Image of Domain of Relation is Image Set

Theorem
Let $S$ and $T$ be sets.

Let $\mathcal R \subseteq S \times T$ be a relation.

The image of the domain of $\mathcal R$ is the image set of $\mathcal R$:


 * $\mathcal R \left [{\operatorname{Dom} \left({\mathcal R}\right)}\right] = \operatorname{Im} \left ({\mathcal R}\right)$

where $\operatorname{Im} \left ({\mathcal R}\right)$ is the image of $\mathcal R$.

Proof
Let $y \in \mathcal R \left [{\operatorname{Dom} \left({\mathcal R}\right)}\right]$.

Let $y \in \operatorname{Im} \left ({\mathcal R}\right)$.

The result follows by definition of set equality.