Pressing-Down-Lemma for Jech's notion of stationary sets

Question

So, apparently there is a variant of the Pressing-Down-Lemma (or Fodor's Lemma) for Jech's notion of stationarity, i.e. for sets in $[X]^\lambda$. Does anybody know a citable source for this?

Answer 1

Following the references in his Handbook article, the following should include the theorem you're looking for.

Namely, if $S$ is a stationary subset of $[X]^\lambda$ and $f(x)\in x$ for all $x\in S\setminus\{\varnothing\}$, then there is a stationary $S'$ on which $f$ is constant.

Thomas J. Jech, Some combinatorial problems concerning uncountable cardinals, Ann. Math. Logic 5 (1972-73), 165--198.

The theorem is Theorem 3.2 on p.179 (this is actually part (d) of the theorem, which is a spillover to page 180).