Theorem restopn3

Description: If A is open, then A is open in the restriction to itself. (Contributed by Glauco Siliprandi, 21-Dec-2024)

	Ref	Expression
Assertion	restopn3	$$⊢\left(J\in \mathrm{Top}\wedge A\in J\right)\to A\in \left(J{\upharpoonright }_{𝑡}A\right)$$

Proof

Step	Hyp	Ref	Expression
1		simpr	$$⊢\left(J\in \mathrm{Top}\wedge A\in J\right)\to A\in J$$
2		ssidd	$$⊢\left(J\in \mathrm{Top}\wedge A\in J\right)\to A\subseteq A$$
3		restopn2	$$⊢\left(J\in \mathrm{Top}\wedge A\in J\right)\to \left(A\in \left(J{\upharpoonright }_{𝑡}A\right)\leftrightarrow \left(A\in J\wedge A\subseteq A\right)\right)$$
4	1 2 3	mpbir2and	$$⊢\left(J\in \mathrm{Top}\wedge A\in J\right)\to A\in \left(J{\upharpoonright }_{𝑡}A\right)$$