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Theorem acsfiel 14905
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 14903 . . . . 5  |-  ( C  e.  (ACS `  X
)  ->  C  e.  (Moore `  X ) )
2 mress 14844 . . . . 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 5890 . . . . . 6  |-  ( C  e.  (ACS `  X
)  ->  X  e.  dom ACS )
7 elpw2g 4610 . . . . . 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 14904 . . . . . 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 2539 . . . . . 6  |-  ( s  =  S  ->  (
s  e.  C  <->  S  e.  C ) )
15 pweq 4013 . . . . . . . 8  |-  ( s  =  S  ->  ~P s  =  ~P S
)
1615ineq1d 3699 . . . . . . 7  |-  ( s  =  S  ->  ( ~P s  i^i  Fin )  =  ( ~P S  i^i  Fin ) )
17 sseq2 3526 . . . . . . 7  |-  ( s  =  S  ->  (
( F `  y
)  C_  s  <->  ( F `  y )  C_  S
) )
1816, 17raleqbidv 3072 . . . . . 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 3212 . . . 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 1379    e. wcel 1767   A.wral 2814    i^i cin 3475    C_ wss 3476   ~Pcpw 4010   dom cdm 4999   ` cfv 5586   Fincfn 7513  Moorecmre 14833  mrClscmrc 14834  ACScacs 14836
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-fv 5594  df-mre 14837  df-mrc 14838  df-acs 14840
This theorem is referenced by:  acsfiel2  14906  isacs3lem  15649
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