Why is the definition of an image of a subset use existential quantifier rather than universal? According to my textbook (Discrete Mathematics and Its Applications by Rosen), the definition of the image under function $f $ $(f:A\rightarrow B)$ of the subset $S$ $(S\subseteq A)$ is $$f(S) = \{t | \exists s \in S (t = f(s)) \}$$
What I'm confused about this is that I think it should be a universal quantification since we're talking about the set containing all the images of elements in set $S$, right?
Just because a definition in spoken English uses the word 'all' doesn't mean there should be a universal quantifier. Here's another example: suppose you have a bunch of sets $X_1, X_2, \dots$. The set of all elements appearing in these sets is
$$\bigcup_{n \in \mathbb{N}} X_n = \{ x \mid \exists n \in \mathbb{N},\, x \in X_n \}$$
which is defined using an existential quantifier, not a universal quantifer. This is because the set of 'all the elements' of these sets is precisely the set of elements which appear in 'at least one' of these sets.

In this case, $f(S)$ is the set of all the elements in the image of $S$ under $f$. That is, it's the set of elements which are of the form $f(s)$ 'for some' $s$... that's why we have an existential quantifier.

If we had a universal quantifier instead, the set would be
$$\forall_f(S) = \{ t \mid \forall s \in S,\, t = f(s) \}$$
That is, $t \in \forall_f(S)$ if every element of $S$ maps to $t$ under $f$. If $S \ne \varnothing$ then this set is either empty (if $f$ is non-constant on $S$) or has one element (if $f$ is constant on $S$)... so in general not very interesting.
I've never seen this notation ($s\in S(t = f(s))$), is it really exactly like this in your textbook? It simply means, that the image of $S$ under $f$ is the set of all points $t \in B$, such that there is a $s \in S$ with $f(s) = t$. You can also write is like this:
$$ f(S) = \{ f(s) \; | \; s \in S \} \subset B $$