Definition df-ms 12509

Description: Define the (proper) class of metric spaces. (Contributed by NM, 27-Aug-2006.)

Assertion

Ref	Expression
df-ms	|- MetSp = { f e. *MetSp | ( ( dist ` f ) |` ( ( Base ` f ) X. ( Base ` f ) ) ) e. ( Met ` ( Base ` f ) ) }

Detailed syntax breakdown of Definition df-ms

Step	Hyp	Ref	Expression
1		cms 12506	. 2 class MetSp
2		vf	. . . . . . 7 setvar f
3	2	cv 1330	. . . . . 6 class f
4		cds 12030	. . . . . 6 class dist
5	3, 4	cfv 5123	. . . . 5 class ( dist ` f )
6		cbs 11959	. . . . . . 7 class Base
7	3, 6	cfv 5123	. . . . . 6 class ( Base ` f )
8	7, 7	cxp 4537	. . . . 5 class ( ( Base ` f ) X. ( Base ` f ) )
9	5, 8	cres 4541	. . . 4 class ( ( dist ` f ) |` ( ( Base ` f ) X. ( Base ` f ) ) )
10		cmet 12150	. . . . 5 class Met
11	7, 10	cfv 5123	. . . 4 class ( Met ` ( Base ` f ) )
12	9, 11	wcel 1480	. . 3 wff ( ( dist ` f ) |` ( ( Base ` f ) X. ( Base ` f ) ) ) e. ( Met ` ( Base ` f ) )
13		cxms 12505	. . 3 class *MetSp
14	12, 2, 13	crab 2420	. 2 class { f e. *MetSp | ( ( dist ` f ) |` ( ( Base ` f ) X. ( Base ` f ) ) ) e. ( Met ` ( Base ` f ) ) }
15	1, 14	wceq 1331	1 wff MetSp = { f e. *MetSp | ( ( dist ` f ) |` ( ( Base ` f ) X. ( Base ` f ) ) ) e. ( Met ` ( Base ` f ) ) }

This definition is referenced by:  isms  12622