Assume that the universe is homogenous and isotropic, and the following equation holds:
\begin{equation}R_{00}-\frac{1}{2}g_{00}R=8\pi GT_{00}; \space \space \nabla_{\mu}T^{\mu 0}=0.\end{equation}
How do I prove that the following equations are identically satisfied provided that the above two are satisfied?
\begin{equation}R_{0i}-\frac{1}{2}g_{0i}R=8\pi GT_{0i}; \space \space R_{ij}-\frac{1}{2}g_{ij}R=8\pi GT_{ij}; \space \space \nabla_{\mu}T^{\mu i}=0.\end{equation}
My approach was to write $g_{00}=1$ and $g_{ij}=-a^2\gamma_{ij}$ and evaluate the Ricci tensors and so on, but I know this is not the way to do it. Can anyone suggest me the way?