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 $t \in T$ iff $f$ is a surjection.

Proof
Follows immediately from the definition of surjection.


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

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.

So, by the Rule of Transposition, if $f$ is a surjection then $f^{-1} \left({t}\right)$ can not be empty.


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

Hence the result.