Definition df-tms 21415

Description: Define the function mapping a metric to a metric space. (Contributed by Mario Carneiro, 2-Sep-2015.)

Assertion

Ref	Expression
df-tms	|- toMetSp = ( d e. U. ran *Met |-> ( { <. ( Base ` ndx ) , dom dom d >. , <. ( dist ` ndx ) , d >. } sSet <. (TopSet ` ndx ) , ( MetOpen ` d ) >. ) )

Detailed syntax breakdown of Definition df-tms

Step	Hyp	Ref	Expression
1		ctmt 21412	. 2 class toMetSp
2		vd	. . 3 setvar d
3		cxmt 19032	. . . . 5 class *Met
4	3	crn 4840	. . . 4 class ran *Met
5	4	cuni 4190	. . 3 class U. ran *Met
6		cnx 15196	. . . . . . 7 class ndx
7		cbs 15199	. . . . . . 7 class Base
8	6, 7	cfv 5589	. . . . . 6 class ( Base ` ndx )
9	2	cv 1451	. . . . . . . 8 class d
10	9	cdm 4839	. . . . . . 7 class dom d
11	10	cdm 4839	. . . . . 6 class dom dom d
12	8, 11	cop 3965	. . . . 5 class <. ( Base ` ndx ) , dom dom d >.
13		cds 15277	. . . . . . 7 class dist
14	6, 13	cfv 5589	. . . . . 6 class ( dist ` ndx )
15	14, 9	cop 3965	. . . . 5 class <. ( dist ` ndx ) , d >.
16	12, 15	cpr 3961	. . . 4 class { <. ( Base ` ndx ) , dom dom d >. , <. ( dist ` ndx ) , d >. }
17		cts 15274	. . . . . 6 class TopSet
18	6, 17	cfv 5589	. . . . 5 class (TopSet ` ndx )
19		cmopn 19037	. . . . . 6 class MetOpen
20	9, 19	cfv 5589	. . . . 5 class ( MetOpen ` d )
21	18, 20	cop 3965	. . . 4 class <. (TopSet ` ndx ) , ( MetOpen ` d ) >.
22		csts 15197	. . . 4 class sSet
23	16, 21, 22	co 6308	. . 3 class ( { <. ( Base ` ndx ) , dom dom d >. , <. ( dist ` ndx ) , d >. } sSet <. (TopSet ` ndx ) , ( MetOpen ` d ) >. )
24	2, 5, 23	cmpt 4454	. 2 class ( d e. U. ran *Met |-> ( { <. ( Base ` ndx ) , dom dom d >. , <. ( dist ` ndx ) , d >. } sSet <. (TopSet ` ndx ) , ( MetOpen ` d ) >. ) )
25	1, 24	wceq 1452	1 wff toMetSp = ( d e. U. ran *Met |-> ( { <. ( Base ` ndx ) , dom dom d >. , <. ( dist ` ndx ) , d >. } sSet <. (TopSet ` ndx ) , ( MetOpen ` d ) >. ) )

This definition is referenced by:  tmsval  21574