Preimages All Exist iff Surjection/Proof 1

Proof
Let $f$ be a surjection.


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

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

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

By Proof by Contradiction, it follows that:
 * $\neg \exists t \in T: f^{-1} \paren t = \O$

Sufficient Condition
Let $f$ be such that
 * $\neg \exists t \in T: f^{-1} \paren t = \O$


 * $f$ is a surjection then $f^{-1} \paren t$ can not be empty.

$f$ is not a surjection.

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

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

which contradicts the hypothesis.

By Proof by Contradiction, it follows that $f$ is a surjection.