Preimages All Exist iff Surjection

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

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

Let $f^{-1} \left({t}\right)$ be the preimage of $t \in T$.

Then $f^{-1} \left({t}\right)$ is empty for no $t \in T$ $f$ is a surjection.

Necessary Condition
We use a Proof by Contraposition.

To that end, suppose:
 * $\exists t \in T: f^{-1} \left({t}\right) = \varnothing$

That is:
 * $\neg \left({\forall t \in T: \exists s \in S: f \left({s}\right) = t}\right)$

So, by definition, $f: S \to T$ is not a surjection.

From Rule of Transposition it follows that if $f$ is a surjection, $\neg \exists t \in T: f^{-1} \left({t}\right) = \varnothing$.

Sufficient Condition
We again use a Proof by Contraposition.

To that end, suppose $f$ is not a surjection.

Then by definition:
 * $\exists t \in T: \neg \left({\exists s \in S: f \left({s}\right) = t}\right)$

That is:
 * $\exists t \in T: f^{-1} \left({t}\right) = \varnothing$

From Rule of Transposition it follows that if $\neg \exists t \in T: f^{-1} \left({t}\right) = \varnothing$, then $f$ is a surjection.