Theorem topgrpbas

Description: The base set of a constructed topological group. (Contributed by Mario Carneiro, 29-Aug-2015)

	Ref	Expression
Hypothesis	topgrpfn.w	|- W = { <. ( Base ` ndx ) , B >. , <. ( +g ` ndx ) , .+ >. , <. ( TopSet ` ndx ) , J >. }
Assertion	topgrpbas	|- ( B e. X -> B = ( Base ` W ) )

Proof

Step	Hyp	Ref	Expression
1		topgrpfn.w	|- W = { <. ( Base ` ndx ) , B >. , <. ( +g ` ndx ) , .+ >. , <. ( TopSet ` ndx ) , J >. }
2	1	topgrpstr	|- W Struct <. 1 , 9 >.
3		baseid	|- Base = Slot ( Base ` ndx )
4		snsstp1	|- { <. ( Base ` ndx ) , B >. } C_ { <. ( Base ` ndx ) , B >. , <. ( +g ` ndx ) , .+ >. , <. ( TopSet ` ndx ) , J >. }
5	4 1	sseqtrri	|- { <. ( Base ` ndx ) , B >. } C_ W
6	2 3 5	strfv	|- ( B e. X -> B = ( Base ` W ) )