Equivalence of Ising and Random Cluster Model Partition Functions

Question

It is well documented that the partition function of the Ising model is equivalent to the partition function of the Random Cluster model. However I cannot seem to find a resource that actually shows this. To be specific, let $G=(V,E)$ be some graph and $$Z_I = \Sigma_{\sigma: V\rightarrow \{-1,1\}}e^{-\beta H(\sigma)},\: H(\sigma) = -J\Sigma_{xy\in E} \sigma(x)\sigma(y),$$ where $\beta, J \in [0,\infty)$ and $$Z_{RC} = \Sigma_{\omega:E \rightarrow \{0,1\}}(\Pi_{e\in E}p^{\omega(e)}(1-p)^{1-\omega(e)})q^{k(\omega)},$$ where $p\in[0,1],q\in (0,\infty)$ and $k(\omega)$ is the number of connected components of the graph $(V,\{e\in E: \omega(e) = 1\})$ . Then how would one go about showing that $Z_{I} = Z_{RC}$ for some suitable constants? Also, direction towards any resources properly showing this would be appreciated too.