Image of Subset under Relation is Subset of Image

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

Let $$A, B \subseteq S$$ such that $$A \subseteq B$$.

Then the image of $$A$$ is a subset of the image of $$B$$:

$$A \subseteq B \Longrightarrow \mathcal{R} \left({A}\right) \subseteq \mathcal{R} \left({B}\right)$$.

Proof
Suppose $$\mathcal{R} \left({A}\right) \not \subseteq \mathcal{R} \left({B}\right)$$.

$$ $$ $$ $$

... and the result follows by the Rule of Transposition.