Coreflexive Relation is Subset of Diagonal Relation

Theorem
A coreflexive relation is a subset of the diagonal relation.

Proof
Let $$\mathcal{R} \subseteq S \times S$$ be a coreflexive relation.

Let $$\left({x, y}\right) \in \mathcal{R}$$.

By definition of coreflexive, it follows that $$x = y$$, and hence $$\left({x, y}\right) = \left({x, x}\right)$$.

So by definition of the diagonal relation, $$\left({x, y}\right) \in \Delta_S$$.

Hence the result.