  A R x )
5 eldmg 5204 . . 3  |-  ( A  e.  C  ->  ( A  e.  dom  R  <->  E. x  A R x ) )
653ad2ant1 1017 . 2  |-  ( ( A  e.  C  /\  B  e.  D  /\  A R B )  -> 
( A  e.  dom  R  <->  E. x  A R x ) )
74, 6mpbird 232 1  |-  ( ( A  e.  C  /\  B  e.  D  /\  A R B )  ->  A  e.  dom  R )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ w3a 973   E.wex 1596    e. wcel 1767   class class class wbr 4453   dom cdm 5005
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 2826  df-v 3120  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-sn 4034  df-pr 4036  df-op 4040  df-br 4454  df-dm 5015
This theorem is referenced by:  brelrng  5238  releldm  5241  sossfld  5460  brtpos  6976  tfrlem9a  7067  perpln1  23935  lmdvg  27760  tz6.12-afv  32048  rlimdmafv  32052
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