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Theorem acsficl 14373
Description: A closure in an algebraic closure system is the union of the closures of finite subsets. (Contributed by Stefan O'Rear, 2-Apr-2015.)
Hypothesis
Ref Expression
acsdrscl.f  |-  F  =  (mrCls `  C )
Assertion
Ref Expression
acsficl  |-  ( ( C  e.  (ACS `  X )  /\  S  C_  X )  ->  ( F `  S )  =  U. ( F "
( ~P S  i^i  Fin ) ) )

Proof of Theorem acsficl
Dummy variables  s 
t are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elfvdm 5637 . . . 4  |-  ( C  e.  (ACS `  X
)  ->  X  e.  dom ACS )
2 elpw2g 4255 . . . 4  |-  ( X  e.  dom ACS  ->  ( S  e.  ~P X  <->  S  C_  X
) )
31, 2syl 15 . . 3  |-  ( C  e.  (ACS `  X
)  ->  ( S  e.  ~P X  <->  S  C_  X
) )
43biimpar 471 . 2  |-  ( ( C  e.  (ACS `  X )  /\  S  C_  X )  ->  S  e.  ~P X )
5 isacs3lem 14368 . . . . 5  |-  ( C  e.  (ACS `  X
)  ->  ( C  e.  (Moore `  X )  /\  A. s  e.  ~P  C ( (toInc `  s )  e. Dirset  ->  U. s  e.  C ) ) )
6 acsdrscl.f . . . . . 6  |-  F  =  (mrCls `  C )
76isacs4lem 14370 . . . . 5  |-  ( ( 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 ) ) ) )
86isacs5lem 14371 . . . . 5  |-  ( ( 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 )
) ) )
95, 7, 83syl 18 . . . 4  |-  ( C  e.  (ACS `  X
)  ->  ( C  e.  (Moore `  X )  /\  A. s  e.  ~P  X ( F `  s )  =  U. ( F " ( ~P s  i^i  Fin )
) ) )
109simprd 449 . . 3  |-  ( C  e.  (ACS `  X
)  ->  A. s  e.  ~P  X ( F `
 s )  = 
U. ( F "
( ~P s  i^i 
Fin ) ) )
1110adantr 451 . 2  |-  ( ( C  e.  (ACS `  X )  /\  S  C_  X )  ->  A. s  e.  ~P  X ( F `
 s )  = 
U. ( F "
( ~P s  i^i 
Fin ) ) )
12 fveq2 5608 . . . 4  |-  ( s  =  S  ->  ( F `  s )  =  ( F `  S ) )
13 pweq 3704 . . . . . . 7  |-  ( s  =  S  ->  ~P s  =  ~P S
)
1413ineq1d 3445 . . . . . 6  |-  ( s  =  S  ->  ( ~P s  i^i  Fin )  =  ( ~P S  i^i  Fin ) )
1514imaeq2d 5094 . . . . 5  |-  ( s  =  S  ->  ( F " ( ~P s  i^i  Fin ) )  =  ( F " ( ~P S  i^i  Fin )
) )
1615unieqd 3919 . . . 4  |-  ( s  =  S  ->  U. ( F " ( ~P s  i^i  Fin ) )  = 
U. ( F "
( ~P S  i^i  Fin ) ) )
1712, 16eqeq12d 2372 . . 3  |-  ( s  =  S  ->  (
( F `  s
)  =  U. ( F " ( ~P s  i^i  Fin ) )  <->  ( F `  S )  =  U. ( F " ( ~P S  i^i  Fin )
) ) )
1817rspcva 2958 . 2  |-  ( ( S  e.  ~P X  /\  A. s  e.  ~P  X ( F `  s )  =  U. ( F " ( ~P s  i^i  Fin )
) )  ->  ( F `  S )  =  U. ( F "
( ~P S  i^i  Fin ) ) )
194, 11, 18syl2anc 642 1  |-  ( ( C  e.  (ACS `  X )  /\  S  C_  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 1642    e. wcel 1710   A.wral 2619    i^i cin 3227    C_ wss 3228   ~Pcpw 3701   U.cuni 3908   dom cdm 4771   "cima 4774   ` cfv 5337   Fincfn 6951  Moorecmre 13583  mrClscmrc 13584  ACScacs 13586  Dirsetcdrs 14160  toInccipo 14353
This theorem is referenced by:  acsficld  14377  isnacs3  26108
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-sep 4222  ax-nul 4230  ax-pow 4269  ax-pr 4295  ax-un 4594  ax-cnex 8883  ax-resscn 8884  ax-1cn 8885  ax-icn 8886  ax-addcl 8887  ax-addrcl 8888  ax-mulcl 8889  ax-mulrcl 8890  ax-mulcom 8891  ax-addass 8892  ax-mulass 8893  ax-distr 8894  ax-i2m1 8895  ax-1ne0 8896  ax-1rid 8897  ax-rnegex 8898  ax-rrecex 8899  ax-cnre 8900  ax-pre-lttri 8901  ax-pre-lttrn 8902  ax-pre-ltadd 8903  ax-pre-mulgt0 8904
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-nel 2524  df-ral 2624  df-rex 2625  df-reu 2626  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-pss 3244  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-tp 3724  df-op 3725  df-uni 3909  df-int 3944  df-iun 3988  df-br 4105  df-opab 4159  df-mpt 4160  df-tr 4195  df-eprel 4387  df-id 4391  df-po 4396  df-so 4397  df-fr 4434  df-we 4436  df-ord 4477  df-on 4478  df-lim 4479  df-suc 4480  df-om 4739  df-xp 4777  df-rel 4778  df-cnv 4779  df-co 4780  df-dm 4781  df-rn 4782  df-res 4783  df-ima 4784  df-iota 5301  df-fun 5339  df-fn 5340  df-f 5341  df-f1 5342  df-fo 5343  df-f1o 5344  df-fv 5345  df-ov 5948  df-oprab 5949  df-mpt2 5950  df-1st 6209  df-2nd 6210  df-riota 6391  df-recs 6475  df-rdg 6510  df-1o 6566  df-oadd 6570  df-er 6747  df-en 6952  df-dom 6953  df-sdom 6954  df-fin 6955  df-pnf 8959  df-mnf 8960  df-xr 8961  df-ltxr 8962  df-le 8963  df-sub 9129  df-neg 9130  df-nn 9837  df-2 9894  df-3 9895  df-4 9896  df-5 9897  df-6 9898  df-7 9899  df-8 9900  df-9 9901  df-10 9902  df-n0 10058  df-z 10117  df-dec 10217  df-uz 10323  df-fz 10875  df-struct 13247  df-ndx 13248  df-slot 13249  df-base 13250  df-tset 13324  df-ple 13325  df-ocomp 13326  df-mre 13587  df-mrc 13588  df-acs 13590  df-preset 14161  df-drs 14162  df-poset 14179  df-ipo 14354
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