Homomorphism Preserves Subsemigroups

Theorem
Let $\left({S, \circ}\right)$ and $\left({T, *}\right)$ be semigroups.

Let $\phi: \left({S, \circ}\right) \to \left({T, *}\right)$ be a homomorphism.

Let $S'$ be a subsemigroup of $S$.

Then $\phi \left({S'}\right)$ is a subsemigroup of $T$.

Proof
By Restriction of Homomorphism to Image is Epimorphism, $\phi$ is an epimorphism onto its image.

Then by Epimorphism Preserves Semigroups, it follows that the image of $S'$ is a semigroup.

The result follows.