Fisher information is non-increasing under well-behaved transformations

Question

I'm looking for a proof of the following fact:

Under assumptions of regularity by Cramer–Rao, if $T(X_1,\ldots,X_n)$ is an statistic whose induced distribution satisfies also de Cramer–Rao hypothesis then the fisher information satisfies $$\mathcal{I}_{T(X)}(\theta) \le I_X(\theta)$$ where $I_V$ is the Fisher information of variable $V$. The equality holds if and only if the statistic $T$ is sufficient.

Wikipedia proves the last part with the equation $\frac{d}{d\theta} \log(f(x\mid \theta) = \frac{d}{d\theta} \log(g(T(x)\mid\theta)$ where $g$ needs to be the density function of $T(X)$ by the factorization theorem by Halmos and Savage.

Answer 1

The proof of this fact appears in the book by Zacks, The Theory of Statistical Inference around page 184.