Theorem brinxp 4058

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 860	. . 3 |- ((A e. C /\ B e. D /\ ARB) <-> ((A e. C /\ B e. D) /\ ARB))
2	1	baibr 750	. 2 |- ((A e. C /\ B e. D) -> (ARB <-> (A e. C /\ B e. D /\ ARB)))
3		brinxp2 4057	. . 3 |- (B e. D -> (A(R i^i (C X. D))B <-> (A e. C /\ B e. D /\ ARB)))
4	3	adantl 424	. 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 590	1 |- ((A e. C /\ B e. D) -> (ARB <-> A(R i^i (C X. D))B))

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   /\ w3a 858   e. wcel 1300   i^i cin 2592   class class class wbr 3338   X. cxp 3984

This theorem is referenced by:  weinxp 4059  exfo 4795

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-br 3339  df-opab 3396  df-xp 4000

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