Theorem breldmg 4929

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.

Step	Hyp	Ref	Expression
1		breq2 4087	. . . . 5 |- ( x = B -> ( A R x <-> A R B ) )
2	1	spcegv 2891	. . . 4 |- ( B e. D -> ( A R B -> E. x A R x ) )
3	2	imp 124	. . 3 |- ( ( B e. D /\ A R B ) -> E. x A R x )
4	3	3adant1 1039	. 2 |- ( ( A e. C /\ B e. D /\ A R B ) -> E. x A R x )
5		eldmg 4918	. . 3 |- ( A e. C -> ( A e. dom R <-> E. x A R x ) )
6	5	3ad2ant1 1042	. 2 |- ( ( A e. C /\ B e. D /\ A R B ) -> ( A e. dom R <-> E. x A R x ) )
7	4, 6	mpbird 167	1 |- ( ( A e. C /\ B e. D /\ A R B ) -> A e. dom R )

Syntax hints:   -> wi 4   <-> wb 105   /\ w3a 1002   E.wex 1538   e. wcel 2200   class class class wbr 4083   dom cdm 4719

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-v 2801  df-un 3201  df-sn 3672  df-pr 3673  df-op 3675  df-br 4084  df-dm 4729

This theorem is referenced by:  brelrng  4955  releldm  4959  brtposg  6400  shftfvalg  11329  shftfval  11332  geolim2  12023  geoisum1c  12031  ntrivcvgap  12059  eftlub  12201  eflegeo  12212  dvcj  15383  dvrecap  15387  dvef  15401  trilpolemisumle  16406  trilpolemeq1  16408