Theorem uniretop

Description: The underlying set of the standard topology on the reals is the reals. (Contributed by FL, 4-Jun-2007)

	Ref	Expression
Assertion	uniretop	$$⊢ℝ=\bigcup \mathrm{topGen}\left(\mathrm{ran}\left(.\right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		unirnioo	$$⊢ℝ=\bigcup \mathrm{ran}\left(.\right)$$
2		retopbas	$$⊢\mathrm{ran}\left(.\right)\in \mathrm{TopBases}$$
3		unitg	$$⊢\mathrm{ran}\left(.\right)\in \mathrm{TopBases}\to \bigcup \mathrm{topGen}\left(\mathrm{ran}\left(.\right)\right)=\bigcup \mathrm{ran}\left(.\right)$$
4	2 3	ax-mp	$$⊢\bigcup \mathrm{topGen}\left(\mathrm{ran}\left(.\right)\right)=\bigcup \mathrm{ran}\left(.\right)$$
5	1 4	eqtr4i	$$⊢ℝ=\bigcup \mathrm{topGen}\left(\mathrm{ran}\left(.\right)\right)$$