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Definition df-acs 13454
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 7312 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 13450 . 2  class ACS
2 vx . . 3  set  x
3 cvv 2763 . . 3  class  _V
42cv 1618 . . . . . . . 8  class  x
54cpw 3599 . . . . . . 7  class  ~P x
6 vf . . . . . . . 8  set  f
76cv 1618 . . . . . . 7  class  f
85, 5, 7wf 4669 . . . . . 6  wff  f : ~P x --> ~P x
9 vs . . . . . . . . 9  set  s
10 vc . . . . . . . . 9  set  c
119, 10wel 1622 . . . . . . . 8  wff  s  e.  c
129cv 1618 . . . . . . . . . . . . 13  class  s
1312cpw 3599 . . . . . . . . . . . 12  class  ~P s
14 cfn 6831 . . . . . . . . . . . 12  class  Fin
1513, 14cin 3126 . . . . . . . . . . 11  class  ( ~P s  i^i  Fin )
167, 15cima 4664 . . . . . . . . . 10  class  ( f
" ( ~P s  i^i  Fin ) )
1716cuni 3801 . . . . . . . . 9  class  U. (
f " ( ~P s  i^i  Fin )
)
1817, 12wss 3127 . . . . . . . 8  wff  U. (
f " ( ~P s  i^i  Fin )
)  C_  s
1911, 18wb 178 . . . . . . 7  wff  ( s  e.  c  <->  U. (
f " ( ~P s  i^i  Fin )
)  C_  s )
2019, 9, 5wral 2518 . . . . . 6  wff  A. s  e.  ~P  x ( s  e.  c  <->  U. (
f " ( ~P s  i^i  Fin )
)  C_  s )
218, 20wa 360 . . . . 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 1537 . . . 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 13447 . . . . 5  class Moore
244, 23cfv 4673 . . . 4  class  (Moore `  x )
2522, 10, 24crab 2522 . . 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 4051 . 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 1619 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  13516
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