the class l[a] is the class of sets whose construction is influenced by a, where a may be a (presumably non-constructible) set or a proper class. the definition of this class uses def (x), which is the same as def (x) except instead of evaluating the truth of formulas φ in the model (x,∈), one uses the model (x,∈,a) where a is a unary predicate. the intended interpretation of a(y) is y∈a. then the definition of l[a] is exactly that of l only with def replaced by def.