Inverse of One-to-One Relation is One-to-One

Theorem

The inverse of a one-to-one relation is a one-to-one relation.

Proof

Let $\RR$ be a one-to-one relation.

Let $\RR^{-1}$ denote its inverse

By definition, $\RR$ is a relation which is both many-to-one and one-to-many.

From Inverse of Many-to-One Relation is One-to-Many:

$\RR^{-1}$ is both one-to-many and many-to-one.

Hence the result by definition of one-to-one relation.

$\blacksquare$