Definition:Closed Set under Progressing Mapping

Definition
Let $x$ and $y$ be sets.

Let $g$ be a progressing mapping.

We say that
 * $y$ is closed under $g$ relative to $x$


 * $\forall z \in y \cap \powerset x: \map g z \in y$
 * $\forall z \in y \cap \powerset x: \map g z \in y$

That is:
 * $z \in y \land z \subseteq x \implies \map g z \in y$