Set of Mappings which map to Same Element induces Equivalence Relation

Theorem
Let $X$ and $Y$ be sets.

Let $E$ be the set of all mappings from $X$ to $Y$.

Let $b \in X$.

Let $\mathcal R \subseteq E \times E$ be the relation on $E$ defined as:
 * $\mathcal R := \left\{{\left({f, g}\right) \in \mathcal R: f \left({b}\right) = g \left({b}\right)}\right\}$

Then $\mathcal R$ is an equivalence relation.

Proof
This is an instance of Induced Equivalence is Equivalence Relation.