Proposition 0.4.5.4 in EGA appears to be a general representability theorem. It reads:
Suppose $F$ is a contravariant functor from the category of locally ringed spaces over $S$ to the category of sets. Suppose given representable sub-functors $F_i$ of $F$, such that the morphisms $F_i \to F$ are representable by open immersions. Suppose furthermore that if $Hom(-, X) \to F$ is a morphism and the functors $F_i \times_F Hom(-,X)$ are representable by $Hom(-,X_i)$, the family $X_i$ forms an open covering of $X$. (That $X_i \to X$ is an open immersion follows from the definitions.) Finally, suppose that if $U$ ranges over the open subsets of a locally ringed space $X$, the functor $U \to F(U)$ is a sheaf. Then, $F$ is representable.
I haven't yet been able to grok the proof, but it appears to be some sort of extended gluing construction. This result appears to be used in proving that fibered products exist in the category of schemes. However, it's fairly easy to directly construct fibered products by gluing open affines.
Are there examples where this result actually makes the life of algebraic geometers easier? Also, I'd appreciate any links to examples (outside of EGA) where this result is used.