The mixed and transitive properties of one-dimensional dynamic systems were generally discussed. Here, the $ (I,f) $ was defined as a topological dynamical system (referred to as a dynamical system or system), where $I$ was a compact metric space (metric was$d$) and $f:X \to X$ was a continuous map. Based on the ${\rm{Furstenberg}}$ family, several properties of the one-dimensional dynamical system $ (I,f) $ and its functional envelope system $({L^1}(I,I),{H_f})$ were studied. Based on the mixed and transitive properties studied by previous people, the concept of $\mathcal{F}$-mixed and $\mathcal{F}$-transitive were introduced. The results can help prove that if $ (I,f) $ is $\mathcal{F}$-mixed, and then $({L^1}(I,I),{H_f})$ is $\mathcal{F}$-transitive.