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Theorem isacs4lem 15671
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 4025 . . . . . . . 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 14892 . . . . . . 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 14880 . . . . . . . . . . . 12  |-  ( C  e.  (Moore `  X
)  ->  F : ~P X --> C )
8 ffn 5737 . . . . . . . . . . . 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 14895 . . . . . . . . . 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 5882 . . . . . . . . . . . 12  |-  (mrCls `  C )  e.  _V
184, 17eqeltri 2551 . . . . . . . . . . 11  |-  F  e. 
_V
19 imaexg 6732 . . . . . . . . . . 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 15668 . . . . . . . . 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 5354 . . . . . . . . . . . 12  |-  ( F
" t )  C_  ran  F
24 frn 5743 . . . . . . . . . . . . 13  |-  ( F : ~P X --> C  ->  ran  F  C_  C )
257, 24syl 16 . . . . . . . . . . . 12  |-  ( C  e.  (Moore `  X
)  ->  ran  F  C_  C )
2623, 25syl5ss 3520 . . . . . . . . . . 11  |-  ( C  e.  (Moore `  X
)  ->  ( F " t )  C_  C
)
2718, 19ax-mp 5 . . . . . . . . . . . 12  |-  ( F
" t )  e. 
_V
2827elpw 4022 . . . . . . . . . . 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 5872 . . . . . . . . . . . 12  |-  ( s  =  ( F "
t )  ->  (toInc `  s )  =  (toInc `  ( F " t
) ) )
3332eleq1d 2536 . . . . . . . . . . 11  |-  ( s  =  ( F "
t )  ->  (
(toInc `  s )  e. Dirset  <-> 
(toInc `  ( F " t ) )  e. Dirset
) )
34 unieq 4259 . . . . . . . . . . . 12  |-  ( s  =  ( F "
t )  ->  U. s  =  U. ( F "
t ) )
3534eleq1d 2536 . . . . . . . . . . 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 3217 . . . . . . . . 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 14884 . . . . . . 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 2508 . . . . 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 2879 . . 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 1379    e. wcel 1767   A.wral 2817   _Vcvv 3118    C_ wss 3481   ~Pcpw 4016   U.cuni 4251   ran crn 5006   "cima 5008    Fn wfn 5589   -->wf 5590   ` cfv 5594  Moorecmre 14853  mrClscmrc 14854  Dirsetcdrs 15430  toInccipo 15654
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-cnex 9560  ax-resscn 9561  ax-1cn 9562  ax-icn 9563  ax-addcl 9564  ax-addrcl 9565  ax-mulcl 9566  ax-mulrcl 9567  ax-mulcom 9568  ax-addass 9569  ax-mulass 9570  ax-distr 9571  ax-i2m1 9572  ax-1ne0 9573  ax-1rid 9574  ax-rnegex 9575  ax-rrecex 9576  ax-cnre 9577  ax-pre-lttri 9578  ax-pre-lttrn 9579  ax-pre-ltadd 9580  ax-pre-mulgt0 9581
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2822  df-rex 2823  df-reu 2824  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-int 4289  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  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 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6696  df-1st 6795  df-2nd 6796  df-recs 7054  df-rdg 7088  df-1o 7142  df-oadd 7146  df-er 7323  df-en 7529  df-dom 7530  df-sdom 7531  df-fin 7532  df-pnf 9642  df-mnf 9643  df-xr 9644  df-ltxr 9645  df-le 9646  df-sub 9819  df-neg 9820  df-nn 10549  df-2 10606  df-3 10607  df-4 10608  df-5 10609  df-6 10610  df-7 10611  df-8 10612  df-9 10613  df-10 10614  df-n0 10808  df-z 10877  df-dec 10989  df-uz 11095  df-fz 11685  df-struct 14508  df-ndx 14509  df-slot 14510  df-base 14511  df-tset 14590  df-ple 14591  df-ocomp 14592  df-mre 14857  df-mrc 14858  df-preset 15431  df-drs 15432  df-poset 15449  df-ipo 15655
This theorem is referenced by:  acsdrscl  15673  acsficl  15674  isacs5  15675  isacs4  15676  isacs3  15677
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