Why is the index set of all partial computable functions computable

Question

While reading the statement of Rice's Theorem I noticed that it says "Let $A$ be the index set of any set of partial computable functions $C$, if $A$ is computable then $C$ is $\emptyset$ or the set of all partial computable functions". The theorem itself, and its proof are rather easy to understand, and it is trivial that $A$ is computable when $C=\emptyset$, but how can we show that $A$ is computable when $C$ is the set of all partial computable functions? It is known that the set of indices of all partial computable functions is c.e. But it seems that this doesn't lead to the computability of $A$?