Restriction is Subset of Relation

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

Then the restriction of $$\mathcal{R}$$ to $$X$$ is a subset of $$\mathcal{R}$$.

Proof
From the definition of restriction, $$\forall x \in X: \mathcal{R}|_X \left({x}\right) = \mathcal{R} \left({x}\right)$$.

Thus, $$\forall x \in X: \exists t \in T: \left({x, t}\right) \in \mathcal{R}|_X$$.

But $$\left({x, t}\right)$$ is also (by definition) in $$\mathcal{R}$$.

It follows that $$\mathcal{R}|_X \subseteq \mathcal{R}$$.