Can we write the definition of image of set as below?
Solution 1:
No you can't.
Your definition would imply that for a specific $y$, $f(x)$ is always equal to $y$ no matter what value of $x$ we pick.
If you can read the definition, $f(A)=\{y\in F \;|\; \exists x \in E, y = f(x)\}$, left-to-right the order in which you say things is the order in which you choose your variables.
So, in this case we first pick some $y\in F$ and we assert that there must exist at least one value of $x \in E$ such that $y=f(x)$. If we change $\exists$ with $\forall$ it changes the definition, and we would need $y=f(x)$ to be true $\forall x$.