Renaming Mapping is Bijection

Theorem
Let $f: S \to T$ be a mapping.

Let $r: S / \RR_f \to \Img f$ be the renaming mapping, defined as:


 * $r: S / \RR_f \to \Img f: \map r {\eqclass x {\RR_f} } = \map f x$

where:
 * $\RR_f$ is the equivalence induced by the mapping $f$
 * $S / \RR_f$ is the quotient set of $S$ determined by $\RR_f$
 * $\eqclass x {\RR_f}$ is the equivalence class of $x$ under $\RR_f$.

The renaming mapping is a bijection.