Theorem brinxp2 4727

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 4082	. 2 |- ( A ( R i^i ( C X. D ) ) B <-> ( A R B /\ A ( C X. D ) B ) )
2		ancom 266	. 2 |- ( ( A R B /\ A ( C X. D ) B ) <-> ( A ( C X. D ) B /\ A R B ) )
3		brxp 4691	. . . 4 |- ( A ( C X. D ) B <-> ( A e. C /\ B e. D ) )
4	3	anbi1i 458	. . 3 |- ( ( A ( C X. D ) B /\ A R B ) <-> ( ( A e. C /\ B e. D ) /\ A R B ) )
5		df-3an 982	. . 3 |- ( ( A e. C /\ B e. D /\ A R B ) <-> ( ( A e. C /\ B e. D ) /\ A R B ) )
6	4, 5	bitr4i 187	. 2 |- ( ( A ( C X. D ) B /\ A R B ) <-> ( A e. C /\ B e. D /\ A R B ) )
7	1, 2, 6	3bitri 206	1 |- ( A ( R i^i ( C X. D ) ) B <-> ( A e. C /\ B e. D /\ A R B ) )

Syntax hints:   /\ wa 104   <-> wb 105   /\ w3a 980   e. wcel 2164   i^i cin 3153   class class class wbr 4030   X. cxp 4658

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2167  ax-ext 2175  ax-sep 4148  ax-pow 4204  ax-pr 4239

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-v 2762  df-un 3158  df-in 3160  df-ss 3167  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-br 4031  df-opab 4092  df-xp 4666

This theorem is referenced by:  brinxp  4728  fncnv  5321  erinxp  6665  isstructim  12635  isstructr  12636