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Theorem isacs1i 14598
Description: A closure system determined by a function is a closure system and algebraic. (Contributed by Stefan O'Rear, 3-Apr-2015.)
Assertion
Ref Expression
isacs1i  |-  ( ( X  e.  V  /\  F : ~P X --> ~P X
)  ->  { s  e.  ~P X  |  U. ( F " ( ~P s  i^i  Fin )
)  C_  s }  e.  (ACS `  X )
)
Distinct variable groups:    F, s    X, s
Allowed substitution hint:    V( s)

Proof of Theorem isacs1i
Dummy variables  a 
t  f are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ssrab2 3440 . . . 4  |-  { s  e.  ~P X  |  U. ( F " ( ~P s  i^i  Fin )
)  C_  s }  C_ 
~P X
21a1i 11 . . 3  |-  ( ( X  e.  V  /\  F : ~P X --> ~P X
)  ->  { s  e.  ~P X  |  U. ( F " ( ~P s  i^i  Fin )
)  C_  s }  C_ 
~P X )
3 inss1 3573 . . . . . 6  |-  ( X  i^i  |^| t )  C_  X
4 elpw2g 4458 . . . . . 6  |-  ( X  e.  V  ->  (
( X  i^i  |^| t )  e.  ~P X 
<->  ( X  i^i  |^| t )  C_  X
) )
53, 4mpbiri 233 . . . . 5  |-  ( X  e.  V  ->  ( X  i^i  |^| t )  e. 
~P X )
65ad2antrr 725 . . . 4  |-  ( ( ( X  e.  V  /\  F : ~P X --> ~P X )  /\  t  C_ 
{ s  e.  ~P X  |  U. ( F " ( ~P s  i^i  Fin ) )  C_  s } )  ->  ( X  i^i  |^| t )  e. 
~P X )
7 imassrn 5183 . . . . . . . . 9  |-  ( F
" ( ~P ( X  i^i  |^| t )  i^i 
Fin ) )  C_  ran  F
8 frn 5568 . . . . . . . . . 10  |-  ( F : ~P X --> ~P X  ->  ran  F  C_  ~P X )
98adantl 466 . . . . . . . . 9  |-  ( ( X  e.  V  /\  F : ~P X --> ~P X
)  ->  ran  F  C_  ~P X )
107, 9syl5ss 3370 . . . . . . . 8  |-  ( ( X  e.  V  /\  F : ~P X --> ~P X
)  ->  ( F " ( ~P ( X  i^i  |^| t )  i^i 
Fin ) )  C_  ~P X )
1110unissd 4118 . . . . . . 7  |-  ( ( X  e.  V  /\  F : ~P X --> ~P X
)  ->  U. ( F " ( ~P ( X  i^i  |^| t )  i^i 
Fin ) )  C_  U. ~P X )
12 unipw 4545 . . . . . . 7  |-  U. ~P X  =  X
1311, 12syl6sseq 3405 . . . . . 6  |-  ( ( X  e.  V  /\  F : ~P X --> ~P X
)  ->  U. ( F " ( ~P ( X  i^i  |^| t )  i^i 
Fin ) )  C_  X )
1413adantr 465 . . . . 5  |-  ( ( ( X  e.  V  /\  F : ~P X --> ~P X )  /\  t  C_ 
{ s  e.  ~P X  |  U. ( F " ( ~P s  i^i  Fin ) )  C_  s } )  ->  U. ( F " ( ~P ( X  i^i  |^| t )  i^i 
Fin ) )  C_  X )
15 inss2 3574 . . . . . . . . . . . . . 14  |-  ( X  i^i  |^| t )  C_  |^| t
16 intss1 4146 . . . . . . . . . . . . . 14  |-  ( a  e.  t  ->  |^| t  C_  a )
1715, 16syl5ss 3370 . . . . . . . . . . . . 13  |-  ( a  e.  t  ->  ( X  i^i  |^| t )  C_  a )
1817adantl 466 . . . . . . . . . . . 12  |-  ( ( ( ( X  e.  V  /\  F : ~P X --> ~P X )  /\  t  C_  { s  e.  ~P X  |  U. ( F " ( ~P s  i^i  Fin )
)  C_  s }
)  /\  a  e.  t )  ->  ( X  i^i  |^| t )  C_  a )
19 sspwb 4544 . . . . . . . . . . . 12  |-  ( ( X  i^i  |^| t
)  C_  a  <->  ~P ( X  i^i  |^| t )  C_  ~P a )
2018, 19sylib 196 . . . . . . . . . . 11  |-  ( ( ( ( X  e.  V  /\  F : ~P X --> ~P X )  /\  t  C_  { s  e.  ~P X  |  U. ( F " ( ~P s  i^i  Fin )
)  C_  s }
)  /\  a  e.  t )  ->  ~P ( X  i^i  |^| t
)  C_  ~P a
)
21 ssrin 3578 . . . . . . . . . . 11  |-  ( ~P ( X  i^i  |^| t )  C_  ~P a  ->  ( ~P ( X  i^i  |^| t )  i^i 
Fin )  C_  ( ~P a  i^i  Fin )
)
2220, 21syl 16 . . . . . . . . . 10  |-  ( ( ( ( X  e.  V  /\  F : ~P X --> ~P X )  /\  t  C_  { s  e.  ~P X  |  U. ( F " ( ~P s  i^i  Fin )
)  C_  s }
)  /\  a  e.  t )  ->  ( ~P ( X  i^i  |^| t )  i^i  Fin )  C_  ( ~P a  i^i  Fin ) )
23 imass2 5207 . . . . . . . . . 10  |-  ( ( ~P ( X  i^i  |^| t )  i^i  Fin )  C_  ( ~P a  i^i  Fin )  ->  ( F " ( ~P ( X  i^i  |^| t )  i^i 
Fin ) )  C_  ( F " ( ~P a  i^i  Fin )
) )
2422, 23syl 16 . . . . . . . . 9  |-  ( ( ( ( X  e.  V  /\  F : ~P X --> ~P X )  /\  t  C_  { s  e.  ~P X  |  U. ( F " ( ~P s  i^i  Fin )
)  C_  s }
)  /\  a  e.  t )  ->  ( F " ( ~P ( X  i^i  |^| t )  i^i 
Fin ) )  C_  ( F " ( ~P a  i^i  Fin )
) )
2524unissd 4118 . . . . . . . 8  |-  ( ( ( ( X  e.  V  /\  F : ~P X --> ~P X )  /\  t  C_  { s  e.  ~P X  |  U. ( F " ( ~P s  i^i  Fin )
)  C_  s }
)  /\  a  e.  t )  ->  U. ( F " ( ~P ( X  i^i  |^| t )  i^i 
Fin ) )  C_  U. ( F " ( ~P a  i^i  Fin )
) )
26 ssel2 3354 . . . . . . . . . 10  |-  ( ( t  C_  { s  e.  ~P X  |  U. ( F " ( ~P s  i^i  Fin )
)  C_  s }  /\  a  e.  t
)  ->  a  e.  { s  e.  ~P X  |  U. ( F "
( ~P s  i^i 
Fin ) )  C_  s } )
27 pweq 3866 . . . . . . . . . . . . . . . 16  |-  ( s  =  a  ->  ~P s  =  ~P a
)
2827ineq1d 3554 . . . . . . . . . . . . . . 15  |-  ( s  =  a  ->  ( ~P s  i^i  Fin )  =  ( ~P a  i^i  Fin ) )
2928imaeq2d 5172 . . . . . . . . . . . . . 14  |-  ( s  =  a  ->  ( F " ( ~P s  i^i  Fin ) )  =  ( F " ( ~P a  i^i  Fin )
) )
3029unieqd 4104 . . . . . . . . . . . . 13  |-  ( s  =  a  ->  U. ( F " ( ~P s  i^i  Fin ) )  = 
U. ( F "
( ~P a  i^i 
Fin ) ) )
31 id 22 . . . . . . . . . . . . 13  |-  ( s  =  a  ->  s  =  a )
3230, 31sseq12d 3388 . . . . . . . . . . . 12  |-  ( s  =  a  ->  ( U. ( F " ( ~P s  i^i  Fin )
)  C_  s  <->  U. ( F " ( ~P a  i^i  Fin ) )  C_  a ) )
3332elrab 3120 . . . . . . . . . . 11  |-  ( a  e.  { s  e. 
~P X  |  U. ( F " ( ~P s  i^i  Fin )
)  C_  s }  <->  ( a  e.  ~P X  /\  U. ( F "
( ~P a  i^i 
Fin ) )  C_  a ) )
3433simprbi 464 . . . . . . . . . 10  |-  ( a  e.  { s  e. 
~P X  |  U. ( F " ( ~P s  i^i  Fin )
)  C_  s }  ->  U. ( F "
( ~P a  i^i 
Fin ) )  C_  a )
3526, 34syl 16 . . . . . . . . 9  |-  ( ( t  C_  { s  e.  ~P X  |  U. ( F " ( ~P s  i^i  Fin )
)  C_  s }  /\  a  e.  t
)  ->  U. ( F " ( ~P a  i^i  Fin ) )  C_  a )
3635adantll 713 . . . . . . . 8  |-  ( ( ( ( X  e.  V  /\  F : ~P X --> ~P X )  /\  t  C_  { s  e.  ~P X  |  U. ( F " ( ~P s  i^i  Fin )
)  C_  s }
)  /\  a  e.  t )  ->  U. ( F " ( ~P a  i^i  Fin ) )  C_  a )
3725, 36sstrd 3369 . . . . . . 7  |-  ( ( ( ( X  e.  V  /\  F : ~P X --> ~P X )  /\  t  C_  { s  e.  ~P X  |  U. ( F " ( ~P s  i^i  Fin )
)  C_  s }
)  /\  a  e.  t )  ->  U. ( F " ( ~P ( X  i^i  |^| t )  i^i 
Fin ) )  C_  a )
3837ralrimiva 2802 . . . . . 6  |-  ( ( ( X  e.  V  /\  F : ~P X --> ~P X )  /\  t  C_ 
{ s  e.  ~P X  |  U. ( F " ( ~P s  i^i  Fin ) )  C_  s } )  ->  A. a  e.  t  U. ( F " ( ~P ( X  i^i  |^| t )  i^i 
Fin ) )  C_  a )
39 ssint 4147 . . . . . 6  |-  ( U. ( F " ( ~P ( X  i^i  |^| t )  i^i  Fin ) )  C_  |^| t  <->  A. a  e.  t  U. ( F " ( ~P ( X  i^i  |^| t )  i^i  Fin ) )  C_  a
)
4038, 39sylibr 212 . . . . 5  |-  ( ( ( X  e.  V  /\  F : ~P X --> ~P X )  /\  t  C_ 
{ s  e.  ~P X  |  U. ( F " ( ~P s  i^i  Fin ) )  C_  s } )  ->  U. ( F " ( ~P ( X  i^i  |^| t )  i^i 
Fin ) )  C_  |^| t )
4114, 40ssind 3577 . . . 4  |-  ( ( ( X  e.  V  /\  F : ~P X --> ~P X )  /\  t  C_ 
{ s  e.  ~P X  |  U. ( F " ( ~P s  i^i  Fin ) )  C_  s } )  ->  U. ( F " ( ~P ( X  i^i  |^| t )  i^i 
Fin ) )  C_  ( X  i^i  |^| t
) )
42 pweq 3866 . . . . . . . . 9  |-  ( s  =  ( X  i^i  |^| t )  ->  ~P s  =  ~P ( X  i^i  |^| t ) )
4342ineq1d 3554 . . . . . . . 8  |-  ( s  =  ( X  i^i  |^| t )  ->  ( ~P s  i^i  Fin )  =  ( ~P ( X  i^i  |^| t )  i^i 
Fin ) )
4443imaeq2d 5172 . . . . . . 7  |-  ( s  =  ( X  i^i  |^| t )  ->  ( F " ( ~P s  i^i  Fin ) )  =  ( F " ( ~P ( X  i^i  |^| t )  i^i  Fin ) ) )
4544unieqd 4104 . . . . . 6  |-  ( s  =  ( X  i^i  |^| t )  ->  U. ( F " ( ~P s  i^i  Fin ) )  = 
U. ( F "
( ~P ( X  i^i  |^| t )  i^i 
Fin ) ) )
46 id 22 . . . . . 6  |-  ( s  =  ( X  i^i  |^| t )  ->  s  =  ( X  i^i  |^| t ) )
4745, 46sseq12d 3388 . . . . 5  |-  ( s  =  ( X  i^i  |^| t )  ->  ( U. ( F " ( ~P s  i^i  Fin )
)  C_  s  <->  U. ( F " ( ~P ( X  i^i  |^| t )  i^i 
Fin ) )  C_  ( X  i^i  |^| t
) ) )
4847elrab 3120 . . . 4  |-  ( ( X  i^i  |^| t
)  e.  { s  e.  ~P X  |  U. ( F " ( ~P s  i^i  Fin )
)  C_  s }  <->  ( ( X  i^i  |^| t )  e.  ~P X  /\  U. ( F
" ( ~P ( X  i^i  |^| t )  i^i 
Fin ) )  C_  ( X  i^i  |^| t
) ) )
496, 41, 48sylanbrc 664 . . 3  |-  ( ( ( X  e.  V  /\  F : ~P X --> ~P X )  /\  t  C_ 
{ s  e.  ~P X  |  U. ( F " ( ~P s  i^i  Fin ) )  C_  s } )  ->  ( X  i^i  |^| t )  e. 
{ s  e.  ~P X  |  U. ( F " ( ~P s  i^i  Fin ) )  C_  s } )
502, 49ismred2 14544 . 2  |-  ( ( X  e.  V  /\  F : ~P X --> ~P X
)  ->  { s  e.  ~P X  |  U. ( F " ( ~P s  i^i  Fin )
)  C_  s }  e.  (Moore `  X )
)
51 fssxp 5573 . . . 4  |-  ( F : ~P X --> ~P X  ->  F  C_  ( ~P X  X.  ~P X ) )
52 pwexg 4479 . . . . 5  |-  ( X  e.  V  ->  ~P X  e.  _V )
53 xpexg 6510 . . . . 5  |-  ( ( ~P X  e.  _V  /\ 
~P X  e.  _V )  ->  ( ~P X  X.  ~P X )  e. 
_V )
5452, 52, 53syl2anc 661 . . . 4  |-  ( X  e.  V  ->  ( ~P X  X.  ~P X
)  e.  _V )
55 ssexg 4441 . . . 4  |-  ( ( F  C_  ( ~P X  X.  ~P X )  /\  ( ~P X  X.  ~P X )  e. 
_V )  ->  F  e.  _V )
5651, 54, 55syl2anr 478 . . 3  |-  ( ( X  e.  V  /\  F : ~P X --> ~P X
)  ->  F  e.  _V )
57 simpr 461 . . . 4  |-  ( ( X  e.  V  /\  F : ~P X --> ~P X
)  ->  F : ~P X --> ~P X )
58 pweq 3866 . . . . . . . . . 10  |-  ( s  =  t  ->  ~P s  =  ~P t
)
5958ineq1d 3554 . . . . . . . . 9  |-  ( s  =  t  ->  ( ~P s  i^i  Fin )  =  ( ~P t  i^i  Fin ) )
6059imaeq2d 5172 . . . . . . . 8  |-  ( s  =  t  ->  ( F " ( ~P s  i^i  Fin ) )  =  ( F " ( ~P t  i^i  Fin )
) )
6160unieqd 4104 . . . . . . 7  |-  ( s  =  t  ->  U. ( F " ( ~P s  i^i  Fin ) )  = 
U. ( F "
( ~P t  i^i 
Fin ) ) )
62 id 22 . . . . . . 7  |-  ( s  =  t  ->  s  =  t )
6361, 62sseq12d 3388 . . . . . 6  |-  ( s  =  t  ->  ( U. ( F " ( ~P s  i^i  Fin )
)  C_  s  <->  U. ( F " ( ~P t  i^i  Fin ) )  C_  t ) )
6463elrab3 3121 . . . . 5  |-  ( t  e.  ~P X  -> 
( t  e.  {
s  e.  ~P X  |  U. ( F "
( ~P s  i^i 
Fin ) )  C_  s }  <->  U. ( F "
( ~P t  i^i 
Fin ) )  C_  t ) )
6564rgen 2784 . . . 4  |-  A. t  e.  ~P  X ( t  e.  { s  e. 
~P X  |  U. ( F " ( ~P s  i^i  Fin )
)  C_  s }  <->  U. ( F " ( ~P t  i^i  Fin )
)  C_  t )
6657, 65jctir 538 . . 3  |-  ( ( X  e.  V  /\  F : ~P X --> ~P X
)  ->  ( F : ~P X --> ~P X  /\  A. t  e.  ~P  X ( t  e. 
{ s  e.  ~P X  |  U. ( F " ( ~P s  i^i  Fin ) )  C_  s }  <->  U. ( F "
( ~P t  i^i 
Fin ) )  C_  t ) ) )
67 feq1 5545 . . . . 5  |-  ( f  =  F  ->  (
f : ~P X --> ~P X  <->  F : ~P X --> ~P X ) )
68 imaeq1 5167 . . . . . . . . 9  |-  ( f  =  F  ->  (
f " ( ~P t  i^i  Fin )
)  =  ( F
" ( ~P t  i^i  Fin ) ) )
6968unieqd 4104 . . . . . . . 8  |-  ( f  =  F  ->  U. (
f " ( ~P t  i^i  Fin )
)  =  U. ( F " ( ~P t  i^i  Fin ) ) )
7069sseq1d 3386 . . . . . . 7  |-  ( f  =  F  ->  ( U. ( f " ( ~P t  i^i  Fin )
)  C_  t  <->  U. ( F " ( ~P t  i^i  Fin ) )  C_  t ) )
7170bibi2d 318 . . . . . 6  |-  ( f  =  F  ->  (
( t  e.  {
s  e.  ~P X  |  U. ( F "
( ~P s  i^i 
Fin ) )  C_  s }  <->  U. ( f "
( ~P t  i^i 
Fin ) )  C_  t )  <->  ( t  e.  { s  e.  ~P X  |  U. ( F " ( ~P s  i^i  Fin ) )  C_  s }  <->  U. ( F "
( ~P t  i^i 
Fin ) )  C_  t ) ) )
7271ralbidv 2738 . . . . 5  |-  ( f  =  F  ->  ( A. t  e.  ~P  X ( t  e. 
{ s  e.  ~P X  |  U. ( F " ( ~P s  i^i  Fin ) )  C_  s }  <->  U. ( f "
( ~P t  i^i 
Fin ) )  C_  t )  <->  A. t  e.  ~P  X ( t  e.  { s  e. 
~P X  |  U. ( F " ( ~P s  i^i  Fin )
)  C_  s }  <->  U. ( F " ( ~P t  i^i  Fin )
)  C_  t )
) )
7367, 72anbi12d 710 . . . 4  |-  ( f  =  F  ->  (
( f : ~P X
--> ~P X  /\  A. t  e.  ~P  X
( t  e.  {
s  e.  ~P X  |  U. ( F "
( ~P s  i^i 
Fin ) )  C_  s }  <->  U. ( f "
( ~P t  i^i 
Fin ) )  C_  t ) )  <->  ( F : ~P X --> ~P X  /\  A. t  e.  ~P  X ( t  e. 
{ s  e.  ~P X  |  U. ( F " ( ~P s  i^i  Fin ) )  C_  s }  <->  U. ( F "
( ~P t  i^i 
Fin ) )  C_  t ) ) ) )
7473spcegv 3061 . . 3  |-  ( F  e.  _V  ->  (
( F : ~P X
--> ~P X  /\  A. t  e.  ~P  X
( t  e.  {
s  e.  ~P X  |  U. ( F "
( ~P s  i^i 
Fin ) )  C_  s }  <->  U. ( F "
( ~P t  i^i 
Fin ) )  C_  t ) )  ->  E. f ( f : ~P X --> ~P X  /\  A. t  e.  ~P  X ( t  e. 
{ s  e.  ~P X  |  U. ( F " ( ~P s  i^i  Fin ) )  C_  s }  <->  U. ( f "
( ~P t  i^i 
Fin ) )  C_  t ) ) ) )
7556, 66, 74sylc 60 . 2  |-  ( ( X  e.  V  /\  F : ~P X --> ~P X
)  ->  E. f
( f : ~P X
--> ~P X  /\  A. t  e.  ~P  X
( t  e.  {
s  e.  ~P X  |  U. ( F "
( ~P s  i^i 
Fin ) )  C_  s }  <->  U. ( f "
( ~P t  i^i 
Fin ) )  C_  t ) ) )
76 isacs 14592 . 2  |-  ( { s  e.  ~P X  |  U. ( F "
( ~P s  i^i 
Fin ) )  C_  s }  e.  (ACS `  X )  <->  ( {
s  e.  ~P X  |  U. ( F "
( ~P s  i^i 
Fin ) )  C_  s }  e.  (Moore `  X )  /\  E. f ( f : ~P X --> ~P X  /\  A. t  e.  ~P  X ( t  e. 
{ s  e.  ~P X  |  U. ( F " ( ~P s  i^i  Fin ) )  C_  s }  <->  U. ( f "
( ~P t  i^i 
Fin ) )  C_  t ) ) ) )
7750, 75, 76sylanbrc 664 1  |-  ( ( X  e.  V  /\  F : ~P X --> ~P X
)  ->  { s  e.  ~P X  |  U. ( F " ( ~P s  i^i  Fin )
)  C_  s }  e.  (ACS `  X )
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1369   E.wex 1586    e. wcel 1756   A.wral 2718   {crab 2722   _Vcvv 2975    i^i cin 3330    C_ wss 3331   ~Pcpw 3863   U.cuni 4094   |^|cint 4131    X. cxp 4841   ran crn 4844   "cima 4846   -->wf 5417   ` cfv 5421   Fincfn 7313  Moorecmre 14523  ACScacs 14526
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4416  ax-nul 4424  ax-pow 4473  ax-pr 4534  ax-un 6375
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2571  df-ne 2611  df-ral 2723  df-rex 2724  df-rab 2727  df-v 2977  df-sbc 3190  df-dif 3334  df-un 3336  df-in 3338  df-ss 3345  df-nul 3641  df-if 3795  df-pw 3865  df-sn 3881  df-pr 3883  df-op 3887  df-uni 4095  df-int 4132  df-br 4296  df-opab 4354  df-mpt 4355  df-id 4639  df-xp 4849  df-rel 4850  df-cnv 4851  df-co 4852  df-dm 4853  df-rn 4854  df-res 4855  df-ima 4856  df-iota 5384  df-fun 5423  df-fn 5424  df-f 5425  df-fv 5429  df-mre 14527  df-acs 14530
This theorem is referenced by:  acsfn  14600
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