Intersection of Applications of Down Mappings at Element equals Way Below Closure of Element

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

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

Let for all $I \in \operatorname {Ids}$: $m_I: S \to \operatorname {Ids}$ be a mapping:
 * $\forall x \in S: x \preceq \sup I \implies \map {m_I} x = \set {x \wedge i: i \in I}$

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

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

Let $x \in S$.

Then
 * $\ds \bigcap \set {\map {m_I} x: I \in \operatorname {Ids} } = x^\ll$

where $x^\ll$ denotes the way below closure of $x$.

Proof
Thus