Easton's theorem in the presence of Woodin cardinals

Under the assumption that δ is a Woodin cardinal and GCH holds, I show that if F is any class function from the regular cardinals to the cardinals such that (1) κ < cf(F (κ)), (2) κ < λ implies F (κ) ≤ F (λ), and (3) δ is closed under F , then there is a cofinality-preserving forcing extension in which 2 = F (γ) for each regular cardinal γ < δ, and in which δ remains Woodin. Unlike the analogous results for supercompact cardinals [Men76] and strong cardinals [FH08], there is no requirement that the function F be locally definable. I deduce a global version of the above result: Assuming GCH, if F is a function satisfying (1) and (2) above, and C is a class of Woodin cardinals, each of which is closed under F , then there is a cofinality-preserving forcing extension in which 2 = F (γ) for all regular cardinals γ and each cardinal in C remains