What is the importance of the condition "| xy | < p" in pumping lemma? [duplicate]

Let L be a language. w $$\in$$ L , and w could be broken in xyz.

Then if L is regular, there exists a pumping length p such that:

1. |y| $$\gt$$ 0
2. |xy| $$\le$$ p
3. $$\forall$$ i $$\ge$$ 0, xy$$^i$$z $$\in$$ L

I understand that this lemma is based on pigeon hole principle and proof by contradiction.

For conditon (1), I understand that the string we are pumping should be $$\ge$$ 1.

But I don't understand the significance of condition (2). Why should the legth of xy be equal to or smaller than p . Will not it work otherwise?

$$p$$ is the number of states in the automaton. As you say, the pumping lemma is about the pigeonhole principle. Suppose that $$q$$ is the first state that's repeated when you read input $$w$$. Then $$x$$ is the string that you read before you reach $$q$$ for the first time, and $$y$$ is the string you read between the first and second visits to $$q$$. Since there are only $$p$$ states including the start state, you must repeat a state after doing at most $$p$$ transitions, i.e., reading a string of length at most $$p$$. So the condition $$|xy|\leq p$$ comes from the proof.
In reality, when an automaton reads a particular string $$w$$, there might be many times when it returns to a state that has already been visited. If you take $$w\in L$$ any split $$w=xyz$$ with $$|y|\geq 1$$, such that the automaton is in the same state at the beginning and end of $$y$$ then $$xy^iz$$ will be in $$L$$ for all $$i$$, regardless of how long $$xy$$ is.