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Theorem isacs5 14291
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 14285 . . 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 14287 . . 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 14288 . . 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 3646 . . . . . . . . 9  |-  ( s  e.  ~P X  -> 
s  C_  X )
82mrcidb2 13536 . . . . . . . . 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 13527 . . . . . . . . . . . 12  |-  ( C  e.  (Moore `  X
)  ->  F : ~P X --> C )
13 ffun 5407 . . . . . . . . . . . 12  |-  ( F : ~P X --> C  ->  Fun  F )
14 funiunfv 5790 . . . . . . . . . . . 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 2331 . . . . . . . . 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 3218 . . . . . . . 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 3959 . . . . . . . 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 2634 . . . 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 13571 . . 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 1632    e. wcel 1696   A.wral 2556    i^i cin 3164    C_ wss 3165   ~Pcpw 3638   U.cuni 3843   U_ciun 3921   "cima 4708   Fun wfun 5265   -->wf 5267   ` cfv 5271   Fincfn 6879  Moorecmre 13500  mrClscmrc 13501  ACScacs 13503  Dirsetcdrs 14077  toInccipo 14270
This theorem is referenced by:  isacs4  14292
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-oadd 6499  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-nn 9763  df-2 9820  df-3 9821  df-4 9822  df-5 9823  df-6 9824  df-7 9825  df-8 9826  df-9 9827  df-10 9828  df-n0 9982  df-z 10041  df-dec 10141  df-uz 10247  df-fz 10799  df-struct 13166  df-ndx 13167  df-slot 13168  df-base 13169  df-tset 13243  df-ple 13244  df-ocomp 13245  df-mre 13504  df-mrc 13505  df-acs 13507  df-preset 14078  df-drs 14079  df-poset 14096  df-ipo 14271
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