Transfinite Recursion Theorem/Formulation 1

Theorem
Let $M$ be a class.

Let $g$ be a strictly progressing mapping on $M$.

Let $M$ be minimally superinductive under $g$.

For an arbitrary ordinal $\alpha$, let $M_\alpha$ be the $\alpha$th element of $M$ under the well-ordered class $\struct {M, \subseteq}$.

Then:

where:
 * $\On$ denotes the class of all ordinals
 * $K_{II}$ denotes the class of all limit ordinals.