ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  df-struct Unicode version

Definition df-struct 11867
Description: Define a structure with components in  M ... N. This is not a requirement for groups, posets, etc., but it is a useful assumption for component extraction theorems.

As mentioned in the section header, an "extensible structure should be implemented as a function (a set of ordered pairs)". The current definition, however, is less restrictive: it allows for classes which contain the empty set 
(/) to be extensible structures. Because of 0nelfun 5109, such classes cannot be functions. Without the empty set, however, a structure must be a function, see structn0fun 11878:  F Struct  X  ->  Fun  ( F  \  { (/)
} ).

Allowing an extensible structure to contain the empty set ensures that expressions like  { <. A ,  B >. ,  <. C ,  D >. } are structures without asserting or implying that  A,  B,  C and  D are sets (if  A or  B is a proper class, then  <. A ,  B >.  =  (/), see opprc 3694). (Contributed by Mario Carneiro, 29-Aug-2015.)

Assertion
Ref Expression
df-struct  |- Struct  =  { <. f ,  x >.  |  ( x  e.  (  <_  i^i  ( NN  X.  NN ) )  /\  Fun  ( f  \  { (/)
} )  /\  dom  f  C_  ( ... `  x
) ) }
Distinct variable group:    x, f

Detailed syntax breakdown of Definition df-struct
StepHypRef Expression
1 cstr 11861 . 2  class Struct
2 vx . . . . . 6  setvar  x
32cv 1313 . . . . 5  class  x
4 cle 7765 . . . . . 6  class  <_
5 cn 8680 . . . . . . 7  class  NN
65, 5cxp 4505 . . . . . 6  class  ( NN 
X.  NN )
74, 6cin 3038 . . . . 5  class  (  <_  i^i  ( NN  X.  NN ) )
83, 7wcel 1463 . . . 4  wff  x  e.  (  <_  i^i  ( NN  X.  NN ) )
9 vf . . . . . . 7  setvar  f
109cv 1313 . . . . . 6  class  f
11 c0 3331 . . . . . . 7  class  (/)
1211csn 3495 . . . . . 6  class  { (/) }
1310, 12cdif 3036 . . . . 5  class  ( f 
\  { (/) } )
1413wfun 5085 . . . 4  wff  Fun  (
f  \  { (/) } )
1510cdm 4507 . . . . 5  class  dom  f
16 cfz 9741 . . . . . 6  class  ...
173, 16cfv 5091 . . . . 5  class  ( ... `  x )
1815, 17wss 3039 . . . 4  wff  dom  f  C_  ( ... `  x
)
198, 14, 18w3a 945 . . 3  wff  ( x  e.  (  <_  i^i  ( NN  X.  NN ) )  /\  Fun  ( f  \  { (/)
} )  /\  dom  f  C_  ( ... `  x
) )
2019, 9, 2copab 3956 . 2  class  { <. f ,  x >.  |  ( x  e.  (  <_  i^i  ( NN  X.  NN ) )  /\  Fun  ( f  \  { (/)
} )  /\  dom  f  C_  ( ... `  x
) ) }
211, 20wceq 1314 1  wff Struct  =  { <. f ,  x >.  |  ( x  e.  (  <_  i^i  ( NN  X.  NN ) )  /\  Fun  ( f  \  { (/)
} )  /\  dom  f  C_  ( ... `  x
) ) }
Colors of variables: wff set class
This definition is referenced by:  brstruct  11874  isstruct2im  11875  isstruct2r  11876
  Copyright terms: Public domain W3C validator