Theorem breldmg 4810

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 3986	. . . . 5 |- ( x = B -> ( A R x <-> A R B ) )
2	1	spcegv 2814	. . . 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 1005	. 2 |- ( ( A e. C /\ B e. D /\ A R B ) -> E. x A R x )
5		eldmg 4799	. . 3 |- ( A e. C -> ( A e. dom R <-> E. x A R x ) )
6	5	3ad2ant1 1008	. 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 968   E.wex 1480   e. wcel 2136   class class class wbr 3982   dom cdm 4604

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 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-v 2728  df-un 3120  df-sn 3582  df-pr 3583  df-op 3585  df-br 3983  df-dm 4614

This theorem is referenced by:  brelrng  4835  releldm  4839  brtposg  6222  shftfvalg  10760  shftfval  10763  geolim2  11453  geoisum1c  11461  ntrivcvgap  11489  eftlub  11631  eflegeo  11642  dvcj  13313  dvrecap  13317  dvef  13328  trilpolemisumle  13917  trilpolemeq1  13919