Here is a useful website.
So apparently for all pairwise disjoint sets S not containing the empty set
I had no problems proving the first version from the second, and assuming the first one and taking f(S) (image of the whole set S under f) for t, then obviously for all z in S f(z) is in z and f(S). The reason I'm posting is because I can't for the life of me prove that it's the only element of z intersection f(S)!
Yeah, in trying to prove stuff, if you don't see an intuitive path, it often helps just to start pulling out all the "information" or "implications" you can from the given premises. Extract everything you can from the premises, then lay it out in front of you to then see whether there might be a way to combine all that stuff back into a proof.