Let $G$ be a connected split reductive group over a field k of characteristic zero with split maximal torus $T$ and Borel $B \supset T$. Additionally let $\Phi$ be the root system corresponding to $T$ with simple roots $\Delta$ corresponding to $B$.
Then there is an inclusion preserving bijection from subsets I of $\Delta$ to parabolic subgroups $P$ of $G$ containing $B$.
I was wondering if $$P_I= \bigcap_{\alpha \in \Delta\backslash I}P_{\Delta\backslash\{\alpha\}}$$ for $I \subseteq \Delta?$
So a proof may go like this:
The inclusion "$\subset$" is obvious by the inclusion preserving bijection. On the other hand all $P_{\Delta\backslash\{\alpha\}}$ contain $B$ hence there intersection is a closed subgroup containing $B$, thus parabolic. Therefore $\bigcap_{\alpha \in \Delta\backslash I}P_{\Delta\backslash\{\alpha\}}$=$P_J$ for some $I \subseteq J \subseteq \Delta$. Then we know that $P_J \subset P_{\Delta\backslash\{\alpha\}}$ for all $\alpha \in \Delta\backslash I$. By the inclusion preserving bijection
$$J\subset \bigcap_{\alpha \in \Delta\backslash I}\Delta\backslash\{\alpha\}=\Delta \backslash \bigcup_{\alpha \in \Delta\backslash I} \alpha =I. $$ Hence $I=J$.
If I have not made a mistake, I am surprised that this is not easy to find in the literature or does someone know a reference? Is this implicitly phrased by "inclusion preserving bijection?