Renaming Mapping is Bijection

Theorem
The Renaming Mapping is a bijection.

Proof

 * To show that $$r: S / \mathcal{R}_f \to \mathrm{Im} \left({f}\right)$$ is an injection:

Thus $$r: S / \mathcal{R}_f \to \mathrm{Im} \left({f}\right)$$ is an injection


 * To show that $$r: S / \mathcal{R}_f \to \mathrm{Im} \left({f}\right)$$ is a surjection:

Note that for all mappings $$f: S \to T$$, $$f: S \to \mathrm{Im} \left({f}\right)$$ is always a surjection from Surjective Restriction by Limiting Range.

Thus by definition $$\forall y \in \mathrm{Im} \left({f}\right): \exists x \in S: f \left({x}\right) = y$$.

Thus $$r: S / \mathcal{R}_f \to \mathrm{Im} \left({f}\right)$$ is a surjection.


 * As $$r: S / \mathcal{R}_f \to \mathrm{Im} \left({f}\right)$$ is both an injection and a surjection, it is by definition a bijection.