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Theorem isacs1i 13809
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 3371 . . . 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 3504 . . . . . 6  |-  ( X  i^i  |^| t )  C_  X
4 elpw2g 4304 . . . . . 6  |-  ( X  e.  V  ->  (
( X  i^i  |^| t )  e.  ~P X 
<->  ( X  i^i  |^| t )  C_  X
) )
53, 4mpbiri 225 . . . . 5  |-  ( X  e.  V  ->  ( X  i^i  |^| t )  e. 
~P X )
65ad2antrr 707 . . . 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 5156 . . . . . . . . 9  |-  ( F
" ( ~P ( X  i^i  |^| t )  i^i 
Fin ) )  C_  ran  F
8 frn 5537 . . . . . . . . . 10  |-  ( F : ~P X --> ~P X  ->  ran  F  C_  ~P X )
98adantl 453 . . . . . . . . 9  |-  ( ( X  e.  V  /\  F : ~P X --> ~P X
)  ->  ran  F  C_  ~P X )
107, 9syl5ss 3302 . . . . . . . 8  |-  ( ( X  e.  V  /\  F : ~P X --> ~P X
)  ->  ( F " ( ~P ( X  i^i  |^| t )  i^i 
Fin ) )  C_  ~P X )
1110unissd 3981 . . . . . . 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 4355 . . . . . . 7  |-  U. ~P X  =  X
1311, 12syl6sseq 3337 . . . . . 6  |-  ( ( X  e.  V  /\  F : ~P X --> ~P X
)  ->  U. ( F " ( ~P ( X  i^i  |^| t )  i^i 
Fin ) )  C_  X )
1413adantr 452 . . . . 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 3505 . . . . . . . . . . . . . 14  |-  ( X  i^i  |^| t )  C_  |^| t
16 intss1 4007 . . . . . . . . . . . . . 14  |-  ( a  e.  t  ->  |^| t  C_  a )
1715, 16syl5ss 3302 . . . . . . . . . . . . 13  |-  ( a  e.  t  ->  ( X  i^i  |^| t )  C_  a )
1817adantl 453 . . . . . . . . . . . 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 4354 . . . . . . . . . . . 12  |-  ( ( X  i^i  |^| t
)  C_  a  <->  ~P ( X  i^i  |^| t )  C_  ~P a )
2018, 19sylib 189 . . . . . . . . . . 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 3509 . . . . . . . . . . 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 5180 . . . . . . . . . 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 3981 . . . . . . . 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 3286 . . . . . . . . . 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 3745 . . . . . . . . . . . . . . . 16  |-  ( s  =  a  ->  ~P s  =  ~P a
)
2827ineq1d 3484 . . . . . . . . . . . . . . 15  |-  ( s  =  a  ->  ( ~P s  i^i  Fin )  =  ( ~P a  i^i  Fin ) )
2928imaeq2d 5143 . . . . . . . . . . . . . 14  |-  ( s  =  a  ->  ( F " ( ~P s  i^i  Fin ) )  =  ( F " ( ~P a  i^i  Fin )
) )
3029unieqd 3968 . . . . . . . . . . . . 13  |-  ( s  =  a  ->  U. ( F " ( ~P s  i^i  Fin ) )  = 
U. ( F "
( ~P a  i^i 
Fin ) ) )
31 id 20 . . . . . . . . . . . . 13  |-  ( s  =  a  ->  s  =  a )
3230, 31sseq12d 3320 . . . . . . . . . . . 12  |-  ( s  =  a  ->  ( U. ( F " ( ~P s  i^i  Fin )
)  C_  s  <->  U. ( F " ( ~P a  i^i  Fin ) )  C_  a ) )
3332elrab 3035 . . . . . . . . . . 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 451 . . . . . . . . . 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 695 . . . . . . . 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 3301 . . . . . . 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 2732 . . . . . 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 4008 . . . . . 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 204 . . . . 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 3508 . . . 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 3745 . . . . . . . . 9  |-  ( s  =  ( X  i^i  |^| t )  ->  ~P s  =  ~P ( X  i^i  |^| t ) )
4342ineq1d 3484 . . . . . . . 8  |-  ( s  =  ( X  i^i  |^| t )  ->  ( ~P s  i^i  Fin )  =  ( ~P ( X  i^i  |^| t )  i^i 
Fin ) )
4443imaeq2d 5143 . . . . . . 7  |-  ( s  =  ( X  i^i  |^| t )  ->  ( F " ( ~P s  i^i  Fin ) )  =  ( F " ( ~P ( X  i^i  |^| t )  i^i  Fin ) ) )
4544unieqd 3968 . . . . . 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 20 . . . . . 6  |-  ( s  =  ( X  i^i  |^| t )  ->  s  =  ( X  i^i  |^| t ) )
4745, 46sseq12d 3320 . . . . 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 3035 . . . 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 646 . . 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 13755 . 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 5542 . . . 4  |-  ( F : ~P X --> ~P X  ->  F  C_  ( ~P X  X.  ~P X ) )
52 pwexg 4324 . . . . 5  |-  ( X  e.  V  ->  ~P X  e.  _V )
53 xpexg 4929 . . . . 5  |-  ( ( ~P X  e.  _V  /\ 
~P X  e.  _V )  ->  ( ~P X  X.  ~P X )  e. 
_V )
5452, 52, 53syl2anc 643 . . . 4  |-  ( X  e.  V  ->  ( ~P X  X.  ~P X
)  e.  _V )
55 ssexg 4290 . . . 4  |-  ( ( F  C_  ( ~P X  X.  ~P X )  /\  ( ~P X  X.  ~P X )  e. 
_V )  ->  F  e.  _V )
5651, 54, 55syl2anr 465 . . 3  |-  ( ( X  e.  V  /\  F : ~P X --> ~P X
)  ->  F  e.  _V )
57 simpr 448 . . . 4  |-  ( ( X  e.  V  /\  F : ~P X --> ~P X
)  ->  F : ~P X --> ~P X )
58 pweq 3745 . . . . . . . . . 10  |-  ( s  =  t  ->  ~P s  =  ~P t
)
5958ineq1d 3484 . . . . . . . . 9  |-  ( s  =  t  ->  ( ~P s  i^i  Fin )  =  ( ~P t  i^i  Fin ) )
6059imaeq2d 5143 . . . . . . . 8  |-  ( s  =  t  ->  ( F " ( ~P s  i^i  Fin ) )  =  ( F " ( ~P t  i^i  Fin )
) )
6160unieqd 3968 . . . . . . 7  |-  ( s  =  t  ->  U. ( F " ( ~P s  i^i  Fin ) )  = 
U. ( F "
( ~P t  i^i 
Fin ) ) )
62 id 20 . . . . . . 7  |-  ( s  =  t  ->  s  =  t )
6361, 62sseq12d 3320 . . . . . 6  |-  ( s  =  t  ->  ( U. ( F " ( ~P s  i^i  Fin )
)  C_  s  <->  U. ( F " ( ~P t  i^i  Fin ) )  C_  t ) )
6463elrab3 3036 . . . . 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 2714 . . . 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 525 . . 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 5516 . . . . 5  |-  ( f  =  F  ->  (
f : ~P X --> ~P X  <->  F : ~P X --> ~P X ) )
68 imaeq1 5138 . . . . . . . . 9  |-  ( f  =  F  ->  (
f " ( ~P t  i^i  Fin )
)  =  ( F
" ( ~P t  i^i  Fin ) ) )
6968unieqd 3968 . . . . . . . 8  |-  ( f  =  F  ->  U. (
f " ( ~P t  i^i  Fin )
)  =  U. ( F " ( ~P t  i^i  Fin ) ) )
7069sseq1d 3318 . . . . . . 7  |-  ( f  =  F  ->  ( U. ( f " ( ~P t  i^i  Fin )
)  C_  t  <->  U. ( F " ( ~P t  i^i  Fin ) )  C_  t ) )
7170bibi2d 310 . . . . . 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 2669 . . . . 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 692 . . . 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 2980 . . 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 58 . 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 13803 . 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 646 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 set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359   E.wex 1547    = wceq 1649    e. wcel 1717   A.wral 2649   {crab 2653   _Vcvv 2899    i^i cin 3262    C_ wss 3263   ~Pcpw 3742   U.cuni 3957   |^|cint 3992    X. cxp 4816   ran crn 4819   "cima 4821   -->wf 5390   ` cfv 5394   Fincfn 7045  Moorecmre 13734  ACScacs 13737
This theorem is referenced by:  acsfn  13811
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-ral 2654  df-rex 2655  df-rab 2658  df-v 2901  df-sbc 3105  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-op 3766  df-uni 3958  df-int 3993  df-br 4154  df-opab 4208  df-mpt 4209  df-id 4439  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-fv 5402  df-mre 13738  df-acs 13741
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