[FOM] PCF theory

[pax0 at seznam.cz]

I'm trying to understand some pcf theory --
the equivalence of the sentence after "In plain words," with (1):
>Let A be a set of regular cardinals. For any cardinal λ define
>
>(1)  J<λ[A] = {X ⊆ A | pcf(X) ⊆ λ}.
>
>In plain words, X ∈ J<λ[A] iff for every ultrafilter D over A such that X ∈ D,
>cf(Product A/D) < λ. That is, X “forces” the cofinalities of its ultraproducts to
>be below λ.

One time we are taking ultrafilter D over A containing X and in (1) over X in pcf(X).
Can someone explain both directions in "iff" in some detail?
Thank you Jan Pax