Infima Preserving Mapping on Filters is Increasing

Theorem
Let $\left({S, \preceq}\right)$, $\left({T, \precsim}\right)$ be ordered sets.

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

Let every filter $F$ in $\left({S, \preceq}\right)$, $f$ preserve the infimum on $F$.

Then $f$ is increasing.