Image of Element is Subset

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

Let $A \subseteq S$.

Then:
 * $s \in A \implies \mathcal R \left({s}\right) \subseteq \mathcal R \left({A}\right)$

Proof
First note that $\mathcal R \left({s}\right) = \mathcal R \left({\left\{{s}\right\}}\right)$ from Image of Singleton.

Also note that $s \in A \implies \left\{{s}\right\} \subseteq A$ from Singleton Subset.

Then the result follows directly from Image of Subset is Subset of Image.