What is known about the sets enumerated by primitive recursive functions?
In "Extensions of some theorems of Gödel and Church" it's shown by Barkley Rosser that these sets are exactly the recursive sets:


  Corollary I. If a class can be enumerated (allowing repetitions) by a general recursive function, it can be enumerated (allowing repetitions) by a primitive recursive function.


Note that the crux here is repetitions. Since you constructed sets, and comparing them with other sets, any repetitions do not violate their equality.