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 unique mapping $F$ on $\On$ such that:


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

where $F \restriction \alpha$ denotes the restriction of $F$ to $\alpha$.

Proof
First we establish the following

Lemma
By Characteristic of Extending Operation, there exists a mapping $F$ on $\On$ such that:
 * $\forall \alpha \in \On: F \restriction \alpha^+ = \map E {F \restriction \alpha}$

where:
 * $F \restriction \alpha$ denotes the restriction of $F$ to $\alpha$
 * $\alpha^+$ denotes the successor ordinal of $\alpha$.

Then:

It remains to demonstrate uniqueness of $F$.