Domain of Relation is Image of Inverse Relation

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

Let $\RR^{-1} \subseteq T \times S$ be the inverse of $\RR$.

Then:
 * $\Dom \RR = \Img {\RR^{-1} }$

That is, the domain of a relation is the image of its inverse.

Proof
By definition:

Also see

 * Image of Relation is Domain of Inverse Relation