G-Tower is Closed under Mapping

Theorem

Let $M$ be a class.

Let $g: M \to M$ be a progressing mapping on $M$.

Let $M$ be a $g$-tower.

Then $M$ is closed under $g$:

$\forall x: \paren {x \in M \implies \map g x \in M}$

Proof

By definition:

a $g$-tower is a class which is minimally superinductive under $g$

a class which is minimally superinductive under $g$ is superinductive under $g$

a superinductive class is inductive under $g$

an inductive class under $g$ is closed under $g$.

Hence the result.

$\blacksquare$