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 $\mathcal R$ be a one-to-one relation.

Let $\mathcal R^{-1}$ denote its inverse

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

From Inverse of Many-to-One Relation is One-to-Many:
 * $\mathcal R^{-1}$ is both one-to-many and many-to-one.

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