Group Epimorphism Preserves Subgroups

Theorem
Let $\struct {G_1, \circ}$ and $\struct {G_2, *}$ be groups.

Let $\phi: \struct {G_1, \circ} \to \struct {G_2, *}$ be a group epimorphism.

Then:
 * $H \le G_1 \implies \phi \sqbrk H \le G_2$

where:
 * $\phi \sqbrk H$ denotes the image of $H$ under $\phi$
 * $\le$ denotes subgroup.

That is, group epimorphism preserves subgroups.

Proof
By definition, $\phi$ is a group homomorphism.

The result then follows from Group Homomorphism Preserves Subgroups.