Group Epimorphism Preserves Generator

Theorem
Let $\struct {G, \circ}$ and $\struct {H, *}$ be groups.

Let $\phi: G \to H$ be a group epimorphism.

Let $A$ be a generator for $\struct {G, \circ}$.

Then $\phi \sqbrk A$ is a generator for $\struct {H, *}$.

Proof
By definition of generator:


 * $A$ is the intersection of all subgroups of $G$ containing $A$.