Kernel of Magma Homomorphism is Submagma

Theorem
Let $\left({S, *}\right)$ and $\left({T, \circ}\right)$ be algebraic structures.

Let $\left({T, \circ}\right)$ have 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:


 * $\left({\phi^{-1} \left({e}\right), *}\right)$ is a submagma of $\left({S, *}\right)$

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

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

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

Thus:

Hence the result.