John and Anne Dye had 3 children bapt at All Saint's, Leicester, in 1801, 03 and 05 (Elizabeth, Mary and Fanny). Then there's a Sarah DYER b All Saints to John and Anne DYER in 1807 (?error in transcription? there are no other Dyer baptisms at all).
These were followed by a further 7 baptisms to John and Ann (no E) Dye (?same couple) at All Saints between 1812-1826 (that is a long gap between Fanny, or Sarah, and the 1812 bapt). Julia, Harriet, John, Ann, Henry, Francis and Francis William. It might be possible to track some of these in the 1841/51 censuses just in case an aged parent is living with them!
There are no other DYE baptisms on the Leic Churches index from 1558-1837, so if this is Fanny's/Frances's family, presumably they moved into the town from elsewhere before 1801.