Theorem brin 4183

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 3412	. 2 |- ( <. A , B >. e. ( R i^i S ) <-> ( <. A , B >. e. R /\ <. A , B >. e. S ) )
2		df-br 4131	. 2 |- ( A ( R i^i S ) B <-> <. A , B >. e. ( R i^i S ) )
3		df-br 4131	. . 3 |- ( A R B <-> <. A , B >. e. R )
4		df-br 4131	. . 3 |- ( A S B <-> <. A , B >. e. S )
5	3, 4	anbi12i 464	. 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 ) )

This proof depends on syntax axioms:   /\ wa 104   <-> wb 105   e. wcel 2209   i^i cin 3219   <.cop 3712   class class class wbr 4130

This proof depends on 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 proof depends on definitions:  df-bi 117  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-in 3226  df-br 4131

This theorem is used by:  brinxp2  4842  trin2  5179  poirr2  5180  cnvin  5195  tpostpos  6535  erinxp  6883  isunitd  14413