TITLE: Proof by `universal receiver'
QUESTION [13 upvotes]: Anyone following the news knows about the major breakthoughs that have taken place recently in $3$-manifold topology. These have come via a route whose big-picture I find to be conceptually interesting. To prove a property $P$ (largeness, linear over $\mathbb{Z}$, virtual fibering, LERF, virtually biorderable etc.) for a class of objects $X$ (e.g. fundamental groups of finite volume hyperbolic $3$-manifolds), embed each object of $X$ nicely in an object of a `universal receiver' class of objects $R$ (Right-Angled Artin Groups, or RAAGs), each of which are simple and has good properties. The existence of such an embedding in itself implies property $P$ which you are interested in, maybe with some additional effort (Agol's fibering theorem, tameness, etc.). 
Proving a mathematical statement in this way makes a lot of sense, but I don't recall having seen this proof pattern before anywhere else in mathematics. Well, that's not entirely true- Cayley's Theorem that a group embeds in a permutation group has some corollaries (e.g.  Given a group $G$ and subgroup $H$ with $[G:H]=n$, there exists a exists a normal subgroup $N$ of $G$, with $N\subseteq H$ such that $[G:N]|n!$).

Question: Which other conjectures, that objects in a class $X$ have a property $P$, have been proven by embedding objects in $X$ nicely as subobjects of objects in a universal receiver $R$ whose good properties imply $P$ for objects in $X$?

For a compelling example, it would have to be difficult to prove $P$ for objects in $X$ in any other way. For an even more compelling example, the universal receiver $R$ would be surprising (RAAGs are a surprising universal receiver, I think).

REPLY [2 votes]: Part of the proof of the Green-Tao theorem involves embedding the primes into the "semiprimes", which are sufficiently pseudorandom for results like Szemerédi's theorem to apply. In fact, a lot of recent progress has been made with regards to the relative Szemerédi theorem, and maybe there is some hope that Erdős' conjecture on arithmetic progressions could be proven by embedding any sufficiently dense set of numbers into some pseudorandom set.