Metamath Proof Explorer


Theorem otpstset

Description: The open sets of a topological ordered space. (Contributed by Mario Carneiro, 12-Nov-2015) (Revised by AV, 9-Sep-2021)

Ref Expression
Hypothesis otpsstr.w
|- K = { <. ( Base ` ndx ) , B >. , <. ( TopSet ` ndx ) , J >. , <. ( le ` ndx ) , .<_ >. }
Assertion otpstset
|- ( J e. V -> J = ( TopSet ` K ) )

Proof

Step Hyp Ref Expression
1 otpsstr.w
 |-  K = { <. ( Base ` ndx ) , B >. , <. ( TopSet ` ndx ) , J >. , <. ( le ` ndx ) , .<_ >. }
2 1 otpsstr
 |-  K Struct <. 1 , ; 1 0 >.
3 tsetid
 |-  TopSet = Slot ( TopSet ` ndx )
4 snsstp2
 |-  { <. ( TopSet ` ndx ) , J >. } C_ { <. ( Base ` ndx ) , B >. , <. ( TopSet ` ndx ) , J >. , <. ( le ` ndx ) , .<_ >. }
5 4 1 sseqtrri
 |-  { <. ( TopSet ` ndx ) , J >. } C_ K
6 2 3 5 strfv
 |-  ( J e. V -> J = ( TopSet ` K ) )