Set of Intersections with Superset is Cover

Theorem

Let $S$ be a set.

Let $\CC$ be a cover of $S$.

Let $T \supseteq S$ be a superset of $S$.

Then:

$\set {C \cap T : C \in \CC}$

is a cover of $S$.

Proof

Let $x \in S$ be arbitrary.

By definition of cover, there is some $C \in \CC$ such that:

$x \in C$

By definition of superset:

$x \in T$

Therefore, by definition of intersection:

$x \in C \cap T$

As $x \in S$ was arbitrary, the result follows.

$\blacksquare$