Down Mapping is Generated by Approximating Relation

Theorem
Let $L = \struct {S, \wedge, \preceq}$ be a bounded below meet-continuous meet semilattice.

Let $\map {\operatorname{Ids} } L$ be the set of all ideals in $L$.

Let $I$ be an ideal in $L$.

Let $f: S \to \map {\operatorname{Ids} } L$ be a mapping such that:
 * $\forall x \in S: x \preceq \sup I \implies \map f x = \set {x \wedge i: i \in I}$

and
 * $\forall x \in S: x \npreceq \sup I \implies \map f x = x^\preceq$

where $x^\preceq$ denotes the lower closure of $x$.

Then
 * there exists an auxiliary approximating relation $\RR$ on $S$ such that
 * $\forall s \in S: \map f s = s^\RR$

where $s^\RR$ denotes the $\RR$-segment of $s$.

Proof
Define relation $\RR$ on $S$:
 * $\forall x, y \in S: \tuple {x, y} \in \RR \iff x \in \map f y$

By definition of $\RR$-segment:
 * $\forall x \in S: \map f x = x^\RR$

We will prove that:
 * $\RR$ is an approximating relation.

Let $x \in S$.

Suppose the case holds that:
 * $x \preceq \sup I$

Thus:

Suppose the case holds that
 * $x \npreceq \sup I$

Thus