explicit bijection between ${[2n+1]\choose n+1}$ and ${[2n+1]\choose n}$ mapping $A$ to a subset of $A$

Question

let ${A\choose k}=\{S\subseteq A: |S|=k\}$ and $[r]=\{1,\dots,r\}$.
Using Hall's marriage theorem one can prove there exists a bijection $f:{[2n+1]\choose n+1}\to{[2n+1]\choose n}$ so that $f(A)\subset A$ for all $A\in {[2n+1]\choose n+1}$.
Question. is there an explicit simple construction of such an $f$ (or its inverse) ?