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:
 * $\Dom \RR := \set {s \in S: \exists t \in T: \tuple {s, t} \in \RR}$
 * $\Img {\RR^{-1} } := \set {s \in S: \exists T \in T: \tuple {t, s} \in \RR^{-1} }$

Also see

 * Image of Relation is Domain of Inverse Relation