Relation is Reflexive and Coreflexive iff Diagonal

Theorem
Let $S$ be a set.

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

Then $\RR$ is reflexive and coreflexive :
 * $\RR = \Delta_S$

where $\Delta_S$ is the diagonal relation.

Necessary Condition
Let $\RR \subseteq S \times S$ be reflexive and coreflexive.

Then:

Necessary Condition
Let $\RR = \Delta_S$

By definition of set equality:
 * $\Delta_S \subseteq \RR$

and
 * $\RR \subseteq \Delta_S$

From $\Delta_S \subseteq \RR$ it follows by definition that $\mathcal R$ is reflexive.

From $\RR \subseteq \Delta_S$ it follows by definition that $\mathcal R$ is coreflexive.