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Definition df-lm 17215
Description: Define a function on topologies whose value is the convergence relation for the space. Although  f is typically a function from upper integers to the topological space, it doesn't have to be. Unfortunately, the value of the function must exist to use fvmpt 5745, and we use the otherwise unnecessary conjunct  dom  f  C_  CC to ensure that. (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 17212 . 2  class  ~~> t
2 vj . . 3  set  j
3 ctop 16881 . . 3  class  Top
4 vf . . . . . . 7  set  f
54cv 1648 . . . . . 6  class  f
62cv 1648 . . . . . . . 8  class  j
76cuni 3957 . . . . . . 7  class  U. j
8 cc 8921 . . . . . . 7  class  CC
9 cpm 6955 . . . . . . 7  class  ^pm
107, 8, 9co 6020 . . . . . 6  class  ( U. j  ^pm  CC )
115, 10wcel 1717 . . . . 5  wff  f  e.  ( U. j  ^pm  CC )
12 vx . . . . . . 7  set  x
1312cv 1648 . . . . . 6  class  x
1413, 7wcel 1717 . . . . 5  wff  x  e. 
U. j
15 vu . . . . . . . 8  set  u
1612, 15wel 1718 . . . . . . 7  wff  x  e.  u
17 vy . . . . . . . . . 10  set  y
1817cv 1648 . . . . . . . . 9  class  y
1915cv 1648 . . . . . . . . 9  class  u
205, 18cres 4820 . . . . . . . . 9  class  ( f  |`  y )
2118, 19, 20wf 5390 . . . . . . . 8  wff  ( f  |`  y ) : y --> u
22 cuz 10420 . . . . . . . . 9  class  ZZ>=
2322crn 4819 . . . . . . . 8  class  ran  ZZ>=
2421, 17, 23wrex 2650 . . . . . . 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 2649 . . . . 5  wff  A. u  e.  j  ( x  e.  u  ->  E. y  e.  ran  ZZ>= ( f  |`  y ) : y --> u )
2711, 14, 26w3a 936 . . . 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 4206 . . 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 4207 . 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 1649 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:  lmrel  17216  lmrcl  17217  lmfval  17218
  Copyright terms: Public domain W3C validator