Originally Posted by
Ryaη How could I word this to fit the definition better?
Let $\displaystyle x \in A$. $\displaystyle \mbox{dom}R=A \Rightarrow \left( {\exists y \in A} \right)\left[ {\left( {x,y} \right) \in R} \right]$.
By symmetry we know $\displaystyle {\left( {y,x} \right) \in R}$.
Since $\displaystyle {\left( {x,y} \right) \in R}$ and $\displaystyle {\left( {y,x} \right) \in R}$, then by transitivity we know (xRy and yRx) $\displaystyle \Rightarrow$ xRx. (definition of being reflexive)
Therefore R is reflexive on A.