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Theorem isacs4lem 14521
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 3750 . . . . . . . 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 13773 . . . . . . 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 13761 . . . . . . . . . . . 12  |-  ( C  e.  (Moore `  X
)  ->  F : ~P X --> C )
8 ffn 5531 . . . . . . . . . . . 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 13776 . . . . . . . . . 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 5682 . . . . . . . . . . . 12  |-  (mrCls `  C )  e.  _V
184, 17eqeltri 2457 . . . . . . . . . . 11  |-  F  e. 
_V
19 imaexg 5157 . . . . . . . . . . 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 14518 . . . . . . . . 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 5156 . . . . . . . . . . . 12  |-  ( F
" t )  C_  ran  F
24 frn 5537 . . . . . . . . . . . . 13  |-  ( F : ~P X --> C  ->  ran  F  C_  C )
257, 24syl 16 . . . . . . . . . . . 12  |-  ( C  e.  (Moore `  X
)  ->  ran  F  C_  C )
2623, 25syl5ss 3302 . . . . . . . . . . 11  |-  ( C  e.  (Moore `  X
)  ->  ( F " t )  C_  C
)
2718, 19ax-mp 8 . . . . . . . . . . . 12  |-  ( F
" t )  e. 
_V
2827elpw 3748 . . . . . . . . . . 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 5668 . . . . . . . . . . . 12  |-  ( s  =  ( F "
t )  ->  (toInc `  s )  =  (toInc `  ( F " t
) ) )
3332eleq1d 2453 . . . . . . . . . . 11  |-  ( s  =  ( F "
t )  ->  (
(toInc `  s )  e. Dirset  <-> 
(toInc `  ( F " t ) )  e. Dirset
) )
34 unieq 3966 . . . . . . . . . . . 12  |-  ( s  =  ( F "
t )  ->  U. s  =  U. ( F "
t ) )
3534eleq1d 2453 . . . . . . . . . . 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 2993 . . . . . . . . 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 13765 . . . . . . 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 2419 . . . . 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 2731 . . 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 1649    e. wcel 1717   A.wral 2649   _Vcvv 2899    C_ wss 3263   ~Pcpw 3742   U.cuni 3957   ran crn 4819   "cima 4821    Fn wfn 5389   -->wf 5390   ` cfv 5394  Moorecmre 13734  mrClscmrc 13735  Dirsetcdrs 14311  toInccipo 14504
This theorem is referenced by:  acsdrscl  14523  acsficl  14524  isacs5  14525  isacs4  14526  isacs3  14527
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641  ax-cnex 8979  ax-resscn 8980  ax-1cn 8981  ax-icn 8982  ax-addcl 8983  ax-addrcl 8984  ax-mulcl 8985  ax-mulrcl 8986  ax-mulcom 8987  ax-addass 8988  ax-mulass 8989  ax-distr 8990  ax-i2m1 8991  ax-1ne0 8992  ax-1rid 8993  ax-rnegex 8994  ax-rrecex 8995  ax-cnre 8996  ax-pre-lttri 8997  ax-pre-lttrn 8998  ax-pre-ltadd 8999  ax-pre-mulgt0 9000
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-nel 2553  df-ral 2654  df-rex 2655  df-reu 2656  df-rab 2658  df-v 2901  df-sbc 3105  df-csb 3195  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-pss 3279  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-tp 3765  df-op 3766  df-uni 3958  df-int 3993  df-iun 4037  df-br 4154  df-opab 4208  df-mpt 4209  df-tr 4244  df-eprel 4435  df-id 4439  df-po 4444  df-so 4445  df-fr 4482  df-we 4484  df-ord 4525  df-on 4526  df-lim 4527  df-suc 4528  df-om 4786  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-f1 5399  df-fo 5400  df-f1o 5401  df-fv 5402  df-ov 6023  df-oprab 6024  df-mpt2 6025  df-1st 6288  df-2nd 6289  df-riota 6485  df-recs 6569  df-rdg 6604  df-1o 6660  df-oadd 6664  df-er 6841  df-en 7046  df-dom 7047  df-sdom 7048  df-fin 7049  df-pnf 9055  df-mnf 9056  df-xr 9057  df-ltxr 9058  df-le 9059  df-sub 9225  df-neg 9226  df-nn 9933  df-2 9990  df-3 9991  df-4 9992  df-5 9993  df-6 9994  df-7 9995  df-8 9996  df-9 9997  df-10 9998  df-n0 10154  df-z 10215  df-dec 10315  df-uz 10421  df-fz 10976  df-struct 13398  df-ndx 13399  df-slot 13400  df-base 13401  df-tset 13475  df-ple 13476  df-ocomp 13477  df-mre 13738  df-mrc 13739  df-preset 14312  df-drs 14313  df-poset 14330  df-ipo 14505
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