Constant Mapping to Identity is Homomorphism/Groups

Theorem
Let $\left({G_1, \circ_1}\right)$ and $\left({G_2, \circ_2}\right)$ be groups with identities $e_1$ and $e_2$ respectively.

Let $\phi_e: \left({G_1, \circ_1}\right) \to \left({G_2, \circ_2}\right)$ be the constant mapping defined as:
 * $\forall x \in G_1: \phi_e \left({x}\right) = e_2$

Then $\phi_e$ is a group homomorphism whose image is $\left\{{e_2}\right\}$ and whose kernel is $G_1$.

Proof
Let $x, y \in G_1$.

Then:

So $\phi_e$ is a group homomorphism.

The results about image and kernel follow directly by definition.