Theorem isms

Description: Express the predicate " <. X , D >. is a metric space" with underlying set X and distance function D . (Contributed by NM, 27-Aug-2006) (Revised by Mario Carneiro, 24-Aug-2015)

	Ref	Expression
Hypotheses	isms.j	|- J = ( TopOpen ` K )
	isms.x	|- X = ( Base ` K )
	isms.d	|- D = ( ( dist ` K ) |` ( X X. X ) )
Assertion	isms	|- ( K e. MetSp <-> ( K e. *MetSp /\ D e. ( Met ` X ) ) )

Proof

Step	Hyp	Ref	Expression
1		isms.j	|- J = ( TopOpen ` K )
2		isms.x	|- X = ( Base ` K )
3		isms.d	|- D = ( ( dist ` K ) |` ( X X. X ) )
4		fveq2	|- ( f = K -> ( dist ` f ) = ( dist ` K ) )
5		fveq2	|- ( f = K -> ( Base ` f ) = ( Base ` K ) )
6	5 2	eqtr4di	|- ( f = K -> ( Base ` f ) = X )
7	6	sqxpeqd	|- ( f = K -> ( ( Base ` f ) X. ( Base ` f ) ) = ( X X. X ) )
8	4 7	reseq12d	|- ( f = K -> ( ( dist ` f ) |` ( ( Base ` f ) X. ( Base ` f ) ) ) = ( ( dist ` K ) |` ( X X. X ) ) )
9	8 3	eqtr4di	|- ( f = K -> ( ( dist ` f ) |` ( ( Base ` f ) X. ( Base ` f ) ) ) = D )
10	6	fveq2d	|- ( f = K -> ( Met ` ( Base ` f ) ) = ( Met ` X ) )
11	9 10	eleq12d	|- ( f = K -> ( ( ( dist ` f ) |` ( ( Base ` f ) X. ( Base ` f ) ) ) e. ( Met ` ( Base ` f ) ) <-> D e. ( Met ` X ) ) )
12		df-ms	|- MetSp = { f e. *MetSp | ( ( dist ` f ) |` ( ( Base ` f ) X. ( Base ` f ) ) ) e. ( Met ` ( Base ` f ) ) }
13	11 12	elrab2	|- ( K e. MetSp <-> ( K e. *MetSp /\ D e. ( Met ` X ) ) )