regular languages – Pumping lemma: why x in ∣xy∣ ≤ p?

Looking at the pumping lemma, I’ve noticed that in the string $xy^pz$, there seems to be no rule explicitly stated for $x$ and $z$. If I understand correctly, $x$ and $y$ are basically anything on the 2 sides of the string $y^p$ that we’re pumping and thus can be anything in $L$.

Rule 2 & 3 of the pumping lemma are:

  • $|y| geq 1$
  • $|xy| leq p$

Since $|x| = 0$ and $|z| = 0$ seem to be allowed, as they only need to be of non-negative length, we shouldn’t need $x$ in rule 2 and it can be rewritten as $1 leq |y| leq p$.

Are $x$ and $y$ not just a substitute for whatever are on the 2 sides of $y^p$ which we’re pumping? Why is $x$ in rule 2 if it doesn’t seem to make a difference? If $x$ is necessary, why is there no $|yz| leq p$?