Centralizer of Subset is Intersection of Centralizers of Elements

Theorem
Let $\struct {G, \circ}$ be a group.

Let $S \subseteq G$.

Let $\map {C_G} S$ be the centralizer of $S$ in $G$.

Then:
 * $\ds \map {C_G} S = \bigcap_{x \mathop \in S} \map {C_G} x$

where $\map {C_G} z$ is the centralizer of $x$ in $G$.