Yes. All the predestined are chosen by God to possess eternal life and this conscription of the predestined is called the book of life because the knowledge of God, by which He firmly remembers that He has predestined some to eternal life, is called the book of life.
The book of life is in God taken in a metaphorical sense, according to a comparison with human affairs: for it is usual among men that they who are chosen for any office should be inscribed in a book; as, for instance, soldiers, or counsellors, who formerly were called "conscript" fathers.
Predestination and the book of life are different aspects of the same thing: for this latter implies the knowledge of predestination.