Even though the formal syntax rules for first order logic talk about $\forall x$ or $\exists x$ without necessarily including any kind of $\in Y$ part for some domain/set $Y$, sometimes we'll see things like $\forall x \in \mathbb{N}$ for example and I was unsure where that syntax "comes in" to FOL.
In a comment on another question I was told that:
$\forall x\in Y, P(x)$ is just convenient shorthand for $\forall x(x\in Y\to P(x))$
$\exists x\in Y, P(x)$ is just convenient shorthand for $\exists x(x\in Y\land P(x))$
However I don't understand how we determine whether we need the $\to$ symbol or the $\land$ symbol. In particular why isn't it $\land$ for both, or $\to$ for both, etc? How do we know which to use in which situation?
Consider as domain of the interpretation the set of integers, and let the naturals defines as the non-negative integers.
We have that "Every natural numbers is non negative" will be : $∀n(\text {Nat}(n) → (n ≥ 0))$ or equivalently : $∀n(n ∈ \mathbb N → (n ≥ 0))$.
What happens with $\land$ instead of $\to$ ?
We have that : $∀n(\text {Nat}(n) \land (n ≥ 0))$ will be false, because e.g. $\text {Nat}(-1) \land (-1 ≥ 0)$ is false.
$\text {Nat}(-1) → (-1 ≥ 0)$ instead, is true.
Consider now "There is a negative natural number", that is obviously false.
It will be : $\exists n(\text {Nat}(n) \land (n \lt 0))$ or equivalently : $\exists n(n ∈ \mathbb N \land (n \lt 0))$.
What happens with $\to$ instead of $\land$ ?
We have that $\exists n(\text {Nat}(n) \to (n \lt 0))$ will be true, because $\text {Nat}(-1) \to (-1 \lt 0)$ is true.