Theorem breldmg 4817

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 3993	. . . . 5 |- ( x = B -> ( A R x <-> A R B ) )
2	1	spcegv 2818	. . . 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 1010	. 2 |- ( ( A e. C /\ B e. D /\ A R B ) -> E. x A R x )
5		eldmg 4806	. . 3 |- ( A e. C -> ( A e. dom R <-> E. x A R x ) )
6	5	3ad2ant1 1013	. 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 973   E.wex 1485   e. wcel 2141   class class class wbr 3989   dom cdm 4611

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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732  df-un 3125  df-sn 3589  df-pr 3590  df-op 3592  df-br 3990  df-dm 4621

This theorem is referenced by:  brelrng  4842  releldm  4846  brtposg  6233  shftfvalg  10782  shftfval  10785  geolim2  11475  geoisum1c  11483  ntrivcvgap  11511  eftlub  11653  eflegeo  11664  dvcj  13467  dvrecap  13471  dvef  13482  trilpolemisumle  14070  trilpolemeq1  14072