Image under Subset of Relation is Subset of Image under Relation

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

Let $\RR_1 \subseteq S \times T$ be a relation in $S \times T$.

Let $\RR_2 \subseteq \RR_1$.

Let $A \subseteq S$.

Then:
 * $\RR_2 \sqbrk A \subseteq \RR_1 \sqbrk A$

where $\RR_1 \sqbrk A$ denotes the image of $A$ under $\RR_1$.