Any such city which adopts the mayor-council form of government shall continue its existence as a body corporate under the name of City of _________ (inserting the name of such city). The word city as hereinafter used shall mean and refer to any city which has adopted the mayor-council form of government. The city shall continue as a municipal corporation, within the corporate limits as then established, and as thereafter fixed in the manner prescribed by law, subject to all the duties and obligations then pertaining to or incumbent upon it as a municipal corporation and shall continue to enjoy all the rights, immunities, powers, and franchises then enjoyed by it, as well as those that may thereafter or hereinafter be granted to it.