Definition df-mopn 19742

Description: Define a function whose value is the family of open sets of a metric space. See elmopn 22247 for its main property. (Contributed by NM, 1-Sep-2006.)

Assertion

Ref	Expression
df-mopn	|- MetOpen = ( d e. U. ran *Met |-> ( topGen ` ran ( ball ` d ) ) )

Detailed syntax breakdown of Definition df-mopn

Step	Hyp	Ref	Expression
1		cmopn 19736	. 2 class MetOpen
2		vd	. . 3 setvar d
3		cxmt 19731	. . . . 5 class *Met
4	3	crn 5115	. . . 4 class ran *Met
5	4	cuni 4436	. . 3 class U. ran *Met
6	2	cv 1482	. . . . . 6 class d
7		cbl 19733	. . . . . 6 class ball
8	6, 7	cfv 5888	. . . . 5 class ( ball ` d )
9	8	crn 5115	. . . 4 class ran ( ball ` d )
10		ctg 16098	. . . 4 class topGen
11	9, 10	cfv 5888	. . 3 class ( topGen ` ran ( ball ` d ) )
12	2, 5, 11	cmpt 4729	. 2 class ( d e. U. ran *Met |-> ( topGen ` ran ( ball ` d ) ) )
13	1, 12	wceq 1483	1 wff MetOpen = ( d e. U. ran *Met |-> ( topGen ` ran ( ball ` d ) ) )

This definition is referenced by:  mopnval  22243  isxms2  22253  setsmstopn  22283  tngtopn  22454