mathematical accurate definition of the binary independence model
It's better to talk about x, q and R as (random) events - set of outcomes of the random experiment. x and q will be one element sets but R is an event denoting x is relevant to q and thus it is a subset of cartesian product X x Q (pair of a document and a query).

Comma is then set conjunction and P(A,B) = P(A ^ B) which equals to P(A) * P(B) when A is independent of B.

The wiki statement 'by using Bayesian rule' is a bit of a shortcut since you need to apply it several times. To derive the above formula for P(R|x,q), I would start with conditional probability definition (which is the root of Bayesian rule):

P(A|B) = P(A ^ B) / P(B)

Then:

P(R|x,q) = P(R ^ x ^ q) / P(x,q) = P(x|R,q) * P(R,q) / [ P(x|q) * P(q) ] =

= P(x|R,q) * P(R|q) / P(q) / [ P(x|q) * P(q) ]

When you divide nominator and denominator by P(q), you obtain 

P(R|x,q) = P(x|R,q) * P(R|q) / P(x|q)