Relation is Reflexive and Coreflexive iff Diagonal

Theorem
Let $S$ be a set.

Let $\mathcal R \subseteq S \times S$ be a relation on $S$.

Then $\mathcal R$ is reflexive and coreflexive iff:
 * $\mathcal R = \Delta_S$

where $\Delta_S$ is the diagonal relation.

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

By definition of reflexive:
 * $\Delta_S \subseteq \mathcal R$

By definition of coreflexive:
 * $\mathcal R \subseteq \Delta_S$

By definition of set equality:
 * $\mathcal R = \Delta_S$

Necessary Condition
Let $\mathcal R = \Delta_S$

By definition of set equality:
 * $\Delta_S \subseteq \mathcal R$

and
 * $\mathcal R \subseteq \Delta_S$

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

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