I'm attempting a true/false question from a past exam paper and not sure if my approach/understanding is valid:
If $\phi:G \to H$ is a group homomorphism and $N \trianglelefteq G$, then $G/N \cong \phi(G) / \phi(N)$
My attempt: True, since using the First Isomorphism Theorem, we know that $\phi(G) = im(\phi) \cong G/\sim$, and $\phi(N) = im(\phi) \cong N/\sim$. So $\phi(G) / \phi(N) \cong (G/\sim )/(N/\sim)$. Then using the Third Isomorphism theorem, we have an isomorphsim from $(G/\sim )/(N/\sim)$ to $G/N$, i.e. $G/N \cong \phi(G) / \phi(N)$.
Is this a valid use of these theorems, or have I misunderstood how they can be applied?