I realize that, but I still can't seem to solve it.

Given a collection of closed subsets in X that has the FIP, its image (of p) also has the FIP since p is closed. Now, since Y is compact, there exists some

where

is the collection of images of p. But we may have that

contains more than one element "spread" over the collection of closed subsets in X, and then I don't see how we are sure that there is an element in the intersection of that collection.

Sorry for being a bother, I just can't seem to see how to get the desired result. I, of course, appreciate your help.