Transfinite Recursion Theorem/Formulation 3

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

Let $h$ and $f$ be mappings defined for all sets.

Let $c$ be a set.

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

where $K_{II}$ denotes the class of all limit ordinals.

Proof
For an arbitrary ordinal sequence $\theta$, let us define $\map g \theta$ by the following conditions:

By the Transfinite Recursion Theorem: Formulation 2, there exists a unique mapping $F$ on $\On$ such that:


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

Hence we have: