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Theorem breldmg 4884
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 4027 . . . . 5  |-  ( x  =  B  ->  ( A R x  <->  A R B ) )
21spcegv 2869 . . . 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 4874 . . 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 1528    e. wcel 1684   class class class wbr 4023   dom cdm 4689
This theorem is referenced by:  brelrng  4908  releldm  4911  sossfld  5120  brtpos  6243  tfrlem9a  6402  lmdvg  23376  domintreflemb  25062  tz6.12-afv  28035
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-br 4024  df-dm 4699
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