Inversion Mapping is Mapping

Theorem
Let $\struct {G, \circ}$ 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:


 * $\map \iota a \ne \map \iota b \implies a \ne b$: