Theorem otpsbas

Description: The base set 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	otpsbas	|- ( B e. V -> B = ( Base ` 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		baseid	|- Base = Slot ( Base ` ndx )
4		snsstp1	|- { <. ( Base ` ndx ) , B >. } C_ { <. ( Base ` ndx ) , B >. , <. ( TopSet ` ndx ) , J >. , <. ( le ` ndx ) , .<_ >. }
5	4 1	sseqtrri	|- { <. ( Base ` ndx ) , B >. } C_ K
6	2 3 5	strfv	|- ( B e. V -> B = ( Base ` K ) )