Theorem restid

Description: The subspace topology of the base set is the original topology. (Contributed by Jeff Hankins, 9-Jul-2009) (Revised by Mario Carneiro, 13-Aug-2015)

	Ref	Expression
Hypothesis	restid.1	$$⊢X=\bigcup J$$
Assertion	restid	$$⊢J\in V\to J{\upharpoonright }_{𝑡}X=J$$

Proof

Step	Hyp	Ref	Expression
1		restid.1	$$⊢X=\bigcup J$$
2		uniexg	$$⊢J\in V\to \bigcup J\in \mathrm{V}$$
3	1 2	eqeltrid	$$⊢J\in V\to X\in \mathrm{V}$$
4	1	eqimss2i	$$⊢\bigcup J\subseteq X$$
5		sspwuni	$$⊢J\subseteq 𝒫X\leftrightarrow \bigcup J\subseteq X$$
6	4 5	mpbir	$$⊢J\subseteq 𝒫X$$
7		restid2	$$⊢\left(X\in \mathrm{V}\wedge J\subseteq 𝒫X\right)\to J{\upharpoonright }_{𝑡}X=J$$
8	3 6 7	sylancl	$$⊢J\in V\to J{\upharpoonright }_{𝑡}X=J$$