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Theorem acsfiel 13881
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 13879 . . . . 5  |-  ( C  e.  (ACS `  X
)  ->  C  e.  (Moore `  X ) )
2 mress 13820 . . . . 5  |-  ( ( C  e.  (Moore `  X )  /\  S  e.  C )  ->  S  C_  X )
31, 2sylan 459 . . . 4  |-  ( ( C  e.  (ACS `  X )  /\  S  e.  C )  ->  S  C_  X )
43ex 425 . . 3  |-  ( C  e.  (ACS `  X
)  ->  ( S  e.  C  ->  S  C_  X ) )
54pm4.71rd 618 . 2  |-  ( C  e.  (ACS `  X
)  ->  ( S  e.  C  <->  ( S  C_  X  /\  S  e.  C
) ) )
6 elfvdm 5759 . . . . . 6  |-  ( C  e.  (ACS `  X
)  ->  X  e.  dom ACS )
7 elpw2g 4365 . . . . . 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 473 . . . 4  |-  ( ( C  e.  (ACS `  X )  /\  S  C_  X )  ->  S  e.  ~P X )
10 isacs2.f . . . . . . 7  |-  F  =  (mrCls `  C )
1110isacs2 13880 . . . . . 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 452 . . . . 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 453 . . . 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 2498 . . . . . 6  |-  ( s  =  S  ->  (
s  e.  C  <->  S  e.  C ) )
15 pweq 3804 . . . . . . . 8  |-  ( s  =  S  ->  ~P s  =  ~P S
)
1615ineq1d 3543 . . . . . . 7  |-  ( s  =  S  ->  ( ~P s  i^i  Fin )  =  ( ~P S  i^i  Fin ) )
17 sseq2 3372 . . . . . . 7  |-  ( s  =  S  ->  (
( F `  y
)  C_  s  <->  ( F `  y )  C_  S
) )
1816, 17raleqbidv 2918 . . . . . 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 314 . . . . 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 3052 . . . 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 644 . . 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 624 . 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 246 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 178    /\ wa 360    = wceq 1653    e. wcel 1726   A.wral 2707    i^i cin 3321    C_ wss 3322   ~Pcpw 3801   dom cdm 4880   ` cfv 5456   Fincfn 7111  Moorecmre 13809  mrClscmrc 13810  ACScacs 13812
This theorem is referenced by:  acsfiel2  13882  isacs3lem  14594
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-int 4053  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-fv 5464  df-mre 13813  df-mrc 13814  df-acs 13816
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