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Theorem acsfiel 13556
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 13554 . . . . 5  |-  ( C  e.  (ACS `  X
)  ->  C  e.  (Moore `  X ) )
2 mress 13495 . . . . 5  |-  ( ( C  e.  (Moore `  X )  /\  S  e.  C )  ->  S  C_  X )
31, 2sylan 457 . . . 4  |-  ( ( C  e.  (ACS `  X )  /\  S  e.  C )  ->  S  C_  X )
43ex 423 . . 3  |-  ( C  e.  (ACS `  X
)  ->  ( S  e.  C  ->  S  C_  X ) )
54pm4.71rd 616 . 2  |-  ( C  e.  (ACS `  X
)  ->  ( S  e.  C  <->  ( S  C_  X  /\  S  e.  C
) ) )
6 elfvdm 5554 . . . . . 6  |-  ( C  e.  (ACS `  X
)  ->  X  e.  dom ACS )
7 elpw2g 4174 . . . . . 6  |-  ( X  e.  dom ACS  ->  ( S  e.  ~P X  <->  S  C_  X
) )
86, 7syl 15 . . . . 5  |-  ( C  e.  (ACS `  X
)  ->  ( S  e.  ~P X  <->  S  C_  X
) )
98biimpar 471 . . . 4  |-  ( ( C  e.  (ACS `  X )  /\  S  C_  X )  ->  S  e.  ~P X )
10 isacs2.f . . . . . . 7  |-  F  =  (mrCls `  C )
1110isacs2 13555 . . . . . 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 450 . . . . 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 451 . . . 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 2343 . . . . . 6  |-  ( s  =  S  ->  (
s  e.  C  <->  S  e.  C ) )
15 pweq 3628 . . . . . . . 8  |-  ( s  =  S  ->  ~P s  =  ~P S
)
1615ineq1d 3369 . . . . . . 7  |-  ( s  =  S  ->  ( ~P s  i^i  Fin )  =  ( ~P S  i^i  Fin ) )
17 sseq2 3200 . . . . . . 7  |-  ( s  =  S  ->  (
( F `  y
)  C_  s  <->  ( F `  y )  C_  S
) )
1816, 17raleqbidv 2748 . . . . . 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 312 . . . . 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 2882 . . . 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 642 . . 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 622 . 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 244 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 set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684   A.wral 2543    i^i cin 3151    C_ wss 3152   ~Pcpw 3625   dom cdm 4689   ` cfv 5255   Fincfn 6863  Moorecmre 13484  mrClscmrc 13485  ACScacs 13487
This theorem is referenced by:  acsfiel2  13557  isacs3lem  14269
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-fv 5263  df-mre 13488  df-mrc 13489  df-acs 13491
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