Theorem breldmg 4559

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 3789	. . . . 5 |- ( x = B -> ( A R x <-> A R B ) )
2	1	spcegv 2686	. . . 4 |- ( B e. D -> ( A R B -> E. x A R x ) )
3	2	imp 122	. . 3 |- ( ( B e. D /\ A R B ) -> E. x A R x )
4	3	3adant1 956	. 2 |- ( ( A e. C /\ B e. D /\ A R B ) -> E. x A R x )
5		eldmg 4548	. . 3 |- ( A e. C -> ( A e. dom R <-> E. x A R x ) )
6	5	3ad2ant1 959	. 2 |- ( ( A e. C /\ B e. D /\ A R B ) -> ( A e. dom R <-> E. x A R x ) )
7	4, 6	mpbird 165	1 |- ( ( A e. C /\ B e. D /\ A R B ) -> A e. dom R )

Syntax hints:   -> wi 4   <-> wb 103   /\ w3a 919   E.wex 1421   e. wcel 1433   class class class wbr 3785   dom cdm 4363

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-v 2603  df-un 2977  df-sn 3404  df-pr 3405  df-op 3407  df-br 3786  df-dm 4373

This theorem is referenced by:  brelrng  4583  releldm  4587  brtposg  5892  shftfvalg  9706  shftfval  9709