What am I missing about the definition presented here for an image f[a] in "Set Theory for the Working Mathematician" by Krzysztof Ciesielski?

Question

In Set Theory for the Working Mathematician by Krzysztof Ciesielski, he gives the definition of an image as follows.

For f : X → Y , A ⊂ X, we define

f[A] = {f(x): x ∈ X} = {y ∈ Y : ∃x ∈ X (y = f(x))}

Isn't this just the whole range of the function? My intuition tells me it should be x ∈ A, but I don't want to assume a typo. If it is the whole range, why introduce the subset A?

Answer 1

Yes, this is a typo, and it should require $x\in A$ instead of $x\in X$.