Equivalence of Definitions of Quasi-Reflexive Relation

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

$(1)$ implies $(3)$
Let $\RR$ be a quasi-reflexive relation by definition $1$.

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

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

and:
 * $\forall x, y \in S: \tuple {x, y} \in \RR \implies \tuple {y, y} \in \RR$

Hence:
 * $\RR$ is a left quasi-reflexive relation by definition

and:
 * $\RR$ is a right quasi-reflexive relation by definition.

Thus $\RR$ is a quasi-reflexive relation by definition $3$.

$(3)$ implies $(2)$
Let $\RR$ be a quasi-reflexive relation by definition $3$.

Then by definition:
 * $\RR$ is a left quasi-reflexive relation

and:
 * $\RR$ is a right quasi-reflexive relation.

Hence by definition of left quasi-reflexive relation:
 * $\forall x \in \Dom \RR: \tuple {x, x} \in \RR$

and by definition of right quasi-reflexive relation:
 * $\forall x \in \Img \RR: \tuple {x, x} \in \RR$

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

Then by definition of field of relation:
 * $x \in \Dom \RR \cup \Img \RR$

That is, either:
 * $x \in \Dom \RR$, in which case $\tuple {x, x} \in \RR$

or:
 * $x \in \Img \RR$, in which case $\tuple {x, x} \in \RR$.

In both cases $\tuple {x, x} \in \RR$.

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

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

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


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

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

By definition of domain of mapping:
 * $x \in \Dom \RR$

and so by definition of field of relation:
 * $x \in \Field \RR$

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

Simlarly, by definition of image set of mapping:
 * $y \in \Img \RR$

and so by definition of field of relation:
 * $y \in \Field \RR$

Thus :
 * $\tuple {y, y} \in \RR$

That is:
 * $\tuple {x, x} \in \RR \land \tuple {y, y} \in \RR$

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

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