Intersection of Closed Sets is Closed/Closure Operator

Theorem
Let $S$ be a set.

Let $f: \mathcal P(S) \to \mathcal P(S)$ be a closure operator on $S$.

Let $\mathcal C$ be the set of all subsets of $S$ that are closed with respect to $f$.

Let $\mathcal A \subseteq \mathcal C$.

Then $\bigcap \mathcal A \in \mathcal C$.

Proof
Let $Q = \bigcap \mathcal A$.

By the definition of closure operator, $f$ is inflationary, order-preserving, and idempotent.

Let $A \in \mathcal A$.

By Intersection Largest, $Q \subseteq A$.

Since $f$ is order-preserving, $f(Q) \subseteq f(A)$.

By the definition of closed set, $f(A) = A$

Thus $f(Q) \subseteq A$.

Since this holds for all $A \in \mathcal A$, $f(Q) \subseteq \bigcap \mathcal A$ by Intersection Largest.

Since $\bigcap \mathcal A = Q$, $f(Q) \subseteq Q$.

Since $f$ is inflationary, $Q \subseteq f(Q)$.

Thus by Equality of Sets, $Q = f(Q)$.

Therefore $Q$ is closed with respect to $f$.