Inverse is Mapping implies Mapping is Surjection/Proof 1

Proof
Let $f^{-1}: T \to S$ be a mapping.

$f$ is not a surjection.

That is:
 * $\exists y \in T: \neg \exists x \in S: \tuple {x, y} \in f$

By definition of inverse of mapping:
 * $\exists y \in T: \neg \exists x \in S: \tuple {y, x} \in f^{-1}$

which would mean that $f^{-1}$ is not a mapping.

From this contradiction it follows that $f$ is a surjection.