I have recently read the following statement which I do not understand:
Each function $w\mapsto h(x,w)$ has range $Y$.
As far as I understand "$\mapsto$" denotes a mapping. How are they defining the function(s)? Is there a function for each $w$, and each of these functions has the same range?
For each fixed $x$ there is a function $f_x$ with variable $w$ defined as $f_x(w)=h(x,w)$.
The statement then says that $\forall x,x': \operatorname{ran}(f_x) = \operatorname{ran}(f_{x'})$.
Or in a more formal way:
$$\forall x,x': \forall w : \exists w': h(x,w)=h(x',w')$$
which comes down to the same thing.