Transfinite Recursion Theorem/Formulation 2

Theorem
Let $\On$ denote the class of all ordinals.

Let $S$ denote the class of all ordinal sequences.

Let $g$ be a mapping such that $S \subseteq \Dom g$.

Then there exists a mapping $F$ on $\On$ such that:


 * $\forall \alpha \in \On: \map F \alpha = \map g {f \restriction \alpha}$