Image of Set Difference under Relation

Theorem
Let $$\mathcal{R} \subseteq S \times T$$ be a relation. Let $$A$$ and $$B$$ be subsets of $$S$$.

Then:
 * $$\mathcal{R} \left({A}\right) \setminus \mathcal{R} \left({B}\right) \subseteq \mathcal{R} \left({A \setminus B}\right)$$.

where $$\setminus$$ denotes set difference.

Proof
$$ $$ $$

Note
Note that equality does not hold in general.

See the note on Mapping Image of Set Difference for an example of a mapping (which is of course a relation) for which it does not.