FT of the $\delta(t)$:
$$\delta(\omega)=\frac 1 {2\pi}\int_{- \infty}^{+ \infty}\delta(t)e^{i\omega t}dt=\frac 1 {2\pi}$$
The RFT:
$$\delta(t)=\int_{- \infty}^{+ \infty} \delta(\omega)e^{-i\omega t}d\omega=\frac 1{2\pi} \int_{- \infty}^{+ \infty} e^{-i\omega t}d\omega $$
Then the following is written:
$$\int_{- \infty}^{+ \infty}e^{i\omega t}dt=2\pi\delta(\omega)$$
Where does this last equation comes from?
The $\delta(\omega)$ in your final formula is not the $\delta(\omega)$ in your first formula, but instead is the $\delta(t)$ in your second formula with $t$ replaced by $\omega$. Just take your second formula and swap every $t$ and $\omega$ and multiply by $2\pi$.
It is not great notation to use the same symbol for a function and its Fourier transform. Instead of $\delta(\omega)$ it might be helpful to write
$$\hat\delta(\omega)=\frac{1}{2\pi}\ ,$$ or $$\mathcal{F}[\delta](\omega)=\frac{1}{2\pi}\ .$$