Functional Completeness of 3-valued logic
Chapters 5 and 6 of the book [Function algebras on finite sets, Dietlinde Lau, 2006] contain an in depth treatment of functional completeness in many-valued logic (including proofs). In summary: Rosenbergs [1965, 1970] characterization of maximal clones (also called precomplete clones) give a criterion for functional completeness in k-valued logic for any k. 

For the special case of 3-valued logic such a characterization (consisting of 18 maximal/precomplete classes) was given by Jablonskij already in 1954. Hence, in order to verify that your set of 3-valued "operators" are functionally complete, it is enough to check that they don't fall into any of the 18 precomplete classes.