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Theorem isacs4lem 15338
Description: In a closure system in which directed unions of closed sets are closed, closure commutes with directed unions. (Contributed by Stefan O'Rear, 2-Apr-2015.)
Hypothesis
Ref Expression
acsdrscl.f  |-  F  =  (mrCls `  C )
Assertion
Ref Expression
isacs4lem  |-  ( ( 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 ) ) ) )
Distinct variable groups:    C, s,
t    F, s, t    X, s, t

Proof of Theorem isacs4lem
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpll 753 . . . . . . 7  |-  ( ( ( C  e.  (Moore `  X )  /\  A. s  e.  ~P  C
( (toInc `  s
)  e. Dirset  ->  U. s  e.  C ) )  /\  ( t  e.  ~P ~P X  /\  (toInc `  t )  e. Dirset )
)  ->  C  e.  (Moore `  X ) )
2 elpwi 3869 . . . . . . . 8  |-  ( t  e.  ~P ~P X  ->  t  C_  ~P X
)
32ad2antrl 727 . . . . . . 7  |-  ( ( ( C  e.  (Moore `  X )  /\  A. s  e.  ~P  C
( (toInc `  s
)  e. Dirset  ->  U. s  e.  C ) )  /\  ( t  e.  ~P ~P X  /\  (toInc `  t )  e. Dirset )
)  ->  t  C_  ~P X )
4 acsdrscl.f . . . . . . . 8  |-  F  =  (mrCls `  C )
54mrcuni 14559 . . . . . . 7  |-  ( ( C  e.  (Moore `  X )  /\  t  C_ 
~P X )  -> 
( F `  U. t )  =  ( F `  U. ( F " t ) ) )
61, 3, 5syl2anc 661 . . . . . 6  |-  ( ( ( C  e.  (Moore `  X )  /\  A. s  e.  ~P  C
( (toInc `  s
)  e. Dirset  ->  U. s  e.  C ) )  /\  ( t  e.  ~P ~P X  /\  (toInc `  t )  e. Dirset )
)  ->  ( F `  U. t )  =  ( F `  U. ( F " t ) ) )
74mrcf 14547 . . . . . . . . . . . 12  |-  ( C  e.  (Moore `  X
)  ->  F : ~P X --> C )
8 ffn 5559 . . . . . . . . . . . 12  |-  ( F : ~P X --> C  ->  F  Fn  ~P X
)
97, 8syl 16 . . . . . . . . . . 11  |-  ( C  e.  (Moore `  X
)  ->  F  Fn  ~P X )
109adantr 465 . . . . . . . . . 10  |-  ( ( C  e.  (Moore `  X )  /\  (
t  e.  ~P ~P X  /\  (toInc `  t
)  e. Dirset ) )  ->  F  Fn  ~P X
)
11 simpll 753 . . . . . . . . . . 11  |-  ( ( ( C  e.  (Moore `  X )  /\  (
t  e.  ~P ~P X  /\  (toInc `  t
)  e. Dirset ) )  /\  ( x  C_  y  /\  y  C_  X ) )  ->  C  e.  (Moore `  X ) )
12 simprl 755 . . . . . . . . . . 11  |-  ( ( ( C  e.  (Moore `  X )  /\  (
t  e.  ~P ~P X  /\  (toInc `  t
)  e. Dirset ) )  /\  ( x  C_  y  /\  y  C_  X ) )  ->  x  C_  y
)
13 simprr 756 . . . . . . . . . . 11  |-  ( ( ( C  e.  (Moore `  X )  /\  (
t  e.  ~P ~P X  /\  (toInc `  t
)  e. Dirset ) )  /\  ( x  C_  y  /\  y  C_  X ) )  ->  y  C_  X )
1411, 4, 12, 13mrcssd 14562 . . . . . . . . . 10  |-  ( ( ( C  e.  (Moore `  X )  /\  (
t  e.  ~P ~P X  /\  (toInc `  t
)  e. Dirset ) )  /\  ( x  C_  y  /\  y  C_  X ) )  ->  ( F `  x )  C_  ( F `  y )
)
15 simprr 756 . . . . . . . . . 10  |-  ( ( C  e.  (Moore `  X )  /\  (
t  e.  ~P ~P X  /\  (toInc `  t
)  e. Dirset ) )  ->  (toInc `  t )  e. Dirset )
162ad2antrl 727 . . . . . . . . . 10  |-  ( ( C  e.  (Moore `  X )  /\  (
t  e.  ~P ~P X  /\  (toInc `  t
)  e. Dirset ) )  ->  t  C_  ~P X
)
17 fvex 5701 . . . . . . . . . . . 12  |-  (mrCls `  C )  e.  _V
184, 17eqeltri 2513 . . . . . . . . . . 11  |-  F  e. 
_V
19 imaexg 6515 . . . . . . . . . . 11  |-  ( F  e.  _V  ->  ( F " t )  e. 
_V )
2018, 19mp1i 12 . . . . . . . . . 10  |-  ( ( C  e.  (Moore `  X )  /\  (
t  e.  ~P ~P X  /\  (toInc `  t
)  e. Dirset ) )  ->  ( F " t
)  e.  _V )
2110, 14, 15, 16, 20ipodrsima 15335 . . . . . . . . 9  |-  ( ( C  e.  (Moore `  X )  /\  (
t  e.  ~P ~P X  /\  (toInc `  t
)  e. Dirset ) )  ->  (toInc `  ( F " t ) )  e. Dirset
)
2221adantlr 714 . . . . . . . 8  |-  ( ( ( C  e.  (Moore `  X )  /\  A. s  e.  ~P  C
( (toInc `  s
)  e. Dirset  ->  U. s  e.  C ) )  /\  ( t  e.  ~P ~P X  /\  (toInc `  t )  e. Dirset )
)  ->  (toInc `  ( F " t ) )  e. Dirset )
23 imassrn 5180 . . . . . . . . . . . 12  |-  ( F
" t )  C_  ran  F
24 frn 5565 . . . . . . . . . . . . 13  |-  ( F : ~P X --> C  ->  ran  F  C_  C )
257, 24syl 16 . . . . . . . . . . . 12  |-  ( C  e.  (Moore `  X
)  ->  ran  F  C_  C )
2623, 25syl5ss 3367 . . . . . . . . . . 11  |-  ( C  e.  (Moore `  X
)  ->  ( F " t )  C_  C
)
2718, 19ax-mp 5 . . . . . . . . . . . 12  |-  ( F
" t )  e. 
_V
2827elpw 3866 . . . . . . . . . . 11  |-  ( ( F " t )  e.  ~P C  <->  ( F " t )  C_  C
)
2926, 28sylibr 212 . . . . . . . . . 10  |-  ( C  e.  (Moore `  X
)  ->  ( F " t )  e.  ~P C )
3029ad2antrr 725 . . . . . . . . 9  |-  ( ( ( C  e.  (Moore `  X )  /\  A. s  e.  ~P  C
( (toInc `  s
)  e. Dirset  ->  U. s  e.  C ) )  /\  ( t  e.  ~P ~P X  /\  (toInc `  t )  e. Dirset )
)  ->  ( F " t )  e.  ~P C )
31 simplr 754 . . . . . . . . 9  |-  ( ( ( C  e.  (Moore `  X )  /\  A. s  e.  ~P  C
( (toInc `  s
)  e. Dirset  ->  U. s  e.  C ) )  /\  ( t  e.  ~P ~P X  /\  (toInc `  t )  e. Dirset )
)  ->  A. s  e.  ~P  C ( (toInc `  s )  e. Dirset  ->  U. s  e.  C ) )
32 fveq2 5691 . . . . . . . . . . . 12  |-  ( s  =  ( F "
t )  ->  (toInc `  s )  =  (toInc `  ( F " t
) ) )
3332eleq1d 2509 . . . . . . . . . . 11  |-  ( s  =  ( F "
t )  ->  (
(toInc `  s )  e. Dirset  <-> 
(toInc `  ( F " t ) )  e. Dirset
) )
34 unieq 4099 . . . . . . . . . . . 12  |-  ( s  =  ( F "
t )  ->  U. s  =  U. ( F "
t ) )
3534eleq1d 2509 . . . . . . . . . . 11  |-  ( s  =  ( F "
t )  ->  ( U. s  e.  C  <->  U. ( F " t
)  e.  C ) )
3633, 35imbi12d 320 . . . . . . . . . 10  |-  ( s  =  ( F "
t )  ->  (
( (toInc `  s
)  e. Dirset  ->  U. s  e.  C )  <->  ( (toInc `  ( F " t
) )  e. Dirset  ->  U. ( F " t
)  e.  C ) ) )
3736rspcva 3071 . . . . . . . . 9  |-  ( ( ( F " t
)  e.  ~P C  /\  A. s  e.  ~P  C ( (toInc `  s )  e. Dirset  ->  U. s  e.  C ) )  ->  ( (toInc `  ( F " t
) )  e. Dirset  ->  U. ( F " t
)  e.  C ) )
3830, 31, 37syl2anc 661 . . . . . . . 8  |-  ( ( ( C  e.  (Moore `  X )  /\  A. s  e.  ~P  C
( (toInc `  s
)  e. Dirset  ->  U. s  e.  C ) )  /\  ( t  e.  ~P ~P X  /\  (toInc `  t )  e. Dirset )
)  ->  ( (toInc `  ( F " t
) )  e. Dirset  ->  U. ( F " t
)  e.  C ) )
3922, 38mpd 15 . . . . . . 7  |-  ( ( ( C  e.  (Moore `  X )  /\  A. s  e.  ~P  C
( (toInc `  s
)  e. Dirset  ->  U. s  e.  C ) )  /\  ( t  e.  ~P ~P X  /\  (toInc `  t )  e. Dirset )
)  ->  U. ( F " t )  e.  C )
404mrcid 14551 . . . . . . 7  |-  ( ( C  e.  (Moore `  X )  /\  U. ( F " t )  e.  C )  -> 
( F `  U. ( F " t ) )  =  U. ( F " t ) )
411, 39, 40syl2anc 661 . . . . . 6  |-  ( ( ( C  e.  (Moore `  X )  /\  A. s  e.  ~P  C
( (toInc `  s
)  e. Dirset  ->  U. s  e.  C ) )  /\  ( t  e.  ~P ~P X  /\  (toInc `  t )  e. Dirset )
)  ->  ( F `  U. ( F "
t ) )  = 
U. ( F "
t ) )
426, 41eqtrd 2475 . . . . 5  |-  ( ( ( C  e.  (Moore `  X )  /\  A. s  e.  ~P  C
( (toInc `  s
)  e. Dirset  ->  U. s  e.  C ) )  /\  ( t  e.  ~P ~P X  /\  (toInc `  t )  e. Dirset )
)  ->  ( F `  U. t )  = 
U. ( F "
t ) )
4342exp32 605 . . . 4  |-  ( ( C  e.  (Moore `  X )  /\  A. s  e.  ~P  C
( (toInc `  s
)  e. Dirset  ->  U. s  e.  C ) )  -> 
( t  e.  ~P ~P X  ->  ( (toInc `  t )  e. Dirset  ->  ( F `  U. t
)  =  U. ( F " t ) ) ) )
4443ralrimiv 2798 . . 3  |-  ( ( C  e.  (Moore `  X )  /\  A. s  e.  ~P  C
( (toInc `  s
)  e. Dirset  ->  U. s  e.  C ) )  ->  A. t  e.  ~P  ~P X ( (toInc `  t )  e. Dirset  ->  ( F `  U. t
)  =  U. ( F " t ) ) )
4544ex 434 . 2  |-  ( C  e.  (Moore `  X
)  ->  ( A. s  e.  ~P  C
( (toInc `  s
)  e. Dirset  ->  U. s  e.  C )  ->  A. t  e.  ~P  ~P X ( (toInc `  t )  e. Dirset  ->  ( F `  U. t )  =  U. ( F " t ) ) ) )
4645imdistani 690 1  |-  ( ( 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 ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1369    e. wcel 1756   A.wral 2715   _Vcvv 2972    C_ wss 3328   ~Pcpw 3860   U.cuni 4091   ran crn 4841   "cima 4843    Fn wfn 5413   -->wf 5414   ` cfv 5418  Moorecmre 14520  mrClscmrc 14521  Dirsetcdrs 15097  toInccipo 15321
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372  ax-cnex 9338  ax-resscn 9339  ax-1cn 9340  ax-icn 9341  ax-addcl 9342  ax-addrcl 9343  ax-mulcl 9344  ax-mulrcl 9345  ax-mulcom 9346  ax-addass 9347  ax-mulass 9348  ax-distr 9349  ax-i2m1 9350  ax-1ne0 9351  ax-1rid 9352  ax-rnegex 9353  ax-rrecex 9354  ax-cnre 9355  ax-pre-lttri 9356  ax-pre-lttrn 9357  ax-pre-ltadd 9358  ax-pre-mulgt0 9359
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-nel 2609  df-ral 2720  df-rex 2721  df-reu 2722  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-pss 3344  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-tp 3882  df-op 3884  df-uni 4092  df-int 4129  df-iun 4173  df-br 4293  df-opab 4351  df-mpt 4352  df-tr 4386  df-eprel 4632  df-id 4636  df-po 4641  df-so 4642  df-fr 4679  df-we 4681  df-ord 4722  df-on 4723  df-lim 4724  df-suc 4725  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-riota 6052  df-ov 6094  df-oprab 6095  df-mpt2 6096  df-om 6477  df-1st 6577  df-2nd 6578  df-recs 6832  df-rdg 6866  df-1o 6920  df-oadd 6924  df-er 7101  df-en 7311  df-dom 7312  df-sdom 7313  df-fin 7314  df-pnf 9420  df-mnf 9421  df-xr 9422  df-ltxr 9423  df-le 9424  df-sub 9597  df-neg 9598  df-nn 10323  df-2 10380  df-3 10381  df-4 10382  df-5 10383  df-6 10384  df-7 10385  df-8 10386  df-9 10387  df-10 10388  df-n0 10580  df-z 10647  df-dec 10756  df-uz 10862  df-fz 11438  df-struct 14176  df-ndx 14177  df-slot 14178  df-base 14179  df-tset 14257  df-ple 14258  df-ocomp 14259  df-mre 14524  df-mrc 14525  df-preset 15098  df-drs 15099  df-poset 15116  df-ipo 15322
This theorem is referenced by:  acsdrscl  15340  acsficl  15341  isacs5  15342  isacs4  15343  isacs3  15344
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