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:
 * $\forall x \in X: \mathcal R \restriction_X \left({x}\right) = \mathcal R \left({x}\right)$

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

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

It follows that:
 * $\mathcal R \restriction_X \subseteq \mathcal R$