# Using only pseudorandom function to drive many deterministic keys

My question is related to:

Can Pseudorandom values be used as secure keys?

But here I'm only using pseudorandom function.

Assume we have a pseudorandom function, $PRF$. We have a truly random master key: $mk$ for the PRF function.

I want to deterministically generate $n$ keys, so I do:

$\forall i, 1\leq i\leq n: k_i=PRF(mk,i)$.

Question: Is it secure to use each key $k_i$ to generate $m$ pseudorandom values? E.g. $\forall j, 1\leq j\leq m : k_{i,j}=PRF(k_i,j)$.

Is there any protocol using such scheme (if the scheme is secure) ?