How the Irish Saved What?

I don't intend in any way to disparage Mr. Cahill's book, which I haven't read, or the work of great men like St. Patrick, but I have heard it told that without the Irish monks and monasteries, all classical knowledge would have been lost.  I'm trying to square this with the existence of Byzantium and the schools and monasteries of Constantinople which existed at least from 425A.D. until 1453A.D..  Byzantines then brought ancient manuscripts into Europe to help begin the Renaissance.  Maybe I'm missing something, but could it be that the Byzantines saved civilization?

University of Constantinople 425-1453A.D.