Metamath Proof Explorer


Theorem restsn

Description: The only subspace topology induced by the topology { (/) } . (Contributed by FL, 5-Jan-2009) (Revised by Mario Carneiro, 15-Dec-2013)

Ref Expression
Assertion restsn A V 𝑡 A =

Proof

Step Hyp Ref Expression
1 sn0top Top
2 elrest Top A V x 𝑡 A y x = y A
3 1 2 mpan A V x 𝑡 A y x = y A
4 0ex V
5 ineq1 y = y A = A
6 0in A =
7 5 6 syl6eq y = y A =
8 7 eqeq2d y = x = y A x =
9 4 8 rexsn y x = y A x =
10 velsn x x =
11 9 10 bitr4i y x = y A x
12 3 11 syl6bb A V x 𝑡 A x
13 12 eqrdv A V 𝑡 A =