MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  isacs Unicode version

Theorem isacs 13398
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 5408 . 2  |-  ( C  e.  (ACS `  X
)  ->  X  e.  _V )
2 elfvex 5408 . . 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 5377 . . . . . 6  |-  ( x  =  X  ->  (Moore `  x )  =  (Moore `  X ) )
5 pweq 3533 . . . . . . . . 9  |-  ( x  =  X  ->  ~P x  =  ~P X
)
65, 5feq23d 5243 . . . . . . . 8  |-  ( x  =  X  ->  (
f : ~P x --> ~P x  <->  f : ~P X
--> ~P X ) )
75raleqdv 2694 . . . . . . . 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 2005 . . . . . 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 2722 . . . . 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 13363 . . . . 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 5391 . . . . . 6  |-  (Moore `  X )  e.  _V
1312rabex 4061 . . . . 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 5454 . . . 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 2320 . . 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 2314 . . . . . . . 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 2527 . . . . . 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 2005 . . . 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 2860 . . 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 2509   {crab 2512   _Vcvv 2727    i^i cin 3077    C_ wss 3078   ~Pcpw 3530   U.cuni 3727   "cima 4583   -->wf 4588   ` cfv 4592   Fincfn 6749  Moorecmre 13358  ACScacs 13360
This theorem is referenced by:  acsmre  13399  isacs2  13400  isacs1i  13403  mreacs  13404
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 1926  ax-ext 2234  ax-sep 4038  ax-nul 4046  ax-pr 4108  ax-un 4403
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 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-ral 2513  df-rex 2514  df-rab 2516  df-v 2729  df-sbc 2922  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-nul 3363  df-if 3471  df-pw 3532  df-sn 3550  df-pr 3551  df-op 3553  df-uni 3728  df-br 3921  df-opab 3975  df-mpt 3976  df-id 4202  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-fun 4602  df-fn 4603  df-f 4604  df-fv 4608  df-acs 13363
  Copyright terms: Public domain W3C validator