Metamath Proof Explorer

Theorem resthaus

Description: A subspace of a Hausdorff topology is Hausdorff. (Contributed by Mario Carneiro, 2-Mar-2015) (Proof shortened by Mario Carneiro, 25-Aug-2015)

		Ref	Expression
	Assertion	resthaus	$⊢ (J \in Haus \land A \in V) \to J ↾_{𝑡} A \in Haus$

Proof

Step	Hyp	Ref	Expression
1		haustop	$⊢ J \in Haus \to J \in Top$
2		cnhaus	$⊢ (J \in Haus \land ({I ↾}_{(A \cap ⋃ J)}) : A \cap ⋃ J \underset{1-1}{⟶} A \cap ⋃ J \land {I ↾}_{(A \cap ⋃ J)} \in ((J ↾_{𝑡} A) Cn J)) \to J ↾_{𝑡} A \in Haus$
3	1 2	resthauslem	$⊢ (J \in Haus \land A \in V) \to J ↾_{𝑡} A \in Haus$