Theorem brin 3811

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 3126	. 2 |- ( <. A , B >. e. ( R i^i S ) <-> ( <. A , B >. e. R /\ <. A , B >. e. S ) )
2		df-br 3765	. 2 |- ( A ( R i^i S ) B <-> <. A , B >. e. ( R i^i S ) )
3		df-br 3765	. . 3 |- ( A R B <-> <. A , B >. e. R )
4		df-br 3765	. . 3 |- ( A S B <-> <. A , B >. e. S )
5	3, 4	anbi12i 433	. 2 |- ( ( A R B /\ A S B ) <-> ( <. A , B >. e. R /\ <. A , B >. e. S ) )
6	1, 2, 5	3bitr4i 201	1 |- ( A ( R i^i S ) B <-> ( A R B /\ A S B ) )

Syntax hints:   /\ wa 97   <-> wb 98   e. wcel 1393   i^i cin 2916   <.cop 3378   class class class wbr 3764

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-in 2924  df-br 3765

This theorem is referenced by:  brinxp2  4407  trin2  4716  poirr2  4717  cnvin  4731  tpostpos  5879  erinxp  6180