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Definition df-acs 13491
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 7344 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 13487 . 2  class ACS
2 vx . . 3  set  x
3 cvv 2788 . . 3  class  _V
42cv 1622 . . . . . . . 8  class  x
54cpw 3625 . . . . . . 7  class  ~P x
6 vf . . . . . . . 8  set  f
76cv 1622 . . . . . . 7  class  f
85, 5, 7wf 5251 . . . . . 6  wff  f : ~P x --> ~P x
9 vs . . . . . . . . 9  set  s
10 vc . . . . . . . . 9  set  c
119, 10wel 1685 . . . . . . . 8  wff  s  e.  c
129cv 1622 . . . . . . . . . . . . 13  class  s
1312cpw 3625 . . . . . . . . . . . 12  class  ~P s
14 cfn 6863 . . . . . . . . . . . 12  class  Fin
1513, 14cin 3151 . . . . . . . . . . 11  class  ( ~P s  i^i  Fin )
167, 15cima 4692 . . . . . . . . . 10  class  ( f
" ( ~P s  i^i  Fin ) )
1716cuni 3827 . . . . . . . . 9  class  U. (
f " ( ~P s  i^i  Fin )
)
1817, 12wss 3152 . . . . . . . 8  wff  U. (
f " ( ~P s  i^i  Fin )
)  C_  s
1911, 18wb 176 . . . . . . 7  wff  ( s  e.  c  <->  U. (
f " ( ~P s  i^i  Fin )
)  C_  s )
2019, 9, 5wral 2543 . . . . . 6  wff  A. s  e.  ~P  x ( s  e.  c  <->  U. (
f " ( ~P s  i^i  Fin )
)  C_  s )
218, 20wa 358 . . . . 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 1528 . . . 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 13484 . . . . 5  class Moore
244, 23cfv 5255 . . . 4  class  (Moore `  x )
2522, 10, 24crab 2547 . . 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 4077 . 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 1623 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  13553
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