Metamath Proof Explorer


Theorem tsettps

Description: If the topology component is already correctly truncated, then it forms a topological space (with the topology extractor function coming out the same as the component). (Contributed by Mario Carneiro, 13-Aug-2015)

Ref Expression
Hypotheses tsettps.a A=BaseK
tsettps.j J=TopSetK
Assertion tsettps JTopOnAKTopSp

Proof

Step Hyp Ref Expression
1 tsettps.a A=BaseK
2 tsettps.j J=TopSetK
3 1 2 topontopn JTopOnAJ=TopOpenK
4 id JTopOnAJTopOnA
5 3 4 eqeltrrd JTopOnATopOpenKTopOnA
6 eqid TopOpenK=TopOpenK
7 1 6 istps KTopSpTopOpenKTopOnA
8 5 7 sylibr JTopOnAKTopSp