Renaming Mapping from Set of Mappings on Single Element

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:
 * $\mathcal R := \set {\tuple {f, g} \in \mathcal R: \map f b = \map g b}$

Let $e_b: E / \mathcal R \to Y$ be the renaming mapping induced by $\mathcal R$.

Then $e_b$ is a bijection.

Proof
This is an instance of Renaming Mapping is Bijection.