I am looking for a derivation of Friedman's equations from the Cosmological Principle and Einstein's field equations. The text from which I am teaching myself (Schutz) does it all in two massive leaps, whose reasoning are respectively "it is easy to show that ..." (it isn't) and "the Einstein equations [in the FLRW metric, assuming Cosmo Principle] are easy to write down" (they are in fact incredibly time consuming, and hence very easy to make the odd error in, which stops things from cancelling out as needed). Other sources I have seen on the internet just skip the messy manipulation of Riemann and Ricci tensors and Christoffel symbols and just present the resul, or key steps in it, as a fait accompli. I'm sure there must be sources out there that work through this properly, but I just haven't been able to find them. Thanks.