That doesn't work, because is defined as a mapping from to the reals; if you change , then you also change , and so you can't define them out of order.

I suspect you're confusing , the individual policies that an agent could adopt, and , the complete collection of policies that the agent could adopt.

Another way to express the theorem is that there is a many-to-one mapping from choices of to Pareto optimal policies that maximize that choice of .

[Edit] It's not strictly many-to-one, since you can choose s that make you indifferent between multiple Pareto optimal basic policies, but you recapture the many-to-one behavior if you massage your definition of "policy," and it's many-to-one for most choices of .