Theorem restuni

Description: The underlying set of a subspace topology. (Contributed by FL, 5-Jan-2009) (Revised by Mario Carneiro, 13-Aug-2015)

	Ref	Expression
Hypothesis	restuni.1	$$⊢X=\bigcup J$$
Assertion	restuni	$$⊢\left(J\in \mathrm{Top}\wedge A\subseteq X\right)\to A=\bigcup \left(J{\upharpoonright }_{𝑡}A\right)$$

Proof

Step	Hyp	Ref	Expression
1		restuni.1	$$⊢X=\bigcup J$$
2	1	toptopon	$$⊢J\in \mathrm{Top}\leftrightarrow J\in \mathrm{TopOn}\left(X\right)$$
3		resttopon	$$⊢\left(J\in \mathrm{TopOn}\left(X\right)\wedge A\subseteq X\right)\to J{\upharpoonright }_{𝑡}A\in \mathrm{TopOn}\left(A\right)$$
4	2 3	sylanb	$$⊢\left(J\in \mathrm{Top}\wedge A\subseteq X\right)\to J{\upharpoonright }_{𝑡}A\in \mathrm{TopOn}\left(A\right)$$
5		toponuni	$$⊢J{\upharpoonright }_{𝑡}A\in \mathrm{TopOn}\left(A\right)\to A=\bigcup \left(J{\upharpoonright }_{𝑡}A\right)$$
6	4 5	syl	$$⊢\left(J\in \mathrm{Top}\wedge A\subseteq X\right)\to A=\bigcup \left(J{\upharpoonright }_{𝑡}A\right)$$