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Theorem isacs 13881
Description: A set is an algebraic closure system iff it is specified by some function of the finite subsets, such that a set is closed iff it does not expand under the operation. (Contributed by Stefan O'Rear, 2-Apr-2015.)
Assertion
Ref Expression
isacs  |-  ( C  e.  (ACS `  X
)  <->  ( 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 groups:    C, f,
s    f, X, s

Proof of Theorem isacs
Dummy variables  c  x are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elfvex 5761 . 2  |-  ( C  e.  (ACS `  X
)  ->  X  e.  _V )
2 elfvex 5761 . . 3  |-  ( C  e.  (Moore `  X
)  ->  X  e.  _V )
32adantr 453 . 2  |-  ( ( 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 ) ) )  ->  X  e.  _V )
4 fveq2 5731 . . . . . 6  |-  ( x  =  X  ->  (Moore `  x )  =  (Moore `  X ) )
5 pweq 3804 . . . . . . . . 9  |-  ( x  =  X  ->  ~P x  =  ~P X
)
65, 5feq23d 5591 . . . . . . . 8  |-  ( x  =  X  ->  (
f : ~P x --> ~P x  <->  f : ~P X
--> ~P X ) )
75raleqdv 2912 . . . . . . . 8  |-  ( x  =  X  ->  ( A. s  e.  ~P  x ( s  e.  c  <->  U. ( f "
( ~P s  i^i 
Fin ) )  C_  s )  <->  A. s  e.  ~P  X ( s  e.  c  <->  U. (
f " ( ~P s  i^i  Fin )
)  C_  s )
) )
86, 7anbi12d 693 . . . . . . 7  |-  ( x  =  X  ->  (
( f : ~P x
--> ~P x  /\  A. s  e.  ~P  x
( s  e.  c  <->  U. ( f " ( ~P s  i^i  Fin )
)  C_  s )
)  <->  ( f : ~P X --> ~P X  /\  A. s  e.  ~P  X ( s  e.  c  <->  U. ( f "
( ~P s  i^i 
Fin ) )  C_  s ) ) ) )
98exbidv 1637 . . . . . 6  |-  ( x  =  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 ) )  <->  E. f
( f : ~P X
--> ~P X  /\  A. s  e.  ~P  X
( s  e.  c  <->  U. ( f " ( ~P s  i^i  Fin )
)  C_  s )
) ) )
104, 9rabeqbidv 2953 . . . . 5  |-  ( x  =  X  ->  { 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 )
) }  =  {
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 ) ) } )
11 df-acs 13819 . . . . 5  |- 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 ) ) } )
12 fvex 5745 . . . . . 6  |-  (Moore `  X )  e.  _V
1312rabex 4357 . . . . 5  |-  { 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 )
) }  e.  _V
1410, 11, 13fvmpt 5809 . . . 4  |-  ( X  e.  _V  ->  (ACS `  X )  =  {
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 ) ) } )
1514eleq2d 2505 . . 3  |-  ( X  e.  _V  ->  ( C  e.  (ACS `  X
)  <->  C  e.  { 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 )
) } ) )
16 eleq2 2499 . . . . . . . 8  |-  ( c  =  C  ->  (
s  e.  c  <->  s  e.  C ) )
1716bibi1d 312 . . . . . . 7  |-  ( c  =  C  ->  (
( s  e.  c  <->  U. ( f " ( ~P s  i^i  Fin )
)  C_  s )  <->  ( s  e.  C  <->  U. (
f " ( ~P s  i^i  Fin )
)  C_  s )
) )
1817ralbidv 2727 . . . . . 6  |-  ( c  =  C  ->  ( A. s  e.  ~P  X ( s  e.  c  <->  U. ( f "
( ~P s  i^i 
Fin ) )  C_  s )  <->  A. s  e.  ~P  X ( s  e.  C  <->  U. (
f " ( ~P s  i^i  Fin )
)  C_  s )
) )
1918anbi2d 686 . . . . 5  |-  ( c  =  C  ->  (
( f : ~P X
--> ~P X  /\  A. s  e.  ~P  X
( s  e.  c  <->  U. ( f " ( ~P s  i^i  Fin )
)  C_  s )
)  <->  ( f : ~P X --> ~P X  /\  A. s  e.  ~P  X ( s  e.  C  <->  U. ( f "
( ~P s  i^i 
Fin ) )  C_  s ) ) ) )
2019exbidv 1637 . . . 4  |-  ( c  =  C  ->  ( E. f ( f : ~P X --> ~P X  /\  A. s  e.  ~P  X ( s  e.  c  <->  U. ( f "
( ~P s  i^i 
Fin ) )  C_  s ) )  <->  E. f
( f : ~P X
--> ~P X  /\  A. s  e.  ~P  X
( s  e.  C  <->  U. ( f " ( ~P s  i^i  Fin )
)  C_  s )
) ) )
2120elrab 3094 . . 3  |-  ( C  e.  { 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 ) ) }  <-> 
( 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 ) ) ) )
2215, 21syl6bb 254 . 2  |-  ( X  e.  _V  ->  ( C  e.  (ACS `  X
)  <->  ( 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 ) ) ) ) )
231, 3, 22pm5.21nii 344 1  |-  ( C  e.  (ACS `  X
)  <->  ( 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
Syntax hints:    <-> wb 178    /\ wa 360   E.wex 1551    = wceq 1653    e. wcel 1726   A.wral 2707   {crab 2711   _Vcvv 2958    i^i cin 3321    C_ wss 3322   ~Pcpw 3801   U.cuni 4017   "cima 4884   -->wf 5453   ` cfv 5457   Fincfn 7112  Moorecmre 13812  ACScacs 13815
This theorem is referenced by:  acsmre  13882  isacs2  13883  isacs1i  13887  mreacs  13888
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4216  df-opab 4270  df-mpt 4271  df-id 4501  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-fv 5465  df-acs 13819
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