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Definition df-lm 13830
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
StepHypRef Expression
1 clm 13827 . 2  class  ~~> t
2 vj . . 3  setvar  j
3 ctop 13637 . . 3  class  Top
4 vf . . . . . . 7  setvar  f
54cv 1352 . . . . . 6  class  f
62cv 1352 . . . . . . . 8  class  j
76cuni 3811 . . . . . . 7  class  U. j
8 cc 7812 . . . . . . 7  class  CC
9 cpm 6652 . . . . . . 7  class  ^pm
107, 8, 9co 5878 . . . . . 6  class  ( U. j  ^pm  CC )
115, 10wcel 2148 . . . . 5  wff  f  e.  ( U. j  ^pm  CC )
12 vx . . . . . . 7  setvar  x
1312cv 1352 . . . . . 6  class  x
1413, 7wcel 2148 . . . . 5  wff  x  e. 
U. j
15 vu . . . . . . . 8  setvar  u
1612, 15wel 2149 . . . . . . 7  wff  x  e.  u
17 vy . . . . . . . . . 10  setvar  y
1817cv 1352 . . . . . . . . 9  class  y
1915cv 1352 . . . . . . . . 9  class  u
205, 18cres 4630 . . . . . . . . 9  class  ( f  |`  y )
2118, 19, 20wf 5214 . . . . . . . 8  wff  ( f  |`  y ) : y --> u
22 cuz 9531 . . . . . . . . 9  class  ZZ>=
2322crn 4629 . . . . . . . 8  class  ran  ZZ>=
2421, 17, 23wrex 2456 . . . . . . 7  wff  E. y  e.  ran  ZZ>= ( f  |`  y ) : y --> u
2516, 24wi 4 . . . . . 6  wff  ( x  e.  u  ->  E. y  e.  ran  ZZ>= ( f  |`  y ) : y --> u )
2625, 15, 6wral 2455 . . . . 5  wff  A. u  e.  j  ( x  e.  u  ->  E. y  e.  ran  ZZ>= ( f  |`  y ) : y --> u )
2711, 14, 26w3a 978 . . . 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 ) )
2827, 4, 12copab 4065 . . 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 ) ) }
292, 3, 28cmpt 4066 . 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 ) ) } )
301, 29wceq 1353 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 ) ) } )
Colors of variables: wff set class
This definition is referenced by:  lmrcl  13831  lmfval  13832
  Copyright terms: Public domain W3C validator