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Metamath Proof Explorer

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	$⊢ X = ⋃ J$
	Assertion	restid	$⊢ J \in V \to J ↾_{𝑡} X = J$

Proof

Step	Hyp	Ref	Expression
1		restid.1	$⊢ X = ⋃ J$
2		uniexg	$⊢ J \in V \to ⋃ J \in V$
3	1 2	eqeltrid	$⊢ J \in V \to X \in V$
4	1	eqimss2i	$⊢ ⋃ J \subseteq X$
5		sspwuni	$⊢ J \subseteq 𝒫 X \leftrightarrow ⋃ J \subseteq X$
6	4 5	mpbir	$⊢ J \subseteq 𝒫 X$
7		restid2	$⊢ (X \in V \land J \subseteq 𝒫 X) \to J ↾_{𝑡} X = J$
8	3 6 7	sylancl	$⊢ J \in V \to J ↾_{𝑡} X = J$