Mapping Preserves Finite and Filtered Infima

Theorem
Let $\left({S_1, \preceq_1}\right)$, $\left({S_2, \preceq_2}\right)$ be meet semilattices.

Let $f: S_1 \to S_2$ be a mapping.

Let $f$ preserve finite infima and preserve filtered infima.

Then $f$ also preserves all infima

Proof
Assume that
 * $(1): \quad f$ preserves finite infima

and
 * $(2): \quad f$ preserves filtered infima.

Let $X$ be a subset of $S_1$.

Let $X$ admits an infimum in $\left({S_1, \preceq_1}\right)$

Define $Z := \left\{ {\inf A: A \in {\it Fin}\left({X}\right) \land A \ne \varnothing}\right\}$

where
 * $\inf A$ denotes the infimum of $A$ in $\left({S_1, \preceq_1}\right)$
 * ${\it Fin}\left({X}\right)$ denotes the set of all finite subsets of $X$

By Existence of Non-Empty Finite Infima in Meet Semilattice:
 * for every non-empty finite subset $A$ of $X$, $A$ admits an infimum in $\left({S_1, \preceq_1}\right)$

By Infimum of Infima:
 * $\inf Z = \inf X$