 Solution

The perturbative expansion of $Z[J]$ contains arbitrary Feynman diagrams, including disconnected products. The linked-cluster theorem givesbecause the factorials from repeated connected components reproduce the exponential series. Therefore $W[J]=-\hslash \log Z[J]$ is the sum of connected Feynman diagrams.

The Legendre transform removes diagrams that disconnect upon cutting one internal line. Equivalently, every connected diagram is a tree whose vertices are exact one-particle-irreducible vertices and whose edges are exact propagators. Thusand its functional derivatives are the one-particle-irreducible correlation functions. Algebraically, differentiating the Legendre relations givesso an exact quantum field theory propagator joining two proper vertices is precisely the inverse Hessian matrix needed to reconstruct connected diagrams.