The iterated function system $(f_1,f_2,\cdots, f_n)$ satisfies Moran's open set condition iff there exists a nonempty open set $U$ for which we hacve $f_i(U)\cap f_j(U)=\emptyset$ for $i\neq j$ and $U\supset f_i(U)$. Such an open set $U$ will be called a Moran open set for the iterated function system.
I have a question related with some intersection properties when a Moran open set is iterated. Here I write what it seems to be the context of the question.
Le $E$ be an alphabet with $n$ letters. Write ratio list as $(r_e)_{e\in R}$ and the iterated function system as $(f_e)_{e\in E}$. Note that if $U$ is a Moran open set of $(f_e)_{e\in E}$ then $f_\alpha(U)\cap f_\beta(U)=\emptyset$ for two strings $\alpha,\beta \in E^{*}$ unless one is an initial segment of the other. If $\alpha$ has length $n\geq 1$ we set $\alpha^-=\alpha\upharpoonright (n-1)$.
Let $K$ be the invariant set of the iterated system $(f_e)_{e\in E}$. If $A\subset K$ define
$$T=\{\alpha\in E^*:\overline{f_\alpha(U)}\cap A\neq \emptyset,\text{diam }f_\alpha(U)<\text{diam }A\leq \text{diam }\text{diam }f_\alpha^{-}(U)\}$$
I want to know why the inclusion $A\subset \bigcup_{\alpha\in T}f_\alpha (U)$ holds when $A$ is a Borel set of $K$.
This inclusion is used to prove that relation between the Moran's set condition in $\mathbb{R}^n$ and the Hausdorff dimension in the Gerald Edgar's book Measure, Topology and Fractal Geometry.