Extension of Infima Preserving Mapping to Complete Lattice Preserves Infima

Theorem
Let $L_1 = \left({S_1, \preceq_1}\right)$, $L_2 = \left({S_2, \preceq_2}\right)$ be ordered sets.

Let $L_3 = \left({S_3, \preceq_3}\right)$ be a complete lattice.

Suppose that
 * $L_2$ is infima inheriting ordered subset of $L_3$.

Let $f:S_1 \to S_2$ be a mapping such that
 * $f$ preserves infima.

Then $f:S_1 \to S_3$ preserves infima.

Proof
By definition of ordered subset:
 * $S_2 \subseteq S_3$

Then define $g = f:S_1 \to S_3$

Let $X$ be a subset of $S_1$ such that
 * $X$ admits a infimum in $L_1$.

Thus by definition of complete lattice:
 * $g\left[{X}\right]$ admits a infimum in $L_3$.

By definition of mapping preserves infima:
 * $f\left[{X}\right]$ admits a infimum in $L_2$ and $\inf_{L_2} \left({f\left[{X}\right]}\right) = f\left({\inf_{L_1} X}\right)$

Thus by definition of infima inheriting:
 * $\inf_{L_3} \left({g\left[{X}\right]}\right) = g\left({\inf_{L_1} X}\right)$