Elements of Semigroup with Equal Images under Homomorphisms form Subsemigroup

Theorem
Let $\struct {A, \circ}$ and $\struct {B, *}$ be semigroups.

Let $f: A \to B$ and $g: A \to B$ be semigroup homomorphisms.

Then the set:


 * $S = \set {x \in A: \map f x = \map g x}$

is a subsemigroup of $A$.

Proof
Let $x, y \in A$. Then:

Thus $x \circ y \in A$.

So, by the Subsemigroup Closure Test:
 * $S$ is a subsemigroup of $A$.