Theorem brinxp2 4606

Description: Intersection of binary relation with cross product. (Contributed by NM, 3-Mar-2007.) (Revised by Mario Carneiro, 26-Apr-2015.)

Assertion

Ref	Expression
brinxp2	|- ( A ( R i^i ( C X. D ) ) B <-> ( A e. C /\ B e. D /\ A R B ) )

Proof of Theorem brinxp2

Step	Hyp	Ref	Expression
1		brin 3980	. 2 |- ( A ( R i^i ( C X. D ) ) B <-> ( A R B /\ A ( C X. D ) B ) )
2		ancom 264	. 2 |- ( ( A R B /\ A ( C X. D ) B ) <-> ( A ( C X. D ) B /\ A R B ) )
3		brxp 4570	. . . 4 |- ( A ( C X. D ) B <-> ( A e. C /\ B e. D ) )
4	3	anbi1i 453	. . 3 |- ( ( A ( C X. D ) B /\ A R B ) <-> ( ( A e. C /\ B e. D ) /\ A R B ) )
5		df-3an 964	. . 3 |- ( ( A e. C /\ B e. D /\ A R B ) <-> ( ( A e. C /\ B e. D ) /\ A R B ) )
6	4, 5	bitr4i 186	. 2 |- ( ( A ( C X. D ) B /\ A R B ) <-> ( A e. C /\ B e. D /\ A R B ) )
7	1, 2, 6	3bitri 205	1 |- ( A ( R i^i ( C X. D ) ) B <-> ( A e. C /\ B e. D /\ A R B ) )

Syntax hints:   /\ wa 103   <-> wb 104   /\ w3a 962   e. wcel 1480   i^i cin 3070   class class class wbr 3929   X. cxp 4537

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-v 2688  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-br 3930  df-opab 3990  df-xp 4545

This theorem is referenced by:  brinxp  4607  fncnv  5189  erinxp  6503  isstructim  11973  isstructr  11974