Definition:Many-to-One Relation/Defined/Subclass

Definition
Let $f \subseteq S \times T$ be a many-to-one relation.  Let $R \subseteq S$.

Then $f$ is defined on $R$ iff it is defined at all $r \in R$.

Equivalently, iff $R \subseteq \operatorname{dom} f$, the domain of $f$.