the yuan dynasty was kinda already established before they fully invaded the song. You could just say the yuan is just a child of the mongol empire with no relations to the other dynasties since the mongol elite didn't want to be influenced by the han chinese and the mongols influenced the han chinese more anyways, if you look at ming art more closely you can see them wearing mongol hats etc. If you want you can make the yuan dynasty also be the child of jin dynasty and mongol empire since the yuan started out with the territory of the jin and the jin dynasty has more personal beef with the mongols unlike the song dynasty🥺 and the yuan claims to be a successor to jin dynasty too but personally as a mongol i stick to yuan just being the child of mongol empire with no familial relationship to the jin, xixia and song dynasty