Theorem breldmg 4833

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 4007	. . . . 5 |- ( x = B -> ( A R x <-> A R B ) )
2	1	spcegv 2825	. . . 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 1015	. 2 |- ( ( A e. C /\ B e. D /\ A R B ) -> E. x A R x )
5		eldmg 4822	. . 3 |- ( A e. C -> ( A e. dom R <-> E. x A R x ) )
6	5	3ad2ant1 1018	. 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 978   E.wex 1492   e. wcel 2148   class class class wbr 4003   dom cdm 4626

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2739  df-un 3133  df-sn 3598  df-pr 3599  df-op 3601  df-br 4004  df-dm 4636

This theorem is referenced by:  brelrng  4858  releldm  4862  brtposg  6254  shftfvalg  10826  shftfval  10829  geolim2  11519  geoisum1c  11527  ntrivcvgap  11555  eftlub  11697  eflegeo  11708  dvcj  14143  dvrecap  14147  dvef  14158  trilpolemisumle  14756  trilpolemeq1  14758