Theorem breldmg 4943

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 4097	. . . . 5 |- ( x = B -> ( A R x <-> A R B ) )
2	1	spcegv 2895	. . . 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 1042	. 2 |- ( ( A e. C /\ B e. D /\ A R B ) -> E. x A R x )
5		eldmg 4932	. . 3 |- ( A e. C -> ( A e. dom R <-> E. x A R x ) )
6	5	3ad2ant1 1045	. 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 1005   E.wex 1541   e. wcel 2202   class class class wbr 4093   dom cdm 4731

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-v 2805  df-un 3205  df-sn 3679  df-pr 3680  df-op 3682  df-br 4094  df-dm 4741

This theorem is referenced by:  brelrng  4969  releldm  4973  brtposg  6463  shftfvalg  11458  shftfval  11461  geolim2  12153  geoisum1c  12161  ntrivcvgap  12189  eftlub  12331  eflegeo  12342  dvcj  15520  dvrecap  15524  dvef  15538  trilpolemisumle  16770  trilpolemeq1  16772