Theorem breldmg 4982

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 4129	. . . . 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 4971	. . 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 )

Syntax hints:   -> wi 4   <-> wb 105   /\ w3a 1009   E.wex 1545   e. wcel 2209   class class class wbr 4125   dom cdm 4769

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 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 theorem 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 3711  df-pr 3712  df-op 3714  df-br 4126  df-dm 4779

This theorem is referenced by:  brelrng  5008  releldm  5012  brtposg  6515  shftfvalg  11561  shftfval  11564  geolim2  12257  geoisum1c  12265  ntrivcvgap  12293  eftlub  12435  eflegeo  12446  dvcj  15733  dvrecap  15737  dvef  15751  trilpolemisumle  16992  trilpolemeq1  16994