Theorem brin 3832

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 3155	. 2 |- ( <. A , B >. e. ( R i^i S ) <-> ( <. A , B >. e. R /\ <. A , B >. e. S ) )
2		df-br 3786	. 2 |- ( A ( R i^i S ) B <-> <. A , B >. e. ( R i^i S ) )
3		df-br 3786	. . 3 |- ( A R B <-> <. A , B >. e. R )
4		df-br 3786	. . 3 |- ( A S B <-> <. A , B >. e. S )
5	3, 4	anbi12i 447	. 2 |- ( ( A R B /\ A S B ) <-> ( <. A , B >. e. R /\ <. A , B >. e. S ) )
6	1, 2, 5	3bitr4i 210	1 |- ( A ( R i^i S ) B <-> ( A R B /\ A S B ) )

Syntax hints:   /\ wa 102   <-> wb 103   e. wcel 1433   i^i cin 2972   <.cop 3401   class class class wbr 3785

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-v 2603  df-in 2979  df-br 3786

This theorem is referenced by:  brinxp2  4425  trin2  4736  poirr2  4737  cnvin  4751  tpostpos  5902  erinxp  6203