Let's say we have a homomorphism $\phi: G \rightarrow H$ where $G$ and $H$ group. Say $G$ is generated by some generators and relations, for an example $$G = \langle r,s \mid r^{4}=1 , r^2=s^2, rsr=s\rangle.$$ Now, since $\phi$ is a homomorphism then those generators and relations are satisfied by $\phi(r)$ and $\phi(s)$.
Now if we show that $\phi(r) \ne \phi(s), |\phi(r)|=|\phi(s)|= 4 $ is it sufficient to show that the group generated by $\phi(r)$ and $\phi(s)$ is isomorphic to $G$?
Is this true in general? What I mean is if images of each generator have the same order as the generators and images of each generator are different group elements in the image, does it say the group generated by the images of generators is a subgroup of $H$?
Thanks!