Well-Founded Recursion

Theorem
Let $A$ be a class.

Let $\mathcal R$ be a relation foundational on $A$.

Let every $\mathcal R$-initial segment of any element $x$ of $A$ be a small class.

Let $K$ be a class of mappings $f$ that satisfy:


 * The domain of $f$ is some subset $y \subseteq A$ such that $y$ is transitive with respect to $\mathcal R$.
 * $\forall x \in y: f\left({x}\right) = G\left({F\restriction \mathcal R^{-1} x}\right)$ where $F \restriction R^{-1} x$ denotes the restriction of $F$ to $\mathcal R$-initial segment of $x$.

Let $F = \bigcup K$, the union of $K$. Then:


 * $F$ is a function with domain $A$
 * $\forall x \in A: F\left({x}\right) = G\left({F\restriction \mathcal R^{-1} x}\right)$ where $R^{-1} x$
 * $F$ is unique. If another mapping $A$ has the above two properties, then $A = F$.