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
A homomorphism is an epimorphism onto its image by Epimorphism by Restriction of Codomain.

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

The result follows.