Show that $(I+J)/(I\cap J)\cong (I+J)/I\times (I+J)/J$, where $I,J$ are ideals in a commutative ring $R$ with identity.
What I did was first show that $\phi: I+J\to (I+J)/I\times (I+J)/J$ defined by $\phi(r+s)=(r+s+I,r+s+J)$ is a ring homomorphism, and then I applied the first isomorphism theorem.
I only showed that $\phi$ is well-defined and operation-preserving before applying the first isomorphism theorem. I also found that $ker(\phi)=I\cap J$.
Is this sufficient enough to show it is an isomorphism? My teacher said I need to show it is surjective too. Why?