Reflexive Relation is Quasi-Reflexive

Theorem

Let $\RR$ be a reflexive relation on a set $S$.

Then $\RR$ is a quasi-reflexive relation on $S$.

Proof

By definition of reflexive relation:

$\forall x \in S: \tuple {x, x} \in \RR$

Hence by definition of domain:

$x \in \Dom \RR$

and hence by definition of field and Set is Subset of Union:

$x \in \Field \RR$

That is:

$\forall x \in \Field \RR: \tuple {x, x} \in \RR$

Hence the result by definition of quasi-reflexive relation.

$\blacksquare$