\begin{definition}[Definition:Minimally Inductive Class under General Mapping/Definition 1]
Let $A$ be a class.
Let $g: A \to A$ be a mapping on $A$.
$A$ is '''minimally inductive under $g$''' {{iff}}:
{{begin-axiom}}
{{axiom | n = 1
        | t = $A$ is inductive under $g$
}}
{{axiom | n = 2
        | t = No proper subclass of $A$ is inductive under $g$.
}}
{{end-axiom}}
\end{definition}