Preimages All Exist iff Surjection/Proof 2

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Theorem

Let $f: S \to T$ be a mapping.

Let $f^{-1}$ be the inverse of $f$.


Let $\map {f^{-1} } t$ be the preimage of $t \in T$.

Then $\map {f^{-1} } t$ is empty for no $t \in T$ if and only if $f$ is a surjection.


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.

$\blacksquare$