Theorem brin 4085

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 3346	. 2 |- ( <. A , B >. e. ( R i^i S ) <-> ( <. A , B >. e. R /\ <. A , B >. e. S ) )
2		df-br 4034	. 2 |- ( A ( R i^i S ) B <-> <. A , B >. e. ( R i^i S ) )
3		df-br 4034	. . 3 |- ( A R B <-> <. A , B >. e. R )
4		df-br 4034	. . 3 |- ( A S B <-> <. A , B >. e. S )
5	3, 4	anbi12i 460	. 2 |- ( ( A R B /\ A S B ) <-> ( <. A , B >. e. R /\ <. A , B >. e. S ) )
6	1, 2, 5	3bitr4i 212	1 |- ( A ( R i^i S ) B <-> ( A R B /\ A S B ) )

Syntax hints:   /\ wa 104   <-> wb 105   e. wcel 2167   i^i cin 3156   <.cop 3625   class class class wbr 4033

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-v 2765  df-in 3163  df-br 4034

This theorem is referenced by:  brinxp2  4730  trin2  5061  poirr2  5062  cnvin  5077  tpostpos  6322  erinxp  6668  isunitd  13662