Preimages All Exist iff Surjection/Proof 2

Proof
Suppose that there is no $t \in T$ such that $\map {f^{-1} } t$ is empty.

By Denial of Existence, this is equivalent to saying that for all $t \in T$, $\map {f^{-1} } t$ is not empty.

This is equivalent to the statement that $\map {f^{-1} } t$ contains at least one element for each $t \in T$.

In other words, for each $t \in T$, there exists an $s\in S$ such that $\map f s = t$.

This is the definition of $f$ being surjective.

Thus if there is no $t \in T$ such that $\map {f^{-1} } t$ is empty, then $f$ is surjective.

Since this proof only uses statements of equivalence, it also shows that $f$ being surjective implies that there is no $t \in T$ such that $\map {f^{-1} } t$ is empty.