Theorem brinxp2 4678

Description: Intersection of binary relation with Cartesian 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 4041	. 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 4642	. . . 4 |- ( A ( C X. D ) B <-> ( A e. C /\ B e. D ) )
4	3	anbi1i 455	. . 3 |- ( ( A ( C X. D ) B /\ A R B ) <-> ( ( A e. C /\ B e. D ) /\ A R B ) )
5		df-3an 975	. . 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 973   e. wcel 2141   i^i cin 3120   class class class wbr 3989   X. cxp 4609

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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-br 3990  df-opab 4051  df-xp 4617

This theorem is referenced by:  brinxp  4679  fncnv  5264  erinxp  6587  isstructim  12430  isstructr  12431