Definition df-lm 14426

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 14423	. 2 class ~~> t
2		vj	. . 3 setvar j
3		ctop 14233	. . 3 class Top
4		vf	. . . . . . 7 setvar f
5	4	cv 1363	. . . . . 6 class f
6	2	cv 1363	. . . . . . . 8 class j
7	6	cuni 3839	. . . . . . 7 class U. j
8		cc 7877	. . . . . . 7 class CC
9		cpm 6708	. . . . . . 7 class ^pm
10	7, 8, 9	co 5922	. . . . . 6 class ( U. j ^pm CC )
11	5, 10	wcel 2167	. . . . 5 wff f e. ( U. j ^pm CC )
12		vx	. . . . . . 7 setvar x
13	12	cv 1363	. . . . . 6 class x
14	13, 7	wcel 2167	. . . . 5 wff x e. U. j
15		vu	. . . . . . . 8 setvar u
16	12, 15	wel 2168	. . . . . . 7 wff x e. u
17		vy	. . . . . . . . . 10 setvar y
18	17	cv 1363	. . . . . . . . 9 class y
19	15	cv 1363	. . . . . . . . 9 class u
20	5, 18	cres 4665	. . . . . . . . 9 class ( f |` y )
21	18, 19, 20	wf 5254	. . . . . . . 8 wff ( f |` y ) : y --> u
22		cuz 9601	. . . . . . . . 9 class ZZ>=
23	22	crn 4664	. . . . . . . 8 class ran ZZ>=
24	21, 17, 23	wrex 2476	. . . . . . 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 2475	. . . . 5 wff A. u e. j ( x e. u -> E. y e. ran ZZ>= ( f |` y ) : y --> u )
27	11, 14, 26	w3a 980	. . . 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 4093	. . 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 4094	. 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 1364	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  14427  lmfval  14428