Mapping Preserves Finite and Directed Suprema

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

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

Let $f$ preserve finite suprema and preserve directed suprema.

Then $f$ also preserves all suprema

Proof
This follows by mutatis mutandis of the proof of Mapping Preserves Finite and Filtered Infima.