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Theorem isacs 13831
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 5717 . 2  |-  ( C  e.  (ACS `  X
)  ->  X  e.  _V )
2 elfvex 5717 . . 3  |-  ( C  e.  (Moore `  X
)  ->  X  e.  _V )
32adantr 452 . 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 5687 . . . . . 6  |-  ( x  =  X  ->  (Moore `  x )  =  (Moore `  X ) )
5 pweq 3762 . . . . . . . . 9  |-  ( x  =  X  ->  ~P x  =  ~P X
)
65, 5feq23d 5547 . . . . . . . 8  |-  ( x  =  X  ->  (
f : ~P x --> ~P x  <->  f : ~P X
--> ~P X ) )
75raleqdv 2870 . . . . . . . 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 692 . . . . . . 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 1633 . . . . . 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 2911 . . . . 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 13769 . . . . 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 5701 . . . . . 6  |-  (Moore `  X )  e.  _V
1312rabex 4314 . . . . 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 5765 . . . 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 2471 . . 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 2465 . . . . . . . 8  |-  ( c  =  C  ->  (
s  e.  c  <->  s  e.  C ) )
1716bibi1d 311 . . . . . . 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 2686 . . . . . 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 685 . . . . 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 1633 . . . 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 3052 . . 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 253 . 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 343 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 177    /\ wa 359   E.wex 1547    = wceq 1649    e. wcel 1721   A.wral 2666   {crab 2670   _Vcvv 2916    i^i cin 3279    C_ wss 3280   ~Pcpw 3759   U.cuni 3975   "cima 4840   -->wf 5409   ` cfv 5413   Fincfn 7068  Moorecmre 13762  ACScacs 13765
This theorem is referenced by:  acsmre  13832  isacs2  13833  isacs1i  13837  mreacs  13838
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-sbc 3122  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-fv 5421  df-acs 13769
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