Is the Imamate and rule prescribed in the religion? And what is the proof for that?

It has not been established that the Prophet, peace and blessings be upon him, appointed an Imam or a Caliph after him. However, he intended to write down the succession for Abu Bakr, then he left that and said: "Allah and the believers refuse anyone but Abu Bakr." The Companions inferred Abu Bakr's rightful claim to the Caliphate from his leading the people in prayer during the Prophet's illness. So, they chose him for their worldly affairs just as the Prophet approved him for their religious matters, and this was confirmed by Ali ibn Abi Talib.