Transfinite Recursion Theorem/Formulation 4

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

Let $g$ be a mapping defined for all sets.

Let $c$ be a set.

Then there exists a unique $\On$-sequence $S_0, S_1, \dots, S_\alpha, \dots$ such that:

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

Proof
This is a special case of the Transfinite Recursion Theorem: Formulation $3$.

Recall:

Set $\map f x = \bigcup x$, so that:
 * $\map f {F \sqbrk \lambda}$

is then:
 * $\ds \bigcup_{\alpha \mathop < \lambda} \map F \alpha$

Let $S_\alpha$ be the set $\map F \alpha$

and the given form of the Transfinite Recursion Theorem follows.

It remains to demonstrate uniqueness of $S_0, S_1, \dots, S_\alpha, \dots$.