Restriction is Subset of Relation

Theorem

Let $\RR \subseteq S \times T$ be a relation.

Let $X \subseteq S$.

Then the restriction of $\RR$ to $X$ is a subset of $\RR$.

Proof

From the definition of restriction:

$\forall x \in X: \map {\RR \restriction_X} x = \map \RR x$

Thus:

$\forall x \in X: \exists t \in T: \tuple {x, t} \in \RR \restriction_X$

But $\tuple {x, t}$ is also (by definition) in $\RR$.

It follows that:

$\RR \restriction_X \subseteq \RR$

$\blacksquare$