Image under Subset of Relation is Subset of Image under Relation

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

Let $\mathcal R_1 \subseteq S \times T$ be a relation in $S \times T$.

Let $\mathcal R_2 \subseteq \mathcal R_1$.

Let $A \subseteq S$.

Then:
 * $\mathcal R_2 \sqbrk A \subseteq \mathcal R_1 \sqbrk A$

where $\mathcal R_1 \sqbrk A$ denotes the image of $A$ under $\mathcal R_1$.