Image of Mapping from Group Element to Inner Automorphism is Inner Automorphism Group

Theorem
Let $G$ be a group.

Let $\kappa: G \to \Aut G$ be the mapping from $G$ to the automorphism group of $G$ defined as:
 * $\forall x \in G: \map \kappa x := \kappa_x$

where $\kappa_x$ is the inner automorphism on $x$:
 * $\forall g \in G: \map {\kappa_x} g = x g x^{-1}$

Then $\Img \kappa$ is the inner automorphism group of $G$.