Theorem setsmsbas

Description: The base set of a constructed metric space. (Contributed by Mario Carneiro, 28-Aug-2015) (Proof shortened by AV, 12-Nov-2024)

	Ref	Expression
Hypotheses	setsms.x	|- ( ph -> X = ( Base ` M ) )
	setsms.d	|- ( ph -> D = ( ( dist ` M ) |` ( X X. X ) ) )
	setsms.k	|- ( ph -> K = ( M sSet <. ( TopSet ` ndx ) , ( MetOpen ` D ) >. ) )
Assertion	setsmsbas	|- ( ph -> X = ( Base ` K ) )

Proof

Step	Hyp	Ref	Expression
1		setsms.x	|- ( ph -> X = ( Base ` M ) )
2		setsms.d	|- ( ph -> D = ( ( dist ` M ) |` ( X X. X ) ) )
3		setsms.k	|- ( ph -> K = ( M sSet <. ( TopSet ` ndx ) , ( MetOpen ` D ) >. ) )
4		baseid	|- Base = Slot ( Base ` ndx )
5		tsetndxnbasendx	|- ( TopSet ` ndx ) =/= ( Base ` ndx )
6	5	necomi	|- ( Base ` ndx ) =/= ( TopSet ` ndx )
7	4 6	setsnid	|- ( Base ` M ) = ( Base ` ( M sSet <. ( TopSet ` ndx ) , ( MetOpen ` D ) >. ) )
8	3	fveq2d	|- ( ph -> ( Base ` K ) = ( Base ` ( M sSet <. ( TopSet ` ndx ) , ( MetOpen ` D ) >. ) ) )
9	7 1 8	3eqtr4a	|- ( ph -> X = ( Base ` K ) )