(a) When any area in a subdivision as to which a final map has been finally approved by a board of supervisors and filed for record pursuant to this division is thereafter annexed to a city, the final map and any agreements relating to the subdivision shall continue to govern the subdivision.
(b) When any area in a subdivision or proposed subdivision as to which a tentative map or vesting tentative map has been filed but a final map has not been finally approved, or as to which a parcel map is required by this division or local ordinance but the final act required to make the parcel map effective has not been taken, is annexed to a city, all procedures and regulations required by this division or by local ordinance of the annexing city shall be deemed to commence as of the effective date of the annexation and the map shall comply with the requirements of any applicable ordinance of the city to which the area is annexed.