If a township or village in which a free public library has been established pursuant to section 10 is incorporated in its entirety as a city, the library board of the free public library shall continue in office and the free public library shall continue in existence and be governed by the provisions of section 10a as if the free public library had originally been established as a free public library under section 10a. Any tax previously authorized pursuant to this act for the support of the free public library shall continue in effect after the incorporation and shall then be assessed and collected on the taxable property in the city as provided in section 10a.