TITLE: Image of function definition notation
QUESTION [6 upvotes]: In my Linear Algebra and Geometry textbook, it defines the image of a linear transformation $T$ as:
$$\operatorname{Im}\, (T) := \{\; w \in W : \; w=Tv \;\;\text{ for some } v \in V \} $$
As far as I can see, this is just the same as:
$$\operatorname{Im} \, (T) := \{ \;Tv \in W : \;v \in V\}$$
Is there any difference in these definitions? 
If not, why is the first one used?

REPLY [4 votes]: In certain set-theoretic senses, the first one
$$
\{\; w \in W : \; w=Tv \;\;\text{ for some } v \in V \}
$$
is exactly what you get by the Axiom of Separation, while the second one
$$
\{ \;Tv \in W : \;v \in V\}
$$
is taken as a short-hand way of writing it.  So perhaps the author thought the first one would be less confusing to some students.