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Theorem isacs 13501
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
StepHypRef Expression
1 elfvex 5475 . 2  |-  ( C  e.  (ACS `  X
)  ->  X  e.  _V )
2 elfvex 5475 . . 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 5444 . . . . . 6  |-  ( x  =  X  ->  (Moore `  x )  =  (Moore `  X ) )
5 pweq 3588 . . . . . . . . 9  |-  ( x  =  X  ->  ~P x  =  ~P X
)
65, 5feq23d 5310 . . . . . . . 8  |-  ( x  =  X  ->  (
f : ~P x --> ~P x  <->  f : ~P X
--> ~P X ) )
75raleqdv 2713 . . . . . . . 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 694 . . . . . . 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 2006 . . . . . 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 2752 . . . . 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 13439 . . . . 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 5458 . . . . . 6  |-  (Moore `  X )  e.  _V
1312rabex 4125 . . . . 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 5522 . . . 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 2323 . . 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 2317 . . . . . . . 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 2536 . . . . . 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 687 . . . . 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 2006 . . . 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 2891 . . 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 1537    = wceq 1619    e. wcel 1621   A.wral 2516   {crab 2520   _Vcvv 2757    i^i cin 3112    C_ wss 3113   ~Pcpw 3585   U.cuni 3787   "cima 4650   -->wf 4655   ` cfv 4659   Fincfn 6817  Moorecmre 13432  ACScacs 13435
This theorem is referenced by:  acsmre  13502  isacs2  13503  isacs1i  13507  mreacs  13508
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2237  ax-sep 4101  ax-nul 4109  ax-pr 4172  ax-un 4470
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2121  df-mo 2122  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ne 2421  df-ral 2521  df-rex 2522  df-rab 2525  df-v 2759  df-sbc 2953  df-dif 3116  df-un 3118  df-in 3120  df-ss 3127  df-nul 3417  df-if 3526  df-pw 3587  df-sn 3606  df-pr 3607  df-op 3609  df-uni 3788  df-br 3984  df-opab 4038  df-mpt 4039  df-id 4267  df-xp 4661  df-rel 4662  df-cnv 4663  df-co 4664  df-dm 4665  df-rn 4666  df-res 4667  df-ima 4668  df-fun 4669  df-fn 4670  df-f 4671  df-fv 4675  df-acs 13439
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