What is the Pumping Lemma a lemma to?
In the foundational paper of Rabin and Scott, Finite automata and their decision problems, the pumping lemma appears as a lemma (Lemma 8) to the following result (Theorem 9):


  The language accepted by an $n$-state DFA is infinite if and only if it accepts some word whose length lies between $n$ and $2n$.


The pumping lemma implies the $\Longrightarrow$ direction.

It also "gives another proof" that the language $\{0^n 1 0^n : n \geq 0\}$ isn't regular (the original proof in the paper uses Myhill-Nerode theory).