MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  df-ucn Structured version   Unicode version

Definition df-ucn 19873
Description: Define a function on two uniform structures which value is the set of uniformly continuous functions from the first uniform structure to the second. A function  f is uniformly continuous if, roughly speaking, it is possible to guarantee that  ( f `  x
) and  ( f `  y ) be as close to each other as we please by requiring only that  x and  y are sufficiently close to each other; unlike ordinary continuity, the maximum distance between  ( f `  x
) and  ( f `  y ) cannot depend on  x and  y themselves. This formulation is the definition 1 of [BourbakiTop1] p. II.6. (Contributed by Thierry Arnoux, 16-Nov-2017.)
Assertion
Ref Expression
df-ucn  |- Cnu  =  (
u  e.  U. ran UnifOn ,  v  e.  U. ran UnifOn  |->  { f  e.  ( dom  U. v  ^m  dom  U. u )  |  A. s  e.  v  E. r  e.  u  A. x  e.  dom  U. u A. y  e.  dom  U. u ( x r y  ->  ( f `  x ) s ( f `  y ) ) } )
Distinct variable group:    v, u, f, s, r, x, y

Detailed syntax breakdown of Definition df-ucn
StepHypRef Expression
1 cucn 19872 . 2  class Cnu
2 vu . . 3  setvar  u
3 vv . . 3  setvar  v
4 cust 19796 . . . . 5  class UnifOn
54crn 4862 . . . 4  class  ran UnifOn
65cuni 4112 . . 3  class  U. ran UnifOn
7 vx . . . . . . . . . . 11  setvar  x
87cv 1368 . . . . . . . . . 10  class  x
9 vy . . . . . . . . . . 11  setvar  y
109cv 1368 . . . . . . . . . 10  class  y
11 vr . . . . . . . . . . 11  setvar  r
1211cv 1368 . . . . . . . . . 10  class  r
138, 10, 12wbr 4313 . . . . . . . . 9  wff  x r y
14 vf . . . . . . . . . . . 12  setvar  f
1514cv 1368 . . . . . . . . . . 11  class  f
168, 15cfv 5439 . . . . . . . . . 10  class  ( f `
 x )
1710, 15cfv 5439 . . . . . . . . . 10  class  ( f `
 y )
18 vs . . . . . . . . . . 11  setvar  s
1918cv 1368 . . . . . . . . . 10  class  s
2016, 17, 19wbr 4313 . . . . . . . . 9  wff  ( f `
 x ) s ( f `  y
)
2113, 20wi 4 . . . . . . . 8  wff  ( x r y  ->  (
f `  x )
s ( f `  y ) )
222cv 1368 . . . . . . . . . 10  class  u
2322cuni 4112 . . . . . . . . 9  class  U. u
2423cdm 4861 . . . . . . . 8  class  dom  U. u
2521, 9, 24wral 2736 . . . . . . 7  wff  A. y  e.  dom  U. u ( x r y  -> 
( f `  x
) s ( f `
 y ) )
2625, 7, 24wral 2736 . . . . . 6  wff  A. x  e.  dom  U. u A. y  e.  dom  U. u
( x r y  ->  ( f `  x ) s ( f `  y ) )
2726, 11, 22wrex 2737 . . . . 5  wff  E. r  e.  u  A. x  e.  dom  U. u A. y  e.  dom  U. u
( x r y  ->  ( f `  x ) s ( f `  y ) )
283cv 1368 . . . . 5  class  v
2927, 18, 28wral 2736 . . . 4  wff  A. s  e.  v  E. r  e.  u  A. x  e.  dom  U. u A. y  e.  dom  U. u
( x r y  ->  ( f `  x ) s ( f `  y ) )
3028cuni 4112 . . . . . 6  class  U. v
3130cdm 4861 . . . . 5  class  dom  U. v
32 cmap 7235 . . . . 5  class  ^m
3331, 24, 32co 6112 . . . 4  class  ( dom  U. v  ^m  dom  U. u )
3429, 14, 33crab 2740 . . 3  class  { f  e.  ( dom  U. v  ^m  dom  U. u
)  |  A. s  e.  v  E. r  e.  u  A. x  e.  dom  U. u A. y  e.  dom  U. u
( x r y  ->  ( f `  x ) s ( f `  y ) ) }
352, 3, 6, 6, 34cmpt2 6114 . 2  class  ( u  e.  U. ran UnifOn ,  v  e.  U. ran UnifOn  |->  { f  e.  ( dom  U. v  ^m  dom  U. u
)  |  A. s  e.  v  E. r  e.  u  A. x  e.  dom  U. u A. y  e.  dom  U. u
( x r y  ->  ( f `  x ) s ( f `  y ) ) } )
361, 35wceq 1369 1  wff Cnu  =  (
u  e.  U. ran UnifOn ,  v  e.  U. ran UnifOn  |->  { f  e.  ( dom  U. v  ^m  dom  U. u )  |  A. s  e.  v  E. r  e.  u  A. x  e.  dom  U. u A. y  e.  dom  U. u ( x r y  ->  ( f `  x ) s ( f `  y ) ) } )
Colors of variables: wff setvar class
This definition is referenced by:  ucnval  19874
  Copyright terms: Public domain W3C validator