Definition df-cms 23132

Description: Define the class of all complete metric spaces. (Contributed by Mario Carneiro, 15-Oct-2015.)

Assertion

Ref	Expression
df-cms	|- CMetSp = { w e. MetSp | [. ( Base ` w ) / b ]. ( ( dist ` w ) |` ( b X. b ) ) e. ( CMet ` b ) }

Distinct variable group:   w, b

Detailed syntax breakdown of Definition df-cms

Step	Hyp	Ref	Expression
1		ccms 23129	. 2 class CMetSp
2		vw	. . . . . . . 8 setvar w
3	2	cv 1482	. . . . . . 7 class w
4		cds 15950	. . . . . . 7 class dist
5	3, 4	cfv 5888	. . . . . 6 class ( dist ` w )
6		vb	. . . . . . . 8 setvar b
7	6	cv 1482	. . . . . . 7 class b
8	7, 7	cxp 5112	. . . . . 6 class ( b X. b )
9	5, 8	cres 5116	. . . . 5 class ( ( dist ` w ) |` ( b X. b ) )
10		cms 23052	. . . . . 6 class CMet
11	7, 10	cfv 5888	. . . . 5 class ( CMet ` b )
12	9, 11	wcel 1990	. . . 4 wff ( ( dist ` w ) |` ( b X. b ) ) e. ( CMet ` b )
13		cbs 15857	. . . . 5 class Base
14	3, 13	cfv 5888	. . . 4 class ( Base ` w )
15	12, 6, 14	wsbc 3435	. . 3 wff [. ( Base ` w ) / b ]. ( ( dist ` w ) |` ( b X. b ) ) e. ( CMet ` b )
16		cmt 22123	. . 3 class MetSp
17	15, 2, 16	crab 2916	. 2 class { w e. MetSp | [. ( Base ` w ) / b ]. ( ( dist ` w ) |` ( b X. b ) ) e. ( CMet ` b ) }
18	1, 17	wceq 1483	1 wff CMetSp = { w e. MetSp | [. ( Base ` w ) / b ]. ( ( dist ` w ) |` ( b X. b ) ) e. ( CMet ` b ) }

This definition is referenced by:  iscms  23142