I have the following related questions:
$(1)$ Assume that the function symbol "function" has been defined. Is the following sentence $$\forall x_1,...,x_n (x_1,...,x_n \in \mathbb{R} \implies \exists f (function(f)\wedge \forall i \in\{1,...,n\}:f(i)=x_i))$$ a well formed formula and is "$n$" free in this formula? If so, how can I refer to $n$ within this formula by writing down the set $\{1,...,n\}$ and if not, how can I fix $n$ not being free?
$(2)$ Can I even write something like "(...) $f(i)=x_i$" above? Doesn't this assume that the variables behave like a function symbol, so I can "plug in" $i$? Does that mean I would have to write something like $$\forall x_1,...,x_n(x_1,...x_n \in \mathbb{R} \implies \exists f (function(f)\wedge f(1)=x_1 \wedge \cdots \wedge f(n)=x_n))?$$ This would include even more "..." notation, is there a way to write this formally?
$(3)$ In formal languages the "..." notation is not allowed, is there another way to write $\forall x_1,...,x_n$ there that is more formal? For sets one has $\bigcup$ and for sums one has $\sum$, which one could index over and which gets rid of "..." notations. Is there something similar for quantifiers?