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Theorem acsfiel 14703
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 14701 . . . . 5  |-  ( C  e.  (ACS `  X
)  ->  C  e.  (Moore `  X ) )
2 mress 14642 . . . . 5  |-  ( ( C  e.  (Moore `  X )  /\  S  e.  C )  ->  S  C_  X )
31, 2sylan 471 . . . 4  |-  ( ( C  e.  (ACS `  X )  /\  S  e.  C )  ->  S  C_  X )
43ex 434 . . 3  |-  ( C  e.  (ACS `  X
)  ->  ( S  e.  C  ->  S  C_  X ) )
54pm4.71rd 635 . 2  |-  ( C  e.  (ACS `  X
)  ->  ( S  e.  C  <->  ( S  C_  X  /\  S  e.  C
) ) )
6 elfvdm 5818 . . . . . 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 16 . . . . 5  |-  ( C  e.  (ACS `  X
)  ->  ( S  e.  ~P X  <->  S  C_  X
) )
98biimpar 485 . . . 4  |-  ( ( C  e.  (ACS `  X )  /\  S  C_  X )  ->  S  e.  ~P X )
10 isacs2.f . . . . . . 7  |-  F  =  (mrCls `  C )
1110isacs2 14702 . . . . . 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 464 . . . . 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 465 . . . 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 2523 . . . . . 6  |-  ( s  =  S  ->  (
s  e.  C  <->  S  e.  C ) )
15 pweq 3964 . . . . . . . 8  |-  ( s  =  S  ->  ~P s  =  ~P S
)
1615ineq1d 3652 . . . . . . 7  |-  ( s  =  S  ->  ( ~P s  i^i  Fin )  =  ( ~P S  i^i  Fin ) )
17 sseq2 3479 . . . . . . 7  |-  ( s  =  S  ->  (
( F `  y
)  C_  s  <->  ( F `  y )  C_  S
) )
1816, 17raleqbidv 3030 . . . . . 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 321 . . . . 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 3170 . . . 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 661 . . 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 641 . 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 369    = wceq 1370    e. wcel 1758   A.wral 2795    i^i cin 3428    C_ wss 3429   ~Pcpw 3961   dom cdm 4941   ` cfv 5519   Fincfn 7413  Moorecmre 14631  mrClscmrc 14632  ACScacs 14634
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4514  ax-nul 4522  ax-pow 4571  ax-pr 4632  ax-un 6475
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-rab 2804  df-v 3073  df-sbc 3288  df-csb 3390  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-nul 3739  df-if 3893  df-pw 3963  df-sn 3979  df-pr 3981  df-op 3985  df-uni 4193  df-int 4230  df-iun 4274  df-br 4394  df-opab 4452  df-mpt 4453  df-id 4737  df-xp 4947  df-rel 4948  df-cnv 4949  df-co 4950  df-dm 4951  df-rn 4952  df-res 4953  df-ima 4954  df-iota 5482  df-fun 5521  df-fn 5522  df-f 5523  df-fv 5527  df-mre 14635  df-mrc 14636  df-acs 14638
This theorem is referenced by:  acsfiel2  14704  isacs3lem  15447
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