Inverse of Mapping is One-to-Many Relation

Theorem
Let $f$ be a mapping.

Then its inverse $f^{-1}$ is a one-to-many relation.

Hence $f^{-1}$ is not necessarily a mapping itself.

Proof
We have that $f$ is a mapping.

Hence $f$ is a many-to-one relation.

Then from Inverse of Many-to-One Relation is One-to-Many, $f^{-1}$ is one-to-many.