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 e. Fre /\ A e. V ) -> ( J |`t A ) e. Fre )

Proof

Step Hyp Ref Expression
1 t1top
 |-  ( J e. Fre -> J e. Top )
2 cnt1
 |-  ( ( J e. Fre /\ ( _I |` ( A i^i U. J ) ) : ( A i^i U. J ) -1-1-> ( A i^i U. J ) /\ ( _I |` ( A i^i U. J ) ) e. ( ( J |`t A ) Cn J ) ) -> ( J |`t A ) e. Fre )
3 1 2 resthauslem
 |-  ( ( J e. Fre /\ A e. V ) -> ( J |`t A ) e. Fre )