Kernel of Magma Homomorphism is Submagma

Theorem
Let $\struct {S, *}$ be a magma.

Let $\struct {T, \circ}$ be an algebraic structure with an identity $e$.

Let $\phi: S \to T$ be a homomorphism.

Then the kernel of $\phi$ is a submagma of $\struct {S, *}$.

That is:


 * $\struct {\map {\phi^{-1} } e, *}$ is a submagma of $\struct {S, *}$

where $\map {\phi^{-1} } e$ denote the preimage of $e$.

Proof
Let $x, y \in \map {\phi^{-1} } e$.

By the definition of a magma, $S$ is closed under $*$.

That is:
 * $\forall x, y \in S: x * y \in S$

Hence:
 * $x * y \in \Dom \phi$

It is to be shown that:
 * $x * y \in \map {\phi^{-1} } e$

Thus:

Hence the result.