Image of Element is Subset

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

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
From Image of Singleton under Relation:
 * $\mathcal R \left({s}\right) = \mathcal R \left({\left\{{s}\right\}}\right)$

From Singleton of Element is Subset:
 * $s \in A \implies \left\{{s}\right\} \subseteq A$

The result follows from Image of Subset is Subset of Image.