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

Definition df-acs 13734
Description: An important subclass of Moore systems are those which can be interpreted as closure under some collection of operators of finite arity (the collection itself is not required to be finite). These are termed algebraic closure systems; similar to definition (A) of an algebraic closure system in [Schechter] p. 84, but to avoid the complexity of an arbitrary mixed collection of functions of various arities (especially if the axiom of infinity omex 7524 is to be avoided), we consider a single function defined on finite sets instead. (Contributed by Stefan O'Rear, 2-Apr-2015.)
Assertion
Ref Expression
df-acs  |- ACS  =  ( x  e.  _V  |->  { c  e.  (Moore `  x )  |  E. f ( f : ~P x --> ~P x  /\  A. s  e.  ~P  x ( s  e.  c  <->  U. ( f "
( ~P s  i^i 
Fin ) )  C_  s ) ) } )
Distinct variable group:    f, c, s, x

Detailed syntax breakdown of Definition df-acs
StepHypRef Expression
1 cacs 13730 . 2  class ACS
2 vx . . 3  set  x
3 cvv 2892 . . 3  class  _V
42cv 1648 . . . . . . . 8  class  x
54cpw 3735 . . . . . . 7  class  ~P x
6 vf . . . . . . . 8  set  f
76cv 1648 . . . . . . 7  class  f
85, 5, 7wf 5383 . . . . . 6  wff  f : ~P x --> ~P x
9 vs . . . . . . . . 9  set  s
10 vc . . . . . . . . 9  set  c
119, 10wel 1718 . . . . . . . 8  wff  s  e.  c
129cv 1648 . . . . . . . . . . . . 13  class  s
1312cpw 3735 . . . . . . . . . . . 12  class  ~P s
14 cfn 7038 . . . . . . . . . . . 12  class  Fin
1513, 14cin 3255 . . . . . . . . . . 11  class  ( ~P s  i^i  Fin )
167, 15cima 4814 . . . . . . . . . 10  class  ( f
" ( ~P s  i^i  Fin ) )
1716cuni 3950 . . . . . . . . 9  class  U. (
f " ( ~P s  i^i  Fin )
)
1817, 12wss 3256 . . . . . . . 8  wff  U. (
f " ( ~P s  i^i  Fin )
)  C_  s
1911, 18wb 177 . . . . . . 7  wff  ( s  e.  c  <->  U. (
f " ( ~P s  i^i  Fin )
)  C_  s )
2019, 9, 5wral 2642 . . . . . 6  wff  A. s  e.  ~P  x ( s  e.  c  <->  U. (
f " ( ~P s  i^i  Fin )
)  C_  s )
218, 20wa 359 . . . . 5  wff  ( f : ~P x --> ~P x  /\  A. s  e.  ~P  x ( s  e.  c  <->  U. ( f "
( ~P s  i^i 
Fin ) )  C_  s ) )
2221, 6wex 1547 . . . 4  wff  E. f
( f : ~P x
--> ~P x  /\  A. s  e.  ~P  x
( s  e.  c  <->  U. ( f " ( ~P s  i^i  Fin )
)  C_  s )
)
23 cmre 13727 . . . . 5  class Moore
244, 23cfv 5387 . . . 4  class  (Moore `  x )
2522, 10, 24crab 2646 . . 3  class  { c  e.  (Moore `  x
)  |  E. f
( f : ~P x
--> ~P x  /\  A. s  e.  ~P  x
( s  e.  c  <->  U. ( f " ( ~P s  i^i  Fin )
)  C_  s )
) }
262, 3, 25cmpt 4200 . 2  class  ( x  e.  _V  |->  { c  e.  (Moore `  x
)  |  E. f
( f : ~P x
--> ~P x  /\  A. s  e.  ~P  x
( s  e.  c  <->  U. ( f " ( ~P s  i^i  Fin )
)  C_  s )
) } )
271, 26wceq 1649 1  wff ACS  =  ( x  e.  _V  |->  { c  e.  (Moore `  x )  |  E. f ( f : ~P x --> ~P x  /\  A. s  e.  ~P  x ( s  e.  c  <->  U. ( f "
( ~P s  i^i 
Fin ) )  C_  s ) ) } )
Colors of variables: wff set class
This definition is referenced by:  isacs  13796
  Copyright terms: Public domain W3C validator