Monomorphism Image is Isomorphic to Domain

Theorem
The image of a monomorphism is isomorphic to its domain.

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

is an isomorphism.

Proof
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 Range, 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.