Homomorphic Proxy Re-Encryption (HPRE) is an extension of Proxy Re-Encryption (PRE) which combines the advantages of both Homomorphic Encryption (HE) and PRE. A HPRE scheme allows arbitrary evaluations to be performed on ciphertexts under one (the delegator's) public key and, using a re-encryption key, it transforms the resulting ciphertext to a new ciphertext under another (the delegatee's) public key. Prior HPRE schemes are either CPA-secure or CCA-secure but only support partial homomorphic operations. We propose a generic construction of single-hop HPRE scheme which supports fully homomorphic operations. The proposed scheme is proven secure in our new index-based CCA-HPRE model. Our technique is to give a generic transformation that turns any multi-identity identity-based FHE (IBFHE) scheme with key switching into Fully Homomorphic Encryption (FHE) with key switching from which we can obtain the proposed single-hop HPRE scheme. We also present a concrete instantiation of multi-identity IBFHE with key switching from learning with errors (LWE) in the standard model.