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

Definition df-struct 13166
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. (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 13160 . 2  class Struct
2 vx . . . . . 6  set  x
32cv 1631 . . . . 5  class  x
4 cle 8884 . . . . . 6  class  <_
5 cn 9762 . . . . . . 7  class  NN
65, 5cxp 4703 . . . . . 6  class  ( NN 
X.  NN )
74, 6cin 3164 . . . . 5  class  (  <_  i^i  ( NN  X.  NN ) )
83, 7wcel 1696 . . . 4  wff  x  e.  (  <_  i^i  ( NN  X.  NN ) )
9 vf . . . . . . 7  set  f
109cv 1631 . . . . . 6  class  f
11 c0 3468 . . . . . . 7  class  (/)
1211csn 3653 . . . . . 6  class  { (/) }
1310, 12cdif 3162 . . . . 5  class  ( f 
\  { (/) } )
1413wfun 5265 . . . 4  wff  Fun  (
f  \  { (/) } )
1510cdm 4705 . . . . 5  class  dom  f
16 cfz 10798 . . . . . 6  class  ...
173, 16cfv 5271 . . . . 5  class  ( ... `  x )
1815, 17wss 3165 . . . 4  wff  dom  f  C_  ( ... `  x
)
198, 14, 18w3a 934 . . 3  wff  ( x  e.  (  <_  i^i  ( NN  X.  NN ) )  /\  Fun  ( f  \  { (/)
} )  /\  dom  f  C_  ( ... `  x
) )
2019, 9, 2copab 4092 . 2  class  { <. f ,  x >.  |  ( x  e.  (  <_  i^i  ( NN  X.  NN ) )  /\  Fun  ( f  \  { (/)
} )  /\  dom  f  C_  ( ... `  x
) ) }
211, 20wceq 1632 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  13172  isstruct2  13173
  Copyright terms: Public domain W3C validator