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Definition df-struct 13197
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 13191 . 2  class Struct
2 vx . . . . . 6  set  x
32cv 1632 . . . . 5  class  x
4 cle 8913 . . . . . 6  class  <_
5 cn 9791 . . . . . . 7  class  NN
65, 5cxp 4724 . . . . . 6  class  ( NN 
X.  NN )
74, 6cin 3185 . . . . 5  class  (  <_  i^i  ( NN  X.  NN ) )
83, 7wcel 1701 . . . 4  wff  x  e.  (  <_  i^i  ( NN  X.  NN ) )
9 vf . . . . . . 7  set  f
109cv 1632 . . . . . 6  class  f
11 c0 3489 . . . . . . 7  class  (/)
1211csn 3674 . . . . . 6  class  { (/) }
1310, 12cdif 3183 . . . . 5  class  ( f 
\  { (/) } )
1413wfun 5286 . . . 4  wff  Fun  (
f  \  { (/) } )
1510cdm 4726 . . . . 5  class  dom  f
16 cfz 10829 . . . . . 6  class  ...
173, 16cfv 5292 . . . . 5  class  ( ... `  x )
1815, 17wss 3186 . . . 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 4113 . 2  class  { <. f ,  x >.  |  ( x  e.  (  <_  i^i  ( NN  X.  NN ) )  /\  Fun  ( f  \  { (/)
} )  /\  dom  f  C_  ( ... `  x
) ) }
211, 20wceq 1633 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  13203  isstruct2  13204
  Copyright terms: Public domain W3C validator