Theorem isopn3i

Description: An open subset equals its own interior. (Contributed by Mario Carneiro, 30-Dec-2016)

	Ref	Expression
Assertion	isopn3i	$$⊢\left(J\in \mathrm{Top}\wedge S\in J\right)\to \mathrm{int}\left(J\right)\left(S\right)=S$$

Proof

Step	Hyp	Ref	Expression
1		simpr	$$⊢\left(J\in \mathrm{Top}\wedge S\in J\right)\to S\in J$$
2		elssuni	$$⊢S\in J\to S\subseteq \bigcup J$$
3		eqid	$$⊢\bigcup J=\bigcup J$$
4	3	isopn3	$$⊢\left(J\in \mathrm{Top}\wedge S\subseteq \bigcup J\right)\to \left(S\in J\leftrightarrow \mathrm{int}\left(J\right)\left(S\right)=S\right)$$
5	2 4	sylan2	$$⊢\left(J\in \mathrm{Top}\wedge S\in J\right)\to \left(S\in J\leftrightarrow \mathrm{int}\left(J\right)\left(S\right)=S\right)$$
6	1 5	mpbid	$$⊢\left(J\in \mathrm{Top}\wedge S\in J\right)\to \mathrm{int}\left(J\right)\left(S\right)=S$$