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

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

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

Let for all $I \in \mathit{Ids}$: $m_I: S \to \mathit{Ids}$ be a mapping:
 * $\forall x \in S: x \preceq \sup I \implies m_I\left({x}\right) = \left\{ {x \wedge i: i \in I}\right\}$

and
 * $\forall x \in S: x \npreceq \sup I \implies m_I\left({x}\right) = x^\preceq$

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

Let $x \in S$.

Then
 * $\displaystyle \bigcap \left\{ {m_I \left({x}\right): I \in \mathit{Ids} }\right\} = x^\ll$

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

Proof
Thus