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Theorem isacs5 15929
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 15923 . . 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 15925 . . 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 15926 . . 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 20 . 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 457 . . 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 4024 . . . . . . . . 9  |-  ( s  e.  ~P X  -> 
s  C_  X )
82mrcidb2 15035 . . . . . . . . 9  |-  ( ( C  e.  (Moore `  X )  /\  s  C_  X )  ->  (
s  e.  C  <->  ( F `  s )  C_  s
) )
97, 8sylan2 474 . . . . . . . 8  |-  ( ( C  e.  (Moore `  X )  /\  s  e.  ~P X )  -> 
( s  e.  C  <->  ( F `  s ) 
C_  s ) )
109adantr 465 . . . . . . 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 461 . . . . . . . . . 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 15026 . . . . . . . . . . . 12  |-  ( C  e.  (Moore `  X
)  ->  F : ~P X --> C )
13 ffun 5739 . . . . . . . . . . . 12  |-  ( F : ~P X --> C  ->  Fun  F )
14 funiunfv 6161 . . . . . . . . . . . 12  |-  ( Fun 
F  ->  U_ t  e.  ( ~P s  i^i 
Fin ) ( F `
 t )  = 
U. ( F "
( ~P s  i^i 
Fin ) ) )
1512, 13, 143syl 20 . . . . . . . . . . 11  |-  ( C  e.  (Moore `  X
)  ->  U_ t  e.  ( ~P s  i^i 
Fin ) ( F `
 t )  = 
U. ( F "
( ~P s  i^i 
Fin ) ) )
1615ad2antrr 725 . . . . . . . . . 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 2501 . . . . . . . . 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 3526 . . . . . . . 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 4373 . . . . . . . 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 261 . . . . . . 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 253 . . . . . 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 434 . . . . 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 2865 . . . 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 429 . . 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 15070 . . 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 664 . 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 188 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 setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1395    e. wcel 1819   A.wral 2807    i^i cin 3470    C_ wss 3471   ~Pcpw 4015   U.cuni 4251   U_ciun 4332   "cima 5011   Fun wfun 5588   -->wf 5590   ` cfv 5594   Fincfn 7535  Moorecmre 14999  mrClscmrc 15000  ACScacs 15002  Dirsetcdrs 15683  toInccipo 15908
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-int 4289  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6700  df-1st 6799  df-2nd 6800  df-recs 7060  df-rdg 7094  df-1o 7148  df-oadd 7152  df-er 7329  df-en 7536  df-dom 7537  df-sdom 7538  df-fin 7539  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-nn 10557  df-2 10615  df-3 10616  df-4 10617  df-5 10618  df-6 10619  df-7 10620  df-8 10621  df-9 10622  df-10 10623  df-n0 10817  df-z 10886  df-dec 11001  df-uz 11107  df-fz 11698  df-struct 14646  df-ndx 14647  df-slot 14648  df-base 14649  df-tset 14731  df-ple 14732  df-ocomp 14733  df-mre 15003  df-mrc 15004  df-acs 15006  df-preset 15684  df-drs 15685  df-poset 15702  df-ipo 15909
This theorem is referenced by:  isacs4  15930
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