Reference for $\operatorname E\left[f\circ X\mid X^{-1}(\mathcal F)\right]=\operatorname E_{\mathcal L(X)}\left[f\mid\mathcal F\right]\circ X$

Question

Let

• $(\Omega,\mathcal A,\operatorname P)$ be a probability space;
• $(E,\mathcal E)$ be a measurable space;
• $X$ be an $(E,\mathcal E)$-valued random variable on $(\Omega,\mathcal A,\operatorname P)$;
• $\mathcal L(X)$ denote the distribution of $X$ with respect to $\operatorname P$;
• $\operatorname E_{\mathcal L(X)}$ denote the expectation with respect to $\mathcal L(X)$;
• $\mathcal F\subseteq\mathcal E$ be a $\sigma$-algebra on $E$;
• $f:E\to\mathbb R$ be bounded and $\mathcal E$-measurable.

We can easily show that $$\operatorname E\left[f\circ X\mid X^{-1}(\mathcal F)\right]=\operatorname E_{\mathcal L(X)}\left[f\mid\mathcal F\right]\circ X\tag1,$$ but I wasn't able to find this result in any textbook.

Is there a commonly used name for the identity $(1)$ and does anyone have a reference for this result?