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 Range.


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

The result follows.