Principle of Recursive Definition for Well-Ordered Sets

Theorem
Let $J$ be a well-ordered set.

Let $C$ be any set.

Let $\FF$ be the set of all functions that map initial segments $S_a$ of $J$ into $C$.

Then for any function of the form:


 * $\rho: \FF \to C$

there exists a unique function:


 * $h: J \to C$

satisfying:


 * $\forall \alpha \in J: \map h \alpha = \map \rho {h \restriction_{S_\alpha} }$

where $\restriction$ denotes the restriction of a mapping.

Also see

 * Principle of Recursive Definition