We know that $r$ is a partial equivalence relation in set $$Dom(r) = \{x|\exists y.(x,y)\in r\}$$
The problem is to prove that this is an equivalence relation. Here is my proof. Did I do it right?
Let's take $x\in Dom(r)$. Then $x\space r \space y$ for some $y$. $r$ is a partial equivalence relation so we have $y\space r \space x$ from symmetry. So $y \in Dom(r)$. Moreover we have $x \space r \space x$ from transitivity. So for any $x$ in $Dom(r)$, x is also reflexive, so $r$ is an equivalence relation.
Bonus question: what is an example of a relation that is a partial equivalence relation that isn't an equivalence relation?
Take a set $S$ with a equivalence relation $r$. Then take $x \notin S$ and form the set $S' = S \cup \{x\}$. Now $r$ is partial equivalence relation on $S'$, but not an equivalence relation.