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Definition df-lm 17247
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 5765, 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 17244 . 2  class  ~~> t
2 vj . . 3  set  j
3 ctop 16913 . . 3  class  Top
4 vf . . . . . . 7  set  f
54cv 1648 . . . . . 6  class  f
62cv 1648 . . . . . . . 8  class  j
76cuni 3975 . . . . . . 7  class  U. j
8 cc 8944 . . . . . . 7  class  CC
9 cpm 6978 . . . . . . 7  class  ^pm
107, 8, 9co 6040 . . . . . 6  class  ( U. j  ^pm  CC )
115, 10wcel 1721 . . . . 5  wff  f  e.  ( U. j  ^pm  CC )
12 vx . . . . . . 7  set  x
1312cv 1648 . . . . . 6  class  x
1413, 7wcel 1721 . . . . 5  wff  x  e. 
U. j
15 vu . . . . . . . 8  set  u
1612, 15wel 1722 . . . . . . 7  wff  x  e.  u
17 vy . . . . . . . . . 10  set  y
1817cv 1648 . . . . . . . . 9  class  y
1915cv 1648 . . . . . . . . 9  class  u
205, 18cres 4839 . . . . . . . . 9  class  ( f  |`  y )
2118, 19, 20wf 5409 . . . . . . . 8  wff  ( f  |`  y ) : y --> u
22 cuz 10444 . . . . . . . . 9  class  ZZ>=
2322crn 4838 . . . . . . . 8  class  ran  ZZ>=
2421, 17, 23wrex 2667 . . . . . . 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 2666 . . . . 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 4225 . . 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 4226 . 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  17248  lmrcl  17249  lmfval  17250
  Copyright terms: Public domain W3C validator