Definition df-xms 13701

Description: Define the (proper) class of extended metric spaces. (Contributed by Mario Carneiro, 2-Sep-2015.)

Assertion

Ref	Expression
df-xms	|- *MetSp = { f e. TopSp | ( TopOpen ` f ) = ( MetOpen ` ( ( dist ` f ) |` ( ( Base ` f ) X. ( Base ` f ) ) ) ) }

Detailed syntax breakdown of Definition df-xms

Step	Hyp	Ref	Expression
1		cxms 13698	. 2 class *MetSp
2		vf	. . . . . 6 setvar f
3	2	cv 1352	. . . . 5 class f
4		ctopn 12676	. . . . 5 class TopOpen
5	3, 4	cfv 5214	. . . 4 class ( TopOpen ` f )
6		cds 12536	. . . . . . 7 class dist
7	3, 6	cfv 5214	. . . . . 6 class ( dist ` f )
8		cbs 12453	. . . . . . . 8 class Base
9	3, 8	cfv 5214	. . . . . . 7 class ( Base ` f )
10	9, 9	cxp 4623	. . . . . 6 class ( ( Base ` f ) X. ( Base ` f ) )
11	7, 10	cres 4627	. . . . 5 class ( ( dist ` f ) |` ( ( Base ` f ) X. ( Base ` f ) ) )
12		cmopn 13305	. . . . 5 class MetOpen
13	11, 12	cfv 5214	. . . 4 class ( MetOpen ` ( ( dist ` f ) |` ( ( Base ` f ) X. ( Base ` f ) ) ) )
14	5, 13	wceq 1353	. . 3 wff ( TopOpen ` f ) = ( MetOpen ` ( ( dist ` f ) |` ( ( Base ` f ) X. ( Base ` f ) ) ) )
15		ctps 13390	. . 3 class TopSp
16	14, 2, 15	crab 2459	. 2 class { f e. TopSp | ( TopOpen ` f ) = ( MetOpen ` ( ( dist ` f ) |` ( ( Base ` f ) X. ( Base ` f ) ) ) ) }
17	1, 16	wceq 1353	1 wff *MetSp = { f e. TopSp | ( TopOpen ` f ) = ( MetOpen ` ( ( dist ` f ) |` ( ( Base ` f ) X. ( Base ` f ) ) ) ) }

This definition is referenced by:  isxms  13813