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Metamath Proof Explorer

Theorem restt1

Description: A subspace of a T_1 topology is T_1. (Contributed by Mario Carneiro, 25-Aug-2015)

		Ref	Expression
	Assertion	restt1	$⊢ (J \in Fre \land A \in V) \to J ↾_{𝑡} A \in Fre$

Step	Hyp	Ref	Expression
1		t1top	$⊢ J \in Fre \to J \in Top$
2		cnt1	$⊢ (J \in Fre \land ({I ↾}_{(A \cap ⋃ J)}) : A \cap ⋃ J \underset{1-1}{⟶} A \cap ⋃ J \land {I ↾}_{(A \cap ⋃ J)} \in ((J ↾_{𝑡} A) Cn J)) \to J ↾_{𝑡} A \in Fre$
3	1 2	resthauslem	$⊢ (J \in Fre \land A \in V) \to J ↾_{𝑡} A \in Fre$