Preimage of Zero of Homomorphism is Submagma

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

Let $\struct {T, \circ}$ be a magma with a zero element $0$.

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

Then $\struct {\phi^{-1} \sqbrk 0, *}$ is a submagma of $\struct {S, *}$.

Proof
Let $x, y \in \phi^{-1} \sqbrk 0$.

It is to be shown that:
 * $x * y \in \phi^{-1} \sqbrk 0$

Thus:

Hence the result.