MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  isacs5 Unicode version

Theorem isacs5 14275
Description: A closure system is algebraic iff the closure of a generating set is the union of the closures of its finite subsets. (Contributed by Stefan O'Rear, 2-Apr-2015.)
Hypothesis
Ref Expression
acsdrscl.f  |-  F  =  (mrCls `  C )
Assertion
Ref Expression
isacs5  |-  ( C  e.  (ACS `  X
)  <->  ( C  e.  (Moore `  X )  /\  A. s  e.  ~P  X ( F `  s )  =  U. ( F " ( ~P s  i^i  Fin )
) ) )
Distinct variable groups:    C, s    F, s    X, s

Proof of Theorem isacs5
Dummy variable  t is distinct from all other variables.
StepHypRef Expression
1 isacs3lem 14269 . . 3  |-  ( C  e.  (ACS `  X
)  ->  ( C  e.  (Moore `  X )  /\  A. s  e.  ~P  C ( (toInc `  s )  e. Dirset  ->  U. s  e.  C ) ) )
2 acsdrscl.f . . . 4  |-  F  =  (mrCls `  C )
32isacs4lem 14271 . . 3  |-  ( ( C  e.  (Moore `  X )  /\  A. s  e.  ~P  C
( (toInc `  s
)  e. Dirset  ->  U. s  e.  C ) )  -> 
( C  e.  (Moore `  X )  /\  A. t  e.  ~P  ~P X
( (toInc `  t
)  e. Dirset  ->  ( F `
 U. t )  =  U. ( F
" t ) ) ) )
42isacs5lem 14272 . . 3  |-  ( ( C  e.  (Moore `  X )  /\  A. t  e.  ~P  ~P X
( (toInc `  t
)  e. Dirset  ->  ( F `
 U. t )  =  U. ( F
" t ) ) )  ->  ( C  e.  (Moore `  X )  /\  A. s  e.  ~P  X ( F `  s )  =  U. ( F " ( ~P s  i^i  Fin )
) ) )
51, 3, 43syl 18 . 2  |-  ( C  e.  (ACS `  X
)  ->  ( C  e.  (Moore `  X )  /\  A. s  e.  ~P  X ( F `  s )  =  U. ( F " ( ~P s  i^i  Fin )
) ) )
6 simpl 443 . . 3  |-  ( ( C  e.  (Moore `  X )  /\  A. s  e.  ~P  X
( F `  s
)  =  U. ( F " ( ~P s  i^i  Fin ) ) )  ->  C  e.  (Moore `  X ) )
7 elpwi 3633 . . . . . . . . 9  |-  ( s  e.  ~P X  -> 
s  C_  X )
82mrcidb2 13520 . . . . . . . . 9  |-  ( ( C  e.  (Moore `  X )  /\  s  C_  X )  ->  (
s  e.  C  <->  ( F `  s )  C_  s
) )
97, 8sylan2 460 . . . . . . . 8  |-  ( ( C  e.  (Moore `  X )  /\  s  e.  ~P X )  -> 
( s  e.  C  <->  ( F `  s ) 
C_  s ) )
109adantr 451 . . . . . . 7  |-  ( ( ( C  e.  (Moore `  X )  /\  s  e.  ~P X )  /\  ( F `  s )  =  U. ( F
" ( ~P s  i^i  Fin ) ) )  ->  ( s  e.  C  <->  ( F `  s )  C_  s
) )
11 simpr 447 . . . . . . . . . 10  |-  ( ( ( C  e.  (Moore `  X )  /\  s  e.  ~P X )  /\  ( F `  s )  =  U. ( F
" ( ~P s  i^i  Fin ) ) )  ->  ( F `  s )  =  U. ( F " ( ~P s  i^i  Fin )
) )
122mrcf 13511 . . . . . . . . . . . 12  |-  ( C  e.  (Moore `  X
)  ->  F : ~P X --> C )
13 ffun 5391 . . . . . . . . . . . 12  |-  ( F : ~P X --> C  ->  Fun  F )
14 funiunfv 5774 . . . . . . . . . . . 12  |-  ( Fun 
F  ->  U_ t  e.  ( ~P s  i^i 
Fin ) ( F `
 t )  = 
U. ( F "
( ~P s  i^i 
Fin ) ) )
1512, 13, 143syl 18 . . . . . . . . . . 11  |-  ( C  e.  (Moore `  X
)  ->  U_ t  e.  ( ~P s  i^i 
Fin ) ( F `
 t )  = 
U. ( F "
( ~P s  i^i 
Fin ) ) )
1615ad2antrr 706 . . . . . . . . . 10  |-  ( ( ( C  e.  (Moore `  X )  /\  s  e.  ~P X )  /\  ( F `  s )  =  U. ( F
" ( ~P s  i^i  Fin ) ) )  ->  U_ t  e.  ( ~P s  i^i  Fin ) ( F `  t )  =  U. ( F " ( ~P s  i^i  Fin )
) )
1711, 16eqtr4d 2318 . . . . . . . . 9  |-  ( ( ( C  e.  (Moore `  X )  /\  s  e.  ~P X )  /\  ( F `  s )  =  U. ( F
" ( ~P s  i^i  Fin ) ) )  ->  ( F `  s )  =  U_ t  e.  ( ~P s  i^i  Fin ) ( F `  t ) )
1817sseq1d 3205 . . . . . . . 8  |-  ( ( ( C  e.  (Moore `  X )  /\  s  e.  ~P X )  /\  ( F `  s )  =  U. ( F
" ( ~P s  i^i  Fin ) ) )  ->  ( ( F `
 s )  C_  s 
<-> 
U_ t  e.  ( ~P s  i^i  Fin ) ( F `  t )  C_  s
) )
19 iunss 3943 . . . . . . . 8  |-  ( U_ t  e.  ( ~P s  i^i  Fin ) ( F `  t ) 
C_  s  <->  A. t  e.  ( ~P s  i^i 
Fin ) ( F `
 t )  C_  s )
2018, 19syl6bb 252 . . . . . . 7  |-  ( ( ( C  e.  (Moore `  X )  /\  s  e.  ~P X )  /\  ( F `  s )  =  U. ( F
" ( ~P s  i^i  Fin ) ) )  ->  ( ( F `
 s )  C_  s 
<-> 
A. t  e.  ( ~P s  i^i  Fin ) ( F `  t )  C_  s
) )
2110, 20bitrd 244 . . . . . 6  |-  ( ( ( C  e.  (Moore `  X )  /\  s  e.  ~P X )  /\  ( F `  s )  =  U. ( F
" ( ~P s  i^i  Fin ) ) )  ->  ( s  e.  C  <->  A. t  e.  ( ~P s  i^i  Fin ) ( F `  t )  C_  s
) )
2221ex 423 . . . . 5  |-  ( ( C  e.  (Moore `  X )  /\  s  e.  ~P X )  -> 
( ( F `  s )  =  U. ( F " ( ~P s  i^i  Fin )
)  ->  ( s  e.  C  <->  A. t  e.  ( ~P s  i^i  Fin ) ( F `  t )  C_  s
) ) )
2322ralimdva 2621 . . . 4  |-  ( C  e.  (Moore `  X
)  ->  ( A. s  e.  ~P  X
( F `  s
)  =  U. ( F " ( ~P s  i^i  Fin ) )  ->  A. s  e.  ~P  X ( s  e.  C  <->  A. t  e.  ( ~P s  i^i  Fin ) ( F `  t )  C_  s
) ) )
2423imp 418 . . 3  |-  ( ( C  e.  (Moore `  X )  /\  A. s  e.  ~P  X
( F `  s
)  =  U. ( F " ( ~P s  i^i  Fin ) ) )  ->  A. s  e.  ~P  X ( s  e.  C  <->  A. t  e.  ( ~P s  i^i  Fin ) ( F `  t )  C_  s
) )
252isacs2 13555 . . 3  |-  ( C  e.  (ACS `  X
)  <->  ( C  e.  (Moore `  X )  /\  A. s  e.  ~P  X ( s  e.  C  <->  A. t  e.  ( ~P s  i^i  Fin ) ( F `  t )  C_  s
) ) )
266, 24, 25sylanbrc 645 . 2  |-  ( ( C  e.  (Moore `  X )  /\  A. s  e.  ~P  X
( F `  s
)  =  U. ( F " ( ~P s  i^i  Fin ) ) )  ->  C  e.  (ACS
`  X ) )
275, 26impbii 180 1  |-  ( C  e.  (ACS `  X
)  <->  ( C  e.  (Moore `  X )  /\  A. s  e.  ~P  X ( F `  s )  =  U. ( F " ( ~P s  i^i  Fin )
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684   A.wral 2543    i^i cin 3151    C_ wss 3152   ~Pcpw 3625   U.cuni 3827   U_ciun 3905   "cima 4692   Fun wfun 5249   -->wf 5251   ` cfv 5255   Fincfn 6863  Moorecmre 13484  mrClscmrc 13485  ACScacs 13487  Dirsetcdrs 14061  toInccipo 14254
This theorem is referenced by:  isacs4  14276
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-oadd 6483  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-nn 9747  df-2 9804  df-3 9805  df-4 9806  df-5 9807  df-6 9808  df-7 9809  df-8 9810  df-9 9811  df-10 9812  df-n0 9966  df-z 10025  df-dec 10125  df-uz 10231  df-fz 10783  df-struct 13150  df-ndx 13151  df-slot 13152  df-base 13153  df-tset 13227  df-ple 13228  df-ocomp 13229  df-mre 13488  df-mrc 13489  df-acs 13491  df-preset 14062  df-drs 14063  df-poset 14080  df-ipo 14255
  Copyright terms: Public domain W3C validator