Intersection of Class and Set is Set

Theorem
Let $C$ be the class:
 * $C = \set { u : \map \phi {u, p_1, \ldots, p_n} }$

Then for all sets $X$, $C \cap X$ is a set.

Proof
By the definition of class intersection:
 * $a \in C \cap X \implies a \in C \land a \in X$

Thus:
 * $a \in C \cap X \implies a \in X$

The subclass definition gives:
 * $C \cap X \subseteq X$

By Subclass of Set is Set, $C \cap X$ is a set.