Definition df-lm 14777

Description: Define a function on topologies whose value is the convergence relation for sequences into the given topological space. Although f is typically a sequence (a function from an upperset of integers) with values in the topological space, it need not be. Note, however, that the limit property concerns only values at integers, so that the real-valued function ( x e. RR |-> ( sin ` ( pi x. x ) ) ) converges to zero (in the standard topology on the reals) with this definition. (Contributed by NM, 7-Sep-2006.)

Assertion

Ref	Expression
df-lm	|- ~~> t = ( j e. Top |-> { <. f , x >. | ( f e. ( U. j ^pm CC ) /\ x e. U. j /\ A. u e. j ( x e. u -> E. y e. ran ZZ>= ( f |` y ) : y --> u ) ) } )

Distinct variable group:   f, j, x, y, u

Detailed syntax breakdown of Definition df-lm

Step	Hyp	Ref	Expression
1		clm 14774	. 2 class ~~> t
2		vj	. . 3 setvar j
3		ctop 14584	. . 3 class Top
4		vf	. . . . . . 7 setvar f
5	4	cv 1372	. . . . . 6 class f
6	2	cv 1372	. . . . . . . 8 class j
7	6	cuni 3864	. . . . . . 7 class U. j
8		cc 7958	. . . . . . 7 class CC
9		cpm 6759	. . . . . . 7 class ^pm
10	7, 8, 9	co 5967	. . . . . 6 class ( U. j ^pm CC )
11	5, 10	wcel 2178	. . . . 5 wff f e. ( U. j ^pm CC )
12		vx	. . . . . . 7 setvar x
13	12	cv 1372	. . . . . 6 class x
14	13, 7	wcel 2178	. . . . 5 wff x e. U. j
15		vu	. . . . . . . 8 setvar u
16	12, 15	wel 2179	. . . . . . 7 wff x e. u
17		vy	. . . . . . . . . 10 setvar y
18	17	cv 1372	. . . . . . . . 9 class y
19	15	cv 1372	. . . . . . . . 9 class u
20	5, 18	cres 4695	. . . . . . . . 9 class ( f |` y )
21	18, 19, 20	wf 5286	. . . . . . . 8 wff ( f |` y ) : y --> u
22		cuz 9683	. . . . . . . . 9 class ZZ>=
23	22	crn 4694	. . . . . . . 8 class ran ZZ>=
24	21, 17, 23	wrex 2487	. . . . . . 7 wff E. y e. ran ZZ>= ( f |` y ) : y --> u
25	16, 24	wi 4	. . . . . 6 wff ( x e. u -> E. y e. ran ZZ>= ( f |` y ) : y --> u )
26	25, 15, 6	wral 2486	. . . . 5 wff A. u e. j ( x e. u -> E. y e. ran ZZ>= ( f |` y ) : y --> u )
27	11, 14, 26	w3a 981	. . . 4 wff ( f e. ( U. j ^pm CC ) /\ x e. U. j /\ A. u e. j ( x e. u -> E. y e. ran ZZ>= ( f |` y ) : y --> u ) )
28	27, 4, 12	copab 4120	. . 3 class { <. f , x >. | ( f e. ( U. j ^pm CC ) /\ x e. U. j /\ A. u e. j ( x e. u -> E. y e. ran ZZ>= ( f |` y ) : y --> u ) ) }
29	2, 3, 28	cmpt 4121	. 2 class ( j e. Top |-> { <. f , x >. | ( f e. ( U. j ^pm CC ) /\ x e. U. j /\ A. u e. j ( x e. u -> E. y e. ran ZZ>= ( f |` y ) : y --> u ) ) } )
30	1, 29	wceq 1373	1 wff ~~> t = ( j e. Top |-> { <. f , x >. | ( f e. ( U. j ^pm CC ) /\ x e. U. j /\ A. u e. j ( x e. u -> E. y e. ran ZZ>= ( f |` y ) : y --> u ) ) } )

This definition is referenced by:  lmrcl  14778  lmfval  14779