Theorem brinxp 4742

Description: Intersection of binary relation with Cartesian product. (Contributed by NM, 9-Mar-1997.)

Assertion

Ref	Expression
brinxp	|- ( ( A e. C /\ B e. D ) -> ( A R B <-> A ( R i^i ( C X. D ) ) B ) )

Proof of Theorem brinxp

Step	Hyp	Ref	Expression
1		brinxp2 4741	. . 3 |- ( A ( R i^i ( C X. D ) ) B <-> ( A e. C /\ B e. D /\ A R B ) )
2		df-3an 982	. . 3 |- ( ( A e. C /\ B e. D /\ A R B ) <-> ( ( A e. C /\ B e. D ) /\ A R B ) )
3	1, 2	bitri 184	. 2 |- ( A ( R i^i ( C X. D ) ) B <-> ( ( A e. C /\ B e. D ) /\ A R B ) )
4	3	baibr 921	1 |- ( ( A e. C /\ B e. D ) -> ( A R B <-> A ( R i^i ( C X. D ) ) B ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 980   e. wcel 2175   i^i cin 3164   class class class wbr 4043   X. cxp 4672

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 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-14 2178  ax-ext 2186  ax-sep 4161  ax-pow 4217  ax-pr 4252

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1375  df-nf 1483  df-sb 1785  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-ral 2488  df-rex 2489  df-v 2773  df-un 3169  df-in 3171  df-ss 3178  df-pw 3617  df-sn 3638  df-pr 3639  df-op 3641  df-br 4044  df-opab 4105  df-xp 4680

This theorem is referenced by:  poinxp  4743  soinxp  4744  seinxp  4745  isores2  5881  ltpiord  7431