Survivor’s “Eye of the Tiger” wasn’t the original song of Rocky III, “You’re the Best” was written for it by Bill Conti, and sang by Joe Exposito, but Sly rejected it.
The song went to become the unofficial theme song fo The Karate Kid, so I think both movies got the best (pun intended).
Ironically the same thing happened with “Glory of Love” by Peter Cetera, that was written for Rocky IV but found itself as the main theme song of The Karate Kid part II