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

Theorem isacs 13876
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 5758 . 2  |-  ( C  e.  (ACS `  X
)  ->  X  e.  _V )
2 elfvex 5758 . . 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 5728 . . . . . 6  |-  ( x  =  X  ->  (Moore `  x )  =  (Moore `  X ) )
5 pweq 3802 . . . . . . . . 9  |-  ( x  =  X  ->  ~P x  =  ~P X
)
65, 5feq23d 5588 . . . . . . . 8  |-  ( x  =  X  ->  (
f : ~P x --> ~P x  <->  f : ~P X
--> ~P X ) )
75raleqdv 2910 . . . . . . . 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 1636 . . . . . 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 2951 . . . . 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 13814 . . . . 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 5742 . . . . . 6  |-  (Moore `  X )  e.  _V
1312rabex 4354 . . . . 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 5806 . . . 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 2503 . . 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 2497 . . . . . . . 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 2725 . . . . . 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 1636 . . . 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 3092 . . 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 1550    = wceq 1652    e. wcel 1725   A.wral 2705   {crab 2709   _Vcvv 2956    i^i cin 3319    C_ wss 3320   ~Pcpw 3799   U.cuni 4015   "cima 4881   -->wf 5450   ` cfv 5454   Fincfn 7109  Moorecmre 13807  ACScacs 13810
This theorem is referenced by:  acsmre  13877  isacs2  13878  isacs1i  13882  mreacs  13883
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-fv 5462  df-acs 13814
  Copyright terms: Public domain W3C validator