We told you first ... Shareeef was leaning toward signing with the Bruins after decommitting from the University of Arizona following a recruiting scandal involving the head coach.

Well, Tuesday night, Reef -- one of the best high school hoops players in the country -- made it official ... and his mother Shaunie couldn't be happier (he'll be close to home!).

Then ... the family dance party broke out.

Shaq also posted a message about his son's decision -- "My baby @shareefoneal decided he’s gonna be a Bruin."

"Me and your mom @shaunieoneal5 are proud of you dude. This is the first of many steps to fulfill your dreams. #prouddad."