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Theorem isacs4lem 14586
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 731 . . . . . . 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 3799 . . . . . . . 8  |-  ( t  e.  ~P ~P X  ->  t  C_  ~P X
)
32ad2antrl 709 . . . . . . 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 13838 . . . . . . 7  |-  ( ( C  e.  (Moore `  X )  /\  t  C_ 
~P X )  -> 
( F `  U. t )  =  ( F `  U. ( F " t ) ) )
61, 3, 5syl2anc 643 . . . . . 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 13826 . . . . . . . . . . . 12  |-  ( C  e.  (Moore `  X
)  ->  F : ~P X --> C )
8 ffn 5583 . . . . . . . . . . . 12  |-  ( F : ~P X --> C  ->  F  Fn  ~P X
)
97, 8syl 16 . . . . . . . . . . 11  |-  ( C  e.  (Moore `  X
)  ->  F  Fn  ~P X )
109adantr 452 . . . . . . . . . 10  |-  ( ( C  e.  (Moore `  X )  /\  (
t  e.  ~P ~P X  /\  (toInc `  t
)  e. Dirset ) )  ->  F  Fn  ~P X
)
11 simpll 731 . . . . . . . . . . 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 733 . . . . . . . . . . 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 734 . . . . . . . . . . 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 13841 . . . . . . . . . 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 734 . . . . . . . . . 10  |-  ( ( C  e.  (Moore `  X )  /\  (
t  e.  ~P ~P X  /\  (toInc `  t
)  e. Dirset ) )  ->  (toInc `  t )  e. Dirset )
162ad2antrl 709 . . . . . . . . . 10  |-  ( ( C  e.  (Moore `  X )  /\  (
t  e.  ~P ~P X  /\  (toInc `  t
)  e. Dirset ) )  ->  t  C_  ~P X
)
17 fvex 5734 . . . . . . . . . . . 12  |-  (mrCls `  C )  e.  _V
184, 17eqeltri 2505 . . . . . . . . . . 11  |-  F  e. 
_V
19 imaexg 5209 . . . . . . . . . . 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 14583 . . . . . . . . 9  |-  ( ( C  e.  (Moore `  X )  /\  (
t  e.  ~P ~P X  /\  (toInc `  t
)  e. Dirset ) )  ->  (toInc `  ( F " t ) )  e. Dirset
)
2221adantlr 696 . . . . . . . 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 5208 . . . . . . . . . . . 12  |-  ( F
" t )  C_  ran  F
24 frn 5589 . . . . . . . . . . . . 13  |-  ( F : ~P X --> C  ->  ran  F  C_  C )
257, 24syl 16 . . . . . . . . . . . 12  |-  ( C  e.  (Moore `  X
)  ->  ran  F  C_  C )
2623, 25syl5ss 3351 . . . . . . . . . . 11  |-  ( C  e.  (Moore `  X
)  ->  ( F " t )  C_  C
)
2718, 19ax-mp 8 . . . . . . . . . . . 12  |-  ( F
" t )  e. 
_V
2827elpw 3797 . . . . . . . . . . 11  |-  ( ( F " t )  e.  ~P C  <->  ( F " t )  C_  C
)
2926, 28sylibr 204 . . . . . . . . . 10  |-  ( C  e.  (Moore `  X
)  ->  ( F " t )  e.  ~P C )
3029ad2antrr 707 . . . . . . . . 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 732 . . . . . . . . 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 5720 . . . . . . . . . . . 12  |-  ( s  =  ( F "
t )  ->  (toInc `  s )  =  (toInc `  ( F " t
) ) )
3332eleq1d 2501 . . . . . . . . . . 11  |-  ( s  =  ( F "
t )  ->  (
(toInc `  s )  e. Dirset  <-> 
(toInc `  ( F " t ) )  e. Dirset
) )
34 unieq 4016 . . . . . . . . . . . 12  |-  ( s  =  ( F "
t )  ->  U. s  =  U. ( F "
t ) )
3534eleq1d 2501 . . . . . . . . . . 11  |-  ( s  =  ( F "
t )  ->  ( U. s  e.  C  <->  U. ( F " t
)  e.  C ) )
3633, 35imbi12d 312 . . . . . . . . . 10  |-  ( s  =  ( F "
t )  ->  (
( (toInc `  s
)  e. Dirset  ->  U. s  e.  C )  <->  ( (toInc `  ( F " t
) )  e. Dirset  ->  U. ( F " t
)  e.  C ) ) )
3736rspcva 3042 . . . . . . . . 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 643 . . . . . . . 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 13830 . . . . . . 7  |-  ( ( C  e.  (Moore `  X )  /\  U. ( F " t )  e.  C )  -> 
( F `  U. ( F " t ) )  =  U. ( F " t ) )
411, 39, 40syl2anc 643 . . . . . 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 2467 . . . . 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 589 . . . 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 2780 . . 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 424 . 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 672 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 set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725   A.wral 2697   _Vcvv 2948    C_ wss 3312   ~Pcpw 3791   U.cuni 4007   ran crn 4871   "cima 4873    Fn wfn 5441   -->wf 5442   ` cfv 5446  Moorecmre 13799  mrClscmrc 13800  Dirsetcdrs 14376  toInccipo 14569
This theorem is referenced by:  acsdrscl  14588  acsficl  14589  isacs5  14590  isacs4  14591  isacs3  14592
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 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059
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 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-riota 6541  df-recs 6625  df-rdg 6660  df-1o 6716  df-oadd 6720  df-er 6897  df-en 7102  df-dom 7103  df-sdom 7104  df-fin 7105  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-nn 9993  df-2 10050  df-3 10051  df-4 10052  df-5 10053  df-6 10054  df-7 10055  df-8 10056  df-9 10057  df-10 10058  df-n0 10214  df-z 10275  df-dec 10375  df-uz 10481  df-fz 11036  df-struct 13463  df-ndx 13464  df-slot 13465  df-base 13466  df-tset 13540  df-ple 13541  df-ocomp 13542  df-mre 13803  df-mrc 13804  df-preset 14377  df-drs 14378  df-poset 14395  df-ipo 14570
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