Preimages All Exist iff Surjection

Theorem
Let $f: S \to T$ be a mapping.

Let $f^{-1}$ be the inverse of $f$.

Let $\map {f^{-1} } t$ be the preimage of $t \in T$.

Then $\map {f^{-1} } t$ is empty for no $t \in T$ $f$ is a surjection.