Image and preimage definition

Question

I got the following very basic question:

Why is an image - in contrast to a preimage - defined by using the existential quantifier?

Let $f\colon A \to B$ be a mapping and $M$ a subset of $A$ and $N$ a subset of $B$. Then, the image of $M$ can be defined as:

1. $f(M)=\{y\in B: \exists x \in M:(x)=y\}$

and the preimage of N as:

2. $f^{-1}(N)=\{x\in A:() \in N\}$.

Best and thank you very much,

Niki

Answer 1

If $f$ were simply a relation instead of a function, then both definitions would need an existential quantifier. However, the definition of a function is that given $x\in A$, there is one and only one corresponding element of $B$, and that element is given the name $f(x)$. So there's no need for an existential quantifier in the definition of $f^{-1}(N)$, since there's already a notation for "that other object that must exist for $x$ to be in $f^{-1}(N)$".