Inverse of Surjection is Relation both Left-Total and Right-Total

Theorem

Let $f$ be an surjective mapping.

Then its inverse $f^{-1}$ is a relation which is both left-total and right-total.

Proof

We are given that $f$ is a surjective mapping.

By Inverse of Mapping is Right-Total Relation, $f^{-1}$ is a right-total relation.

By definition of surjection, $f$ is itself a right-total relation.

From Inverse of Right-Total Relation is Left-Total, $f^{-1}$ is a left-total relation.

Hence the result.

$\blacksquare$