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Theorem acsfiel 15266
Description: A set is closed in an algebraic closure system iff it contains all closures of finite subsets. (Contributed by Stefan O'Rear, 2-Apr-2015.)
Hypothesis
Ref Expression
isacs2.f  |-  F  =  (mrCls `  C )
Assertion
Ref Expression
acsfiel  |-  ( C  e.  (ACS `  X
)  ->  ( S  e.  C  <->  ( S  C_  X  /\  A. y  e.  ( ~P S  i^i  Fin ) ( F `  y )  C_  S
) ) )
Distinct variable groups:    y, C    y, F    y, S    y, X

Proof of Theorem acsfiel
Dummy variable  s is distinct from all other variables.
StepHypRef Expression
1 acsmre 15264 . . . . 5  |-  ( C  e.  (ACS `  X
)  ->  C  e.  (Moore `  X ) )
2 mress 15205 . . . . 5  |-  ( ( C  e.  (Moore `  X )  /\  S  e.  C )  ->  S  C_  X )
31, 2sylan 469 . . . 4  |-  ( ( C  e.  (ACS `  X )  /\  S  e.  C )  ->  S  C_  X )
43ex 432 . . 3  |-  ( C  e.  (ACS `  X
)  ->  ( S  e.  C  ->  S  C_  X ) )
54pm4.71rd 633 . 2  |-  ( C  e.  (ACS `  X
)  ->  ( S  e.  C  <->  ( S  C_  X  /\  S  e.  C
) ) )
6 elfvdm 5874 . . . . . 6  |-  ( C  e.  (ACS `  X
)  ->  X  e.  dom ACS )
7 elpw2g 4556 . . . . . 6  |-  ( X  e.  dom ACS  ->  ( S  e.  ~P X  <->  S  C_  X
) )
86, 7syl 17 . . . . 5  |-  ( C  e.  (ACS `  X
)  ->  ( S  e.  ~P X  <->  S  C_  X
) )
98biimpar 483 . . . 4  |-  ( ( C  e.  (ACS `  X )  /\  S  C_  X )  ->  S  e.  ~P X )
10 isacs2.f . . . . . . 7  |-  F  =  (mrCls `  C )
1110isacs2 15265 . . . . . 6  |-  ( C  e.  (ACS `  X
)  <->  ( C  e.  (Moore `  X )  /\  A. s  e.  ~P  X ( s  e.  C  <->  A. y  e.  ( ~P s  i^i  Fin ) ( F `  y )  C_  s
) ) )
1211simprbi 462 . . . . 5  |-  ( C  e.  (ACS `  X
)  ->  A. s  e.  ~P  X ( s  e.  C  <->  A. y  e.  ( ~P s  i^i 
Fin ) ( F `
 y )  C_  s ) )
1312adantr 463 . . . 4  |-  ( ( C  e.  (ACS `  X )  /\  S  C_  X )  ->  A. s  e.  ~P  X ( s  e.  C  <->  A. y  e.  ( ~P s  i^i 
Fin ) ( F `
 y )  C_  s ) )
14 eleq1 2474 . . . . . 6  |-  ( s  =  S  ->  (
s  e.  C  <->  S  e.  C ) )
15 pweq 3957 . . . . . . . 8  |-  ( s  =  S  ->  ~P s  =  ~P S
)
1615ineq1d 3639 . . . . . . 7  |-  ( s  =  S  ->  ( ~P s  i^i  Fin )  =  ( ~P S  i^i  Fin ) )
17 sseq2 3463 . . . . . . 7  |-  ( s  =  S  ->  (
( F `  y
)  C_  s  <->  ( F `  y )  C_  S
) )
1816, 17raleqbidv 3017 . . . . . 6  |-  ( s  =  S  ->  ( A. y  e.  ( ~P s  i^i  Fin )
( F `  y
)  C_  s  <->  A. y  e.  ( ~P S  i^i  Fin ) ( F `  y )  C_  S
) )
1914, 18bibi12d 319 . . . . 5  |-  ( s  =  S  ->  (
( s  e.  C  <->  A. y  e.  ( ~P s  i^i  Fin )
( F `  y
)  C_  s )  <->  ( S  e.  C  <->  A. y  e.  ( ~P S  i^i  Fin ) ( F `  y )  C_  S
) ) )
2019rspcva 3157 . . . 4  |-  ( ( S  e.  ~P X  /\  A. s  e.  ~P  X ( s  e.  C  <->  A. y  e.  ( ~P s  i^i  Fin ) ( F `  y )  C_  s
) )  ->  ( S  e.  C  <->  A. y  e.  ( ~P S  i^i  Fin ) ( F `  y )  C_  S
) )
219, 13, 20syl2anc 659 . . 3  |-  ( ( C  e.  (ACS `  X )  /\  S  C_  X )  ->  ( S  e.  C  <->  A. y  e.  ( ~P S  i^i  Fin ) ( F `  y )  C_  S
) )
2221pm5.32da 639 . 2  |-  ( C  e.  (ACS `  X
)  ->  ( ( S  C_  X  /\  S  e.  C )  <->  ( S  C_  X  /\  A. y  e.  ( ~P S  i^i  Fin ) ( F `  y )  C_  S
) ) )
235, 22bitrd 253 1  |-  ( C  e.  (ACS `  X
)  ->  ( S  e.  C  <->  ( S  C_  X  /\  A. y  e.  ( ~P S  i^i  Fin ) ( F `  y )  C_  S
) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    = wceq 1405    e. wcel 1842   A.wral 2753    i^i cin 3412    C_ wss 3413   ~Pcpw 3954   dom cdm 4822   ` cfv 5568   Fincfn 7553  Moorecmre 15194  mrClscmrc 15195  ACScacs 15197
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629  ax-un 6573
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2758  df-rex 2759  df-rab 2762  df-v 3060  df-sbc 3277  df-csb 3373  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-op 3978  df-uni 4191  df-int 4227  df-iun 4272  df-br 4395  df-opab 4453  df-mpt 4454  df-id 4737  df-xp 4828  df-rel 4829  df-cnv 4830  df-co 4831  df-dm 4832  df-rn 4833  df-res 4834  df-ima 4835  df-iota 5532  df-fun 5570  df-fn 5571  df-f 5572  df-fv 5576  df-mre 15198  df-mrc 15199  df-acs 15201
This theorem is referenced by:  acsfiel2  15267  isacs3lem  16118
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