Definition df-lm 15291

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 15288	. 2 class ~~> t
2		vj	. . 3 setvar j
3		ctop 15098	. . 3 class Top
4		vf	. . . . . . 7 setvar f
5	4	cv 1401	. . . . . 6 class f
6	2	cv 1401	. . . . . . . 8 class j
7	6	cuni 3935	. . . . . . 7 class U. j
8		cc 8177	. . . . . . 7 class CC
9		cpm 6923	. . . . . . 7 class ^pm
10	7, 8, 9	co 6085	. . . . . 6 class ( U. j ^pm CC )
11	5, 10	wcel 2209	. . . . 5 wff f e. ( U. j ^pm CC )
12		vx	. . . . . . 7 setvar x
13	12	cv 1401	. . . . . 6 class x
14	13, 7	wcel 2209	. . . . 5 wff x e. U. j
15		vu	. . . . . . . 8 setvar u
16	12, 15	wel 2210	. . . . . . 7 wff x e. u
17		vy	. . . . . . . . . 10 setvar y
18	17	cv 1401	. . . . . . . . 9 class y
19	15	cv 1401	. . . . . . . . 9 class u
20	5, 18	cres 4776	. . . . . . . . 9 class ( f |` y )
21	18, 19, 20	wf 5373	. . . . . . . 8 wff ( f |` y ) : y --> u
22		cuz 9921	. . . . . . . . 9 class ZZ>=
23	22	crn 4775	. . . . . . . 8 class ran ZZ>=
24	21, 17, 23	wrex 2529	. . . . . . 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 2528	. . . . 5 wff A. u e. j ( x e. u -> E. y e. ran ZZ>= ( f |` y ) : y --> u )
27	11, 14, 26	w3a 1009	. . . 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 4191	. . 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 4192	. 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 1402	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 used by:  lmrel  15292  lmrcl  15293  lmfval  15294