Equivalence of Definitions of Left Quasi-Reflexive Relation

Theorem
Let $\RR \subseteq S \times S$ be a relation in $S$.

$(1)$ implies $(2)$
Let $\RR$ be a left quasi-reflexive relation by definition $1$.

Then by definition:
 * $\forall x, y \in S: \tuple {x, y} \in \RR \implies \tuple {x, x} \in \RR$

Let $x \in \Dom \RR$ be arbitrary.

Then by definition of domain:
 * $\exists y \in S: \tuple {x, y} \in \RR$

Hence :
 * $\tuple {x, x} \in \RR$

As $y$ is arbitrary:
 * $\forall x \in \Dom \RR: \tuple {x, x} \in \RR$

Thus $\RR$ is a left quasi-reflexive relation by definition $2$.

$(2)$ implies $(1)$
Let $\RR$ be a left quasi-reflexive relation by definition $2$.

Then by definition:
 * $\forall x \in \Dom \RR: \tuple {x, x} \in \RR$

Let $x, y \in S$ be arbitrary such that $\tuple {x, y} \in \RR$.

Then by definition of image set:
 * $x \in \Dom \RR$

Hence :
 * $\tuple {x, x} \in \RR$

As $\tuple {x, y} \in \RR$ is arbitrary:
 * $\forall x, y \in S: \tuple {x, y} \in \RR \implies \tuple {x, x} \in \RR$

Thus $\RR$ is a left quasi-reflexive relation by definition $1$.