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Definition df-struct 12418
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 5216, such classes cannot be functions. Without the empty set, however, a structure must be a function, see structn0fun 12429:  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 3786). (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 12412 . 2  class Struct
2 vx . . . . . 6  setvar  x
32cv 1347 . . . . 5  class  x
4 cle 7955 . . . . . 6  class  <_
5 cn 8878 . . . . . . 7  class  NN
65, 5cxp 4609 . . . . . 6  class  ( NN 
X.  NN )
74, 6cin 3120 . . . . 5  class  (  <_  i^i  ( NN  X.  NN ) )
83, 7wcel 2141 . . . 4  wff  x  e.  (  <_  i^i  ( NN  X.  NN ) )
9 vf . . . . . . 7  setvar  f
109cv 1347 . . . . . 6  class  f
11 c0 3414 . . . . . . 7  class  (/)
1211csn 3583 . . . . . 6  class  { (/) }
1310, 12cdif 3118 . . . . 5  class  ( f 
\  { (/) } )
1413wfun 5192 . . . 4  wff  Fun  (
f  \  { (/) } )
1510cdm 4611 . . . . 5  class  dom  f
16 cfz 9965 . . . . . 6  class  ...
173, 16cfv 5198 . . . . 5  class  ( ... `  x )
1815, 17wss 3121 . . . 4  wff  dom  f  C_  ( ... `  x
)
198, 14, 18w3a 973 . . 3  wff  ( x  e.  (  <_  i^i  ( NN  X.  NN ) )  /\  Fun  ( f  \  { (/)
} )  /\  dom  f  C_  ( ... `  x
) )
2019, 9, 2copab 4049 . 2  class  { <. f ,  x >.  |  ( x  e.  (  <_  i^i  ( NN  X.  NN ) )  /\  Fun  ( f  \  { (/)
} )  /\  dom  f  C_  ( ... `  x
) ) }
211, 20wceq 1348 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  12425  isstruct2im  12426  isstruct2r  12427
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