Restriction of Equivalence Relation is Equivalence

Theorem
Let $S$ be a sets.

Let $\mathcal R$ be an equivalence relation on $S$.

Let $T \subset S$ and let $\mathcal R \restriction_T$ be the restriction of $\mathcal R$ to $T$.

Then $\mathcal R \restriction_T$ is an equivalence relation on $T$.

Also see

 * Properties of Restriction of Relation‎ for other similar properties of the the restriction of a relation.