Leaving aside the problem of how to compute the inverse of a PRF F, may we use it also as a PRP?
The reciprocal of this statement is true, see for instance Katz and Lindell, Proposition 3.27, when the input length of F is sufficiently large. However, in practice to build a PRP from a PRF people use a Feistel network. This would address the problem of making the PRF invertible.
Leaving aside the invertibility I guess the same proof for PRP $\implies$ PRF is useful here. Am I right?