Proof using Fubini

Solution 1:

Consider a function $ g:\mathbb{R}^{2}\rightarrow\mathbb{R} $ defined as follows : $$ g:\left(x,y\right)\mapsto\left\lbrace\begin{aligned}0,\ \ \ \ \text{If }\ x< y\leq 1\\ f\left(x,y\right),\ \ \ \ \text{If }\ 0\leq y \leq x\end{aligned}\right. $$

Then : $$ \small\int_{0}^{1}{\int_{0}^{x}{f\left(x,y\right)\mathrm{d}y}\,\mathrm{d}x}=\int_{0}^{1}{\int_{0}^{1}{g\left(x,y\right)\mathrm{d}y}\,\mathrm{d}x}=\int_{0}^{1}{\int_{0}^{1}{g\left(x,y\right)\mathrm{d}x}\,\mathrm{d}y}=\int_{0}^{1}{\int_{y}^{1}{f\left(x,y\right)\mathrm{d}x}\,\mathrm{d}y} $$

Solution 2:

\begin{align} \int^1_0\left(\int^x_0f(x,y)dy\right)dx &=\int^1_0\left(\int^1_0 1\{y\le x\}f(x,y)dy\right)dx \\ &=\int^1_0\left(\int^1_0 1\{y\le x\}f(x,y)dx\right)dy \\ &= \int^1_0\left(\int^1_yf(x,y)dx\right)dy. \end{align}