Is $F ↾ a$ a set?

Question

If $F$ is a class function and $a$ is a set, is $F ↾ a$ a set?

Note: It appears in Transfinite induction

Answer 1

Yes. Use the Axiom Schema of Replacement. You may view it as $$F ↾ a =\{\,\langle x,Fx\rangle \mid x\in a\,\}$$

Answer 2

Yes, this is one of the axioms of set theory actually(assuming that by a class function is meant a definable function.).