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 $\RR \subseteq E \times E$ be the relation on $E$ defined as:

$\RR := \set {\tuple {f, g} \in \RR: \map f b = \map g b}$

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

Then $e_b$ is a bijection.

Proof

This is an instance of Renaming Mapping is Bijection.

$\blacksquare$

Sources

• 1975: Bert Mendelson: Introduction to Topology (3rd ed.) ... (previous) ... (next): Chapter $1$: Theory of Sets: $\S 7$: Relations: Exercise $5$