Sino-Tibetan relations during the Ming dynasty
Kublai Khan did not conquer the Song dynasty in South China until 1279, so Tibet was a component of the early Mongol Empire before it was combined into one of its descendant empires with the whole of China under the Yuan dynasty (1271–1368). Van Praag writes that this conquest "marked the end of independent China," which was then incorporated into the Yuan dynasty that ruled China, Tibet, Mongolia, Korea, parts of Siberia and Upper Burma. Morris Rossabi, a professor of Asian history at Queens College, City University of New York, writes that "Khubilai wished to be perceived both as the legitimate Khan of Khans of the Mongols and as the Emperor of China. Though he had, by the early 1260s, become closely identified with China, he still, for a time, claimed universal rule", and yet "despite his successes in China and Korea, Khubilai was unable to have himself accepted as the Great Khan". Thus, with such limited acceptance of his position as Great Khan, Kublai Khan increasingly became identified with China and sought support as Emperor of China.

Q: When did Kublai Khan conquer the song dynasty?
1279