Metamath Proof Explorer


Theorem elrest

Description: The predicate "is an open set of a subspace topology". (Contributed by FL, 5-Jan-2009) (Revised by Mario Carneiro, 15-Dec-2013)

Ref Expression
Assertion elrest J V B W A J 𝑡 B x J A = x B

Proof

Step Hyp Ref Expression
1 restval J V B W J 𝑡 B = ran x J x B
2 1 eleq2d J V B W A J 𝑡 B A ran x J x B
3 eqid x J x B = x J x B
4 vex x V
5 4 inex1 x B V
6 3 5 elrnmpti A ran x J x B x J A = x B
7 2 6 syl6bb J V B W A J 𝑡 B x J A = x B