Theorem breldmg 4987

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 4134	. . . . 5 |- ( x = B -> ( A R x <-> A R B ) )
2	1	spcegv 2913	. . . 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 1046	. 2 |- ( ( A e. C /\ B e. D /\ A R B ) -> E. x A R x )
5		eldmg 4976	. . 3 |- ( A e. C -> ( A e. dom R <-> E. x A R x ) )
6	5	3ad2ant1 1049	. 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 )

This proof depends on syntax axioms:   -> wi 4   <-> wb 105   /\ w3a 1009   E.wex 1545   e. wcel 2209   class class class wbr 4130   dom cdm 4774

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-v 2823  df-un 3224  df-sn 3715  df-pr 3716  df-op 3718  df-br 4131  df-dm 4784

This theorem is used by:  brelrng  5013  releldm  5017  brtposg  6525  shftfvalg  11583  shftfval  11586  geolim2  12279  geoisum1c  12287  ntrivcvgap  12315  eftlub  12457  eflegeo  12468  dvcj  15810  dvrecap  15814  dvef  15828  log2tlbndlog2  16082  trilpolemisumle  17087  trilpolemeq1  17089