Theorem brinxp 3222

Description: Intersection of binary relation with cross product.

Assertion

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

Proof of Theorem brinxp

Step	Hyp	Ref	Expression
1		df-3an 775	. . 3 |- ((A e. C /\ B e. D /\ ARB) <-> ((A e. C /\ B e. D) /\ ARB))
2	1	baibr 684	. 2 |- ((A e. C /\ B e. D) -> (ARB <-> (A e. C /\ B e. D /\ ARB)))
3		brinxp2 3221	. . 3 |- (B e. D -> (A(R i^i (C X. D))B <-> (A e. C /\ B e. D /\ ARB)))
4	3	adantl 388	. 2 |- ((A e. C /\ B e. D) -> (A(R i^i (C X. D))B <-> (A e. C /\ B e. D /\ ARB)))
5	2, 4	bitr4d 529	1 |- ((A e. C /\ B e. D) -> (ARB <-> A(R i^i (C X. D))B))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   /\ w3a 773   e. wcel 955   i^i cin 2036   class class class wbr 2609   X. cxp 3158

This theorem is referenced by:  weinxp 3223  exfo 3807

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-10 963  ax-11 964  ax-12 965  ax-13 966  ax-14 967  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452  ax-sep 2693  ax-pow 2732  ax-pr 2769

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 775  df-ex 978  df-sb 1168  df-eu 1375  df-mo 1376  df-clab 1457  df-cleq 1462  df-clel 1465  df-ne 1579  df-v 1803  df-dif 2039  df-un 2040  df-in 2041  df-ss 2043  df-nul 2271  df-pw 2392  df-sn 2402  df-pr 2403  df-op 2406  df-br 2610  df-opab 2657  df-xp 3174

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