Infima Preserving Mapping on Filters Preserves Filtered Infima

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

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

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

Then $f$ preserves filtered infima.

Proof
Let $F$ be a filtered subset of $S$ such that:
 * $F$ admits an infimum in $\left({S, \preceq}\right)$.

By Filtered iff Upper Closure Filtered:
 * $F^\succeq$ is filtered

where $F^\succeq$ denotes the upper closure of $F$.

By Upper Closure is Upper Set:
 * $F^\succeq$ is upper.

Because filtered is non-empty, by definition:
 * $F^\succeq$ is filter in $\left({S, \preceq}\right)$.

By Infimum of Upper Closure of Set:
 * $F^\succeq$ admits an infimum in $\left({S, \preceq}\right)$

and
 * $\inf \left({F^\succeq}\right) = \inf F$

By assumption and mapping preserves the infimum on subset:
 * $f^\to \left({F^\succeq}\right)$ admits an infimum in $\left({T, \precsim}\right)$

and
 * $\inf \left({f^\to \left({F^\succeq}\right)}\right) = f \left({\inf\left({F^\succeq}\right)}\right)$

By Upper Closure is Closure Operator:
 * $F \subseteq F^\succeq$

By Image of Subset under Relation is Subset of Image/Corollary 2:
 * $f^\to \left({F}\right) \subseteq f^\to \left({F^\succeq}\right)$

By definition of infimum:
 * $f \left({\inf F}\right)$ is lower bound for $f^\to \left({F^\succeq}\right)$

By Lower Bound is Lower Bound for Subset:
 * $f \left({\inf F}\right)$ is a lower bound for $f^\to \left({F}\right)$

We will prove that:
 * for every element $x$ of $T$:
 * if $x$ is lower bound for $f^\to \left({F}\right)$, then $x \precsim f \left({\inf F}\right)$

Let $x \in T$ such that:
 * $x$ is a lower bound for $f^\to \left({F}\right)$

We will prove as a sublemma that:
 * $x$ is a lower bound for $f^\to \left({F^\succeq}\right)$

Let $y \in f^\to \left({F^\succeq}\right)$.

By definition of image of set under mapping:
 * $\exists a \in S: a \in F^\succeq \land y = f \left({a}\right)$

By definition of upper closure of set:
 * $\exists b \in F: b \preceq a$

By Infima Preserving Mapping on Filters is Increasing:
 * $f$ is increasing.

By definition of increasing:
 * $f \left({b}\right) \precsim f \left({a}\right)$

By definition of image of set under mapping:
 * $f \left({b}\right) \in f^\to \left({F}\right)$

By definition of lower bound of set:
 * $x \precsim f \left({b}\right)$

Thus by definition of transitivity:
 * $x \precsim y$

Thus by definition:
 * $x$ is a lower bound for $f^\to \left({F^\succeq}\right)$

This ends the proof of the sublemma.

Thus by definition of infimum:
 * $x \precsim f \left({\inf F}\right)$

Thus again by definition of infimum:
 * $f^\to \left({F}\right)$ admits an infimum in $\left({T, \precsim}\right)$

and
 * $\inf \left({f^\to \left({F}\right)}\right) = f\left({\inf F}\right)$

Thus result follows by definition of mapping preserves filtered infima.