Inductive Set under Progressing Mapping has Minimally Inductive Subset

Theorem
Let $A$ be an inductive class under a mapping $g$.

Let $A$ be a set.

Then there exists some subset $S$ of $A$ such that $S$ is minimally inductive under $g$.

Proof
Let $S$ be the set defined as:
 * $S = \set {s: \exists t \in A: t = \map g s} \cup \set \O$

By definition, we have that:
 * $\O \in S$
 * $s \in S \implies \map g s \in S$

So, by definition, $S$ is inductive under $g$.

Because $A$ is inductive under $g$:
 * $\O \in A$

and so by definition of subset:
 * $S \subseteq A$

It remains to be shown that $S$ is minimally inductive under $g$.