# Fodor theorem on ultrafilter

Reading some materials on set theory, I know the generalisation of Fodor's theorem:

A filter $$\cal F$$ on a regular uncountable cardinal $$\kappa$$ is normal if and only if for every regressive function $$f$$ on a $$\cal F$$-positive set $$S$$, there exists an $$\cal F$$-positive set $$S'\subseteq S$$ on which $$f$$ is constant.

A subsequent lemma states that if follows immediately from this that

An ultrafilter $$\cal U$$ on a regular uncountable cardinal $$\kappa$$ is normal if and only if for every regressive function $$f$$ on a set $$S\in \cal U$$, there exists a set $$S'\in \cal U$$ on which $$f$$ is constant.

I do not see it as "immediate"... There are some differences in the theorem, mainly the lemma does not require the sets $$S$$ ans $$S'$$ to be $$\cal U$$-positive, which make the direct application of the theorem tricky. Would anyone have a detailled source for that?

Assuming "$$\mathcal{F}$$-positive" means "complement is not in $$\mathcal{F}$$," the key point is that for an ultrafilter $$\mathcal{U}$$ the notions "$$\mathcal{U}$$-positive" and "in $$\mathcal{U}$$" coincide.
For any filter $$\mathcal{F}$$, $$X\in\mathcal{F}$$ implies that $$X$$ is $$\mathcal{F}$$-positive. I claim that the converse holds for ultrafilters. Suppose $$\mathcal{U}$$ is an ultrafilter on $$\kappa$$ and suppose $$X$$ is $$\mathcal{U}$$-positive. Then we must have $$\kappa\setminus X\not\in\mathcal{U}$$; but since $$\mathcal{U}$$ is an ultrafilter, this means $$X\in\mathcal{U}$$.