Theorem breldmg 4791

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 3969	. . . . 5 |- ( x = B -> ( A R x <-> A R B ) )
2	1	spcegv 2800	. . . 4 |- ( B e. D -> ( A R B -> E. x A R x ) )
3	2	imp 123	. . 3 |- ( ( B e. D /\ A R B ) -> E. x A R x )
4	3	3adant1 1000	. 2 |- ( ( A e. C /\ B e. D /\ A R B ) -> E. x A R x )
5		eldmg 4780	. . 3 |- ( A e. C -> ( A e. dom R <-> E. x A R x ) )
6	5	3ad2ant1 1003	. 2 |- ( ( A e. C /\ B e. D /\ A R B ) -> ( A e. dom R <-> E. x A R x ) )
7	4, 6	mpbird 166	1 |- ( ( A e. C /\ B e. D /\ A R B ) -> A e. dom R )

Syntax hints:   -> wi 4   <-> wb 104   /\ w3a 963   E.wex 1472   e. wcel 2128   class class class wbr 3965   dom cdm 4585

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-v 2714  df-un 3106  df-sn 3566  df-pr 3567  df-op 3569  df-br 3966  df-dm 4595

This theorem is referenced by:  brelrng  4816  releldm  4820  brtposg  6198  shftfvalg  10711  shftfval  10714  geolim2  11402  geoisum1c  11410  ntrivcvgap  11438  eftlub  11580  eflegeo  11591  dvcj  13044  dvrecap  13048  dvef  13059  trilpolemisumle  13580  trilpolemeq1  13582