Theorem brin 4693

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

Assertion

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

Proof of Theorem brin

Step	Hyp	Ref	Expression
1		elin 3219	. 2 |- (<.A, B>. e. (R i^i S) <-> (<.A, B>. e. R /\ <.A, B>. e. S))
2		df-br 4640	. 2 |- (A(R i^i S)B <-> <.A, B>. e. (R i^i S))
3		df-br 4640	. . 3 |- (ARB <-> <.A, B>. e. R)
4		df-br 4640	. . 3 |- (ASB <-> <.A, B>. e. S)
5	3, 4	anbi12i 678	. 2 |- ((ARB /\ ASB) <-> (<.A, B>. e. R /\ <.A, B>. e. S))
6	1, 2, 5	3bitr4i 268	1 |- (A(R i^i S)B <-> (ARB /\ ASB))

Syntax hints:   <-> wb 176   /\ wa 358   e. wcel 1710   i^i cin 3208  <.cop 4561   class class class wbr 4639

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-br 4640

This theorem is referenced by:  brinxp2  4835  brres  4949  intasym  5028  fncnv  5158  dfid4  5503  trtxp  5781  brtxp  5783  elfix  5787  ersymtr  5932  porta  5933  sopc  5934  weds  5938  enpw1lem1  6061  enmap2lem1  6063  enmap1lem1  6069  nchoicelem8  6296  nchoicelem19  6307