Theorem brin 4034

Description: The intersection of two relations. (Contributed by FL, 7-Oct-2008.)

Assertion

Ref	Expression
brin	|- ( A ( R i^i S ) B <-> ( A R B /\ A S B ) )

Proof of Theorem brin

Step	Hyp	Ref	Expression
1		elin 3305	. 2 |- ( <. A , B >. e. ( R i^i S ) <-> ( <. A , B >. e. R /\ <. A , B >. e. S ) )
2		df-br 3983	. 2 |- ( A ( R i^i S ) B <-> <. A , B >. e. ( R i^i S ) )
3		df-br 3983	. . 3 |- ( A R B <-> <. A , B >. e. R )
4		df-br 3983	. . 3 |- ( A S B <-> <. A , B >. e. S )
5	3, 4	anbi12i 456	. 2 |- ( ( A R B /\ A S B ) <-> ( <. A , B >. e. R /\ <. A , B >. e. S ) )
6	1, 2, 5	3bitr4i 211	1 |- ( A ( R i^i S ) B <-> ( A R B /\ A S B ) )

Syntax hints:   /\ wa 103   <-> wb 104   e. wcel 2136   i^i cin 3115   <.cop 3579   class class class wbr 3982

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 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-v 2728  df-in 3122  df-br 3983

This theorem is referenced by:  brinxp2  4671  trin2  4995  poirr2  4996  cnvin  5011  tpostpos  6232  erinxp  6575