Inversion Mapping is Mapping

Theorem
Let $\left({G, \circ}\right)$ be a group.

Let $\iota: G \to G$ be the inversion mapping on $G$.

Then $\iota$ is indeed a mapping.

Proof
To show that $\iota$ is a mapping, it is sufficient to show that:


 * $\iota \left({a}\right) \ne \iota \left({b}\right) \implies a \ne b$: