Preimages All Exist iff Surjection/Proof 1

Necessary Condition
We use a Proof by Contraposition.

To that end, suppose:
 * $\exists t \in T: \map {f^{-1} } t = \O$

That is:
 * $\neg \paren {\forall t \in T: \exists s \in S: \map f s = t}$

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

From Rule of Transposition it follows that if $f$ is a surjection: $\neg \exists t \in T: \map {f^{-1} } t = \O$

Sufficient Condition
We again use a Proof by Contraposition.

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

Then by definition:
 * $\exists t \in T: \neg \paren {\exists s \in S: \map f s = t}$

That is:
 * $\exists t \in T: \map {f^{-1} } t = \O$

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