I am already struggling right at the beginning.

If we open a book on set theory, we will find a proof of Cantor’s theorem which shows explicitly that for every map e : A → P(A) there is a subset of A outside its image, namely S = { x ∈ A ∣ x ∉ e(x) }

What about e(x) = { x }? In that case S would be the empty set, wouldn't it? Does map imply some additional properties in this case? Am I misunderstanding the statement and it does not try to imply that S is non-empty for every e?

Yes, in that case S is {}. {} ∈ P(A) but is not in the image of e.

Thanks, now it clicked. I misinterpreted the image as the image of x instead of as the image of e. And then did something weired that now makes no longer any sense at all.