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Definition df-acs 13487
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 7340 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 13483 . 2  class ACS
2 vx . . 3  set  x
3 cvv 2789 . . 3  class  _V
42cv 1622 . . . . . . . 8  class  x
54cpw 3626 . . . . . . 7  class  ~P x
6 vf . . . . . . . 8  set  f
76cv 1622 . . . . . . 7  class  f
85, 5, 7wf 5217 . . . . . 6  wff  f : ~P x --> ~P x
9 vs . . . . . . . . 9  set  s
10 vc . . . . . . . . 9  set  c
119, 10wel 1686 . . . . . . . 8  wff  s  e.  c
129cv 1622 . . . . . . . . . . . . 13  class  s
1312cpw 3626 . . . . . . . . . . . 12  class  ~P s
14 cfn 6859 . . . . . . . . . . . 12  class  Fin
1513, 14cin 3152 . . . . . . . . . . 11  class  ( ~P s  i^i  Fin )
167, 15cima 4691 . . . . . . . . . 10  class  ( f
" ( ~P s  i^i  Fin ) )
1716cuni 3828 . . . . . . . . 9  class  U. (
f " ( ~P s  i^i  Fin )
)
1817, 12wss 3153 . . . . . . . 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 2544 . . . . . 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 13480 . . . . 5  class Moore
244, 23cfv 5221 . . . 4  class  (Moore `  x )
2522, 10, 24crab 2548 . . 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 4078 . 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  13549
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