Fourier and its inverse

Question

Im slightly confused as to how to prove the inverse fourier is true, if we have: $$F(\omega)=\mathcal{F}[f(x)]=\int_{-\infty}^\infty f(x)e^{-j\omega x}dx$$ $$f(x)=\mathcal{F}^{-1}[F(\omega)]=\frac1{2\pi}\int_{-\infty}^\infty F(\omega)e^{j\omega x}d\omega$$ then it should be that: $$\mathcal{F}^{-1}[\mathcal{F}[f(x)]]=f(x)$$ and so: $$\frac1{2\pi}\int_{-\infty}^\infty\left(\int_{-\infty}^\infty f(y)e^{-j\omega y}dy\right)e^{j\omega x}d\omega=\frac1{2\pi}\int\limits_{-\infty}^\infty\int\limits_{-\infty}^\infty f(y)e^{j\omega(x-y)}dyd\omega=f(x)$$ but I cannot find a nice way to simplify this or evaluate the general case. Any ideas?