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Theorem isacs5 14598
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 14592 . . 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 14594 . . 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 14595 . . 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 19 . 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 444 . . 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 3807 . . . . . . . . 9  |-  ( s  e.  ~P X  -> 
s  C_  X )
82mrcidb2 13843 . . . . . . . . 9  |-  ( ( C  e.  (Moore `  X )  /\  s  C_  X )  ->  (
s  e.  C  <->  ( F `  s )  C_  s
) )
97, 8sylan2 461 . . . . . . . 8  |-  ( ( C  e.  (Moore `  X )  /\  s  e.  ~P X )  -> 
( s  e.  C  <->  ( F `  s ) 
C_  s ) )
109adantr 452 . . . . . . 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 448 . . . . . . . . . 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 13834 . . . . . . . . . . . 12  |-  ( C  e.  (Moore `  X
)  ->  F : ~P X --> C )
13 ffun 5593 . . . . . . . . . . . 12  |-  ( F : ~P X --> C  ->  Fun  F )
14 funiunfv 5995 . . . . . . . . . . . 12  |-  ( Fun 
F  ->  U_ t  e.  ( ~P s  i^i 
Fin ) ( F `
 t )  = 
U. ( F "
( ~P s  i^i 
Fin ) ) )
1512, 13, 143syl 19 . . . . . . . . . . 11  |-  ( C  e.  (Moore `  X
)  ->  U_ t  e.  ( ~P s  i^i 
Fin ) ( F `
 t )  = 
U. ( F "
( ~P s  i^i 
Fin ) ) )
1615ad2antrr 707 . . . . . . . . . 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 2471 . . . . . . . . 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 3375 . . . . . . . 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 4132 . . . . . . . 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 253 . . . . . . 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 245 . . . . . 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 424 . . . . 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 2784 . . . 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 419 . . 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 13878 . . 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 646 . 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 181 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 177    /\ wa 359    = wceq 1652    e. wcel 1725   A.wral 2705    i^i cin 3319    C_ wss 3320   ~Pcpw 3799   U.cuni 4015   U_ciun 4093   "cima 4881   Fun wfun 5448   -->wf 5450   ` cfv 5454   Fincfn 7109  Moorecmre 13807  mrClscmrc 13808  ACScacs 13810  Dirsetcdrs 14384  toInccipo 14577
This theorem is referenced by:  isacs4  14599
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  ax-un 4701  ax-cnex 9046  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-pre-mulgt0 9067
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  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-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-int 4051  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-riota 6549  df-recs 6633  df-rdg 6668  df-1o 6724  df-oadd 6728  df-er 6905  df-en 7110  df-dom 7111  df-sdom 7112  df-fin 7113  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294  df-nn 10001  df-2 10058  df-3 10059  df-4 10060  df-5 10061  df-6 10062  df-7 10063  df-8 10064  df-9 10065  df-10 10066  df-n0 10222  df-z 10283  df-dec 10383  df-uz 10489  df-fz 11044  df-struct 13471  df-ndx 13472  df-slot 13473  df-base 13474  df-tset 13548  df-ple 13549  df-ocomp 13550  df-mre 13811  df-mrc 13812  df-acs 13814  df-preset 14385  df-drs 14386  df-poset 14403  df-ipo 14578
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