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Definition df-acs 13507
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 7360 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 13503 . 2  class ACS
2 vx . . 3  set  x
3 cvv 2801 . . 3  class  _V
42cv 1631 . . . . . . . 8  class  x
54cpw 3638 . . . . . . 7  class  ~P x
6 vf . . . . . . . 8  set  f
76cv 1631 . . . . . . 7  class  f
85, 5, 7wf 5267 . . . . . 6  wff  f : ~P x --> ~P x
9 vs . . . . . . . . 9  set  s
10 vc . . . . . . . . 9  set  c
119, 10wel 1697 . . . . . . . 8  wff  s  e.  c
129cv 1631 . . . . . . . . . . . . 13  class  s
1312cpw 3638 . . . . . . . . . . . 12  class  ~P s
14 cfn 6879 . . . . . . . . . . . 12  class  Fin
1513, 14cin 3164 . . . . . . . . . . 11  class  ( ~P s  i^i  Fin )
167, 15cima 4708 . . . . . . . . . 10  class  ( f
" ( ~P s  i^i  Fin ) )
1716cuni 3843 . . . . . . . . 9  class  U. (
f " ( ~P s  i^i  Fin )
)
1817, 12wss 3165 . . . . . . . 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 2556 . . . . . 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 1531 . . . 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 13500 . . . . 5  class Moore
244, 23cfv 5271 . . . 4  class  (Moore `  x )
2522, 10, 24crab 2560 . . 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 4093 . 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 1632 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  13569
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