Minimally Closed Class under Progressing Mapping/Statement

Element of Closed Class under Progressing Mapping with no Proper Closed Subclass Lemmata: Statement of Conditions
Let $N$ be a class which is closed under a progressing mapping $g$.

Let $b$ be an element of $N$ such that no proper subclass $A$ of $N$ such that $b \in N$ is closed under $g$.