Meet of Suprema equals Supremum of Meet of Ideals implies Ideal Supremum is Meet Preserving

Theorem
Let $\mathscr S = \struct {S, \wedge, \preceq}$ be an up-complete meet semilattice.

Let $f: \map {\it Ids} {\mathscr S} \to S$ be a mapping such that:
 * $\forall I \in \map {\it Ids} {\mathscr S}: \map f I = \sup_{\mathscr S} I$

where
 * $\map {\it Ids} {\mathscr S}$ denotes the set of all ideals in $\mathscr S$

Let
 * $\forall I_1, I_2 \in \map {\it Ids} {\mathscr S}: \paren {\sup I_1} \wedge \paren {\sup I_2} = \sup \set {i \wedge j: i \in I_1, j \in I_2}$

Then: $f$ preserves meet as a mapping from $\struct {\map {\it Ids} {\mathscr S}, \subseteq}$ into $\mathscr S$

Proof
Let $I, J \in \map {\it Ids} {\mathscr S}$ such that
 * $\set {I, J}$ admits an infimum in $\struct {\map {\it Ids} {\mathscr S}, \subseteq}$.

By definition of image of set:
 * $\map {f^\to} {\set {I, J} } = \set {\map f I, \map f J}$

Thus by definition of meet semilattice:
 * $\map {f^\to} {\set {I, J} }$ admits an infimum in $\mathscr S$.

Thus: