Let $S$ be a scheme and $\mathscr{X}$ an Artin stack over $S$. Let $X$ be a scheme and $P : X \to \mathscr{X}$ a representable morphism, which is smooth and surjective. Then this $P$ is an epimorphism.

(i.e., for any $S$-schemes $U$ and $x \in \mathscr{X}(U)$, there exists an etale surjective morphism $U' \to U$ (of schemes), such that $x|_{U'}$ comes from $X(U')$)

I want to know it in order to show that $\mathscr{X} = [X/ R]$, the quotient stack. ($R = X \times_\mathscr{X} X$)

In 4.3. of Laumon, Moret-Bailly's Champs algébriques, the authors say that this is obvious. But I don't understand.

I've heard that every Artin stack is a stack over fppf topology. If we use it, then trivially $\mathscr{X} = [X/ R]$. But in the proof of this proposition (10.7. of LMB's Champ algebriques), they use the highlighted statement, so it is circular reasoning.

How can I show the highlighted statement without using the stack-ness over fppf topology?