In the proof I'm trying to elaborate I want to write the infimum/minimum as $$\inf\{\alpha,e,f\}=\int_0^\alpha \chi_{(-\infty,\inf(e,f)]}(z)dz,$$ where $\alpha,e,f\in \mathbb{R}$ and $\chi$ is the indicator function, because then it would make the rest easier.
Is it enough to argue that $$\int_0^\alpha \chi_{(-\infty,\inf(e,f)]}(z)dz=\left\{ \begin{array}{lc} \alpha &,\ \text{if} \ \alpha \leq \inf\{e,f\} \\ \inf\{e,f\}&,\ \text{if} \ \alpha > \inf\{e,f\} \end{array}? \right. $$
Could you please answer if it is the case or not?
Thanks in advance!