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.