Exponential transform of a standard Gumbel variable ($Y\sim \mathrm{Gumbel}(0,1)⟹e^{-Y}\sim \mathrm{Exp}(1)$)

If $Y$ has cumulative distribution function $F(y)=\exp (-e^{-y})$, put $Z=e^{-Y}$. For $z>0$, the decreasing transformation gives $\mathbb{P}(Z\le z)=1-F(-\ln z)=1-e^{-z}$, while this probability is zero for $z\le 0$. Hence $Z$ has the rate-one exponential distribution. Conversely, if $Z$ has that exponential distribution, then $-\ln Z$ has the standard Gumbel distribution.