Image of Element is Subset

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

Let $$A \subseteq S$$.

Then $$s \in A \Longrightarrow \mathcal{R} \left({s}\right) \subseteq \mathcal{R} \left({A}\right)$$.

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

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

Then the result follows directly from Subset of Image.