Explain what is meant by a well-founded binary relation on a set.

Given a set $a$, we say that a mapping $f: a \rightarrow \mathcal{P} a$ is recursive if, given any set $b$ equipped with a mapping $g: \mathcal{P} b \rightarrow b$, there exists a unique $h: a \rightarrow b$ such that $h=g \circ h_{*} \circ f$, where $h_{*}: \mathcal{P} a \rightarrow \mathcal{P} b$ denotes the mapping $a^{\prime} \mapsto\left\{h(x) \mid x \in a^{\prime}\right\}$. Show that $f$ is recursive if and only if the relation $\{\langle x, y\rangle \mid x \in f(y)\}$ is well-founded.

[If you need to use any form of the recursion theorem, you should prove it.]