Kernel of Magma Homomorphism is Submagma

Theorem
Let $\left({S, *}\right)$ and $\left({T, \circ}\right)$ be algebraic structures such that $\left({T, \circ}\right)$ has an identity $e$.

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

Then the kernel of $\phi$ is a submagma of $\left({S, *}\right)$.

That is:


 * $\left({\phi^{-1}(e), *}\right)$

is a submagma of $\left({S, *}\right)$

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

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

It must be shown that $x*y \in \phi^{-1}(e)$.

Hence the result.