Renaming Mapping is Bijection/Proof 2

Proof
From Renaming Mapping is Well-Defined, $r: S / \RR_f \to \Img f$ is a well-defined mapping.

By definition, $\RR_f$ is the equivalence relation induced by the mapping $f$.

Hence by definition:
 * $\tuple {s_1, s_2} \in \RR_f \iff \map f {s_1} = \map f {s_2}$

From Condition for Mapping from Quotient Set to be Injection, this is precisely the condition required for $r$ to be an injection.

Next it is noted that the codomain of $r$ is $\Img f$.

Then from Restriction of Mapping to Image is Surjection, we have that $f_{\restriction \Img f}$ is a surjection.

Hence from Condition for Mapping from Quotient Set to be Surjection it follows that $r$ is also a surjection.

Thus $r$ is shown to be both an injection and a surjection, and so by definition is a bijection.