Centralizer in Subgroup is Intersection

Theorem
Let $$G$$ be a group.

Let $$H \le G$$. Then:

$$\forall x \in G: C_H \left({x}\right) = C_G \left({x}\right) \cap H$$

That is, the centralizer of an element in a subgroup is the intersection of that subgroup with the centralizer of the element in the group.

Proof
It is clear that $$g \in C_H \left({x}\right) \iff g \in C_G \left({x}\right) \land g \in H$$.

Thus by definition of set intersection the result follows.