Renaming Mapping is Bijection/Proof 1

Proof of Injectivity
To show that $r: S / \RR_f \to \Img f$ is an injection:

Thus $r: S / \RR_f \to \Img f$ is an injection.

Proof of Surjectivity
To show that $r: S / \RR_f \to \Img f$ is a surjection:

Note that for all mappings $f: S \to T$, $f: S \to \Img f$ is always a surjection from Surjection by Restriction of Codomain.

Thus by definition:
 * $\forall y \in \Img f: \exists x \in S: \map f x = y$

So:

Thus $r: S / \RR_f \to \Img f$ is a surjection.

As $r: S / \RR_f \to \Img f$ is both an injection and a surjection, it is by definition a bijection.