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:
 * $$\mathcal{R} \restriction_X \subseteq \mathcal{R}$$