Monomorphism Image Isomorphic to Domain

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Theorem

That is, if $\phi \left({S_1}\right) \to S_2$ is a monomorphism, then:

$\phi \left({S_1}\right) \to \operatorname{Im} \left({\phi}\right)$

Let $\left({S_1, \circ_1}\right)$ and $\left({S_2, \circ_2}\right)$ be closed algebraic structures.

Let $\phi$ be a monomorphism from $\left({S_1, \circ_1}\right)$ to $\left({S_2, \circ_2}\right)$.

Let $T = \operatorname{Im} \left({\phi}\right)$ be the image of $\phi$.

By Morphism Property Preserves Closure, $\left({T, \circ_2}\right)$ is closed.

As $\phi$ is a monomorphism, it is an injection.

As $\phi \to \operatorname{Im} \left({\phi}\right)$ is a surjection from Surjection iff Image equals Codomain, we see that $\phi \to \operatorname{Im} \left({\phi}\right)$ is a bijection.

Thus $\phi \to \operatorname{Im} \left({\phi}\right)$ is a bijective homomorphism and hence from the definition, an isomorphism.

$\blacksquare$