Preimages All Exist iff Surjection/Proof 2

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

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

This is equivalent to the statement that $f^{-1}\paren t$ 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 $f \paren s = t$.

This is the definition of $f$ being surjective.

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