Theorem breldmg 4753

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 3941	. . . . 5 |- ( x = B -> ( A R x <-> A R B ) )
2	1	spcegv 2777	. . . 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 4742	. . 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 1469   e. wcel 1481   class class class wbr 3937   dom cdm 4547

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 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-v 2691  df-un 3080  df-sn 3538  df-pr 3539  df-op 3541  df-br 3938  df-dm 4557

This theorem is referenced by:  brelrng  4778  releldm  4782  brtposg  6159  shftfvalg  10622  shftfval  10625  geolim2  11313  geoisum1c  11321  ntrivcvgap  11349  eftlub  11433  eflegeo  11444  dvcj  12881  dvrecap  12885  dvef  12896  trilpolemisumle  13406  trilpolemeq1  13408