Theorem brinxp 3227

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 776	. . 3 |- ((A e. C /\ B e. D /\ ARB) <-> ((A e. C /\ B e. D) /\ ARB))
2	1	baibr 685	. 2 |- ((A e. C /\ B e. D) -> (ARB <-> (A e. C /\ B e. D /\ ARB)))
3		brinxp2 3226	. . 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 530	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 774   e. wcel 956   i^i cin 2042   class class class wbr 2614   X. cxp 3163

This theorem is referenced by:  weinxp 3228  exfo 3813

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-10 964  ax-11 965  ax-12 966  ax-13 967  ax-14 968  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457  ax-sep 2698  ax-pow 2737  ax-pr 2774

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 776  df-ex 979  df-sb 1170  df-eu 1380  df-mo 1381  df-clab 1462  df-cleq 1467  df-clel 1470  df-ne 1584  df-v 1808  df-dif 2045  df-un 2046  df-in 2047  df-ss 2049  df-nul 2277  df-pw 2398  df-sn 2408  df-pr 2409  df-op 2412  df-br 2615  df-opab 2662  df-xp 3179

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