MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  isacs1i Unicode version

Theorem isacs1i 13559
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 3258 . . . 4  |-  { s  e.  ~P X  |  U. ( F " ( ~P s  i^i  Fin )
)  C_  s }  C_ 
~P X
21a1i 10 . . 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 3389 . . . . . 6  |-  ( X  i^i  |^| t )  C_  X
4 elpw2g 4174 . . . . . 6  |-  ( X  e.  V  ->  (
( X  i^i  |^| t )  e.  ~P X 
<->  ( X  i^i  |^| t )  C_  X
) )
53, 4mpbiri 224 . . . . 5  |-  ( X  e.  V  ->  ( X  i^i  |^| t )  e. 
~P X )
65ad2antrr 706 . . . 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 5025 . . . . . . . . 9  |-  ( F
" ( ~P ( X  i^i  |^| t )  i^i 
Fin ) )  C_  ran  F
8 frn 5395 . . . . . . . . . 10  |-  ( F : ~P X --> ~P X  ->  ran  F  C_  ~P X )
98adantl 452 . . . . . . . . 9  |-  ( ( X  e.  V  /\  F : ~P X --> ~P X
)  ->  ran  F  C_  ~P X )
107, 9syl5ss 3190 . . . . . . . 8  |-  ( ( X  e.  V  /\  F : ~P X --> ~P X
)  ->  ( F " ( ~P ( X  i^i  |^| t )  i^i 
Fin ) )  C_  ~P X )
11 uniss 3848 . . . . . . . 8  |-  ( ( F " ( ~P ( X  i^i  |^| t )  i^i  Fin ) )  C_  ~P X  ->  U. ( F "
( ~P ( X  i^i  |^| t )  i^i 
Fin ) )  C_  U. ~P X )
1210, 11syl 15 . . . . . . 7  |-  ( ( X  e.  V  /\  F : ~P X --> ~P X
)  ->  U. ( F " ( ~P ( X  i^i  |^| t )  i^i 
Fin ) )  C_  U. ~P X )
13 unipw 4224 . . . . . . 7  |-  U. ~P X  =  X
1412, 13syl6sseq 3224 . . . . . 6  |-  ( ( X  e.  V  /\  F : ~P X --> ~P X
)  ->  U. ( F " ( ~P ( X  i^i  |^| t )  i^i 
Fin ) )  C_  X )
1514adantr 451 . . . . 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 )
16 inss2 3390 . . . . . . . . . . . . . 14  |-  ( X  i^i  |^| t )  C_  |^| t
17 intss1 3877 . . . . . . . . . . . . . 14  |-  ( a  e.  t  ->  |^| t  C_  a )
1816, 17syl5ss 3190 . . . . . . . . . . . . 13  |-  ( a  e.  t  ->  ( X  i^i  |^| t )  C_  a )
1918adantl 452 . . . . . . . . . . . 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 )
20 sspwb 4223 . . . . . . . . . . . 12  |-  ( ( X  i^i  |^| t
)  C_  a  <->  ~P ( X  i^i  |^| t )  C_  ~P a )
2119, 20sylib 188 . . . . . . . . . . 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
)
22 ssrin 3394 . . . . . . . . . . 11  |-  ( ~P ( X  i^i  |^| t )  C_  ~P a  ->  ( ~P ( X  i^i  |^| t )  i^i 
Fin )  C_  ( ~P a  i^i  Fin )
)
2321, 22syl 15 . . . . . . . . . 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 ) )
24 imass2 5049 . . . . . . . . . 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 )
) )
2523, 24syl 15 . . . . . . . . 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 )
) )
26 uniss 3848 . . . . . . . . 9  |-  ( ( F " ( ~P ( X  i^i  |^| t )  i^i  Fin ) )  C_  ( F " ( ~P a  i^i  Fin ) )  ->  U. ( F " ( ~P ( X  i^i  |^| t )  i^i  Fin ) )  C_  U. ( F " ( ~P a  i^i  Fin ) ) )
2725, 26syl 15 . . . . . . . 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 )
) )
28 ssel2 3175 . . . . . . . . . 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 } )
29 pweq 3628 . . . . . . . . . . . . . . . 16  |-  ( s  =  a  ->  ~P s  =  ~P a
)
3029ineq1d 3369 . . . . . . . . . . . . . . 15  |-  ( s  =  a  ->  ( ~P s  i^i  Fin )  =  ( ~P a  i^i  Fin ) )
3130imaeq2d 5012 . . . . . . . . . . . . . 14  |-  ( s  =  a  ->  ( F " ( ~P s  i^i  Fin ) )  =  ( F " ( ~P a  i^i  Fin )
) )
3231unieqd 3838 . . . . . . . . . . . . 13  |-  ( s  =  a  ->  U. ( F " ( ~P s  i^i  Fin ) )  = 
U. ( F "
( ~P a  i^i 
Fin ) ) )
33 id 19 . . . . . . . . . . . . 13  |-  ( s  =  a  ->  s  =  a )
3432, 33sseq12d 3207 . . . . . . . . . . . 12  |-  ( s  =  a  ->  ( U. ( F " ( ~P s  i^i  Fin )
)  C_  s  <->  U. ( F " ( ~P a  i^i  Fin ) )  C_  a ) )
3534elrab 2923 . . . . . . . . . . 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 ) )
3635simprbi 450 . . . . . . . . . 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 )
3728, 36syl 15 . . . . . . . . 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 )
3837adantll 694 . . . . . . . 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 )
3927, 38sstrd 3189 . . . . . . 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 )
4039ralrimiva 2626 . . . . . 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 )
41 ssint 3878 . . . . . 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
)
4240, 41sylibr 203 . . . . 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 )
4315, 42ssind 3393 . . . 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
) )
44 pweq 3628 . . . . . . . . 9  |-  ( s  =  ( X  i^i  |^| t )  ->  ~P s  =  ~P ( X  i^i  |^| t ) )
4544ineq1d 3369 . . . . . . . 8  |-  ( s  =  ( X  i^i  |^| t )  ->  ( ~P s  i^i  Fin )  =  ( ~P ( X  i^i  |^| t )  i^i 
Fin ) )
4645imaeq2d 5012 . . . . . . 7  |-  ( s  =  ( X  i^i  |^| t )  ->  ( F " ( ~P s  i^i  Fin ) )  =  ( F " ( ~P ( X  i^i  |^| t )  i^i  Fin ) ) )
4746unieqd 3838 . . . . . 6  |-  ( s  =  ( X  i^i  |^| t )  ->  U. ( F " ( ~P s  i^i  Fin ) )  = 
U. ( F "
( ~P ( X  i^i  |^| t )  i^i 
Fin ) ) )
48 id 19 . . . . . 6  |-  ( s  =  ( X  i^i  |^| t )  ->  s  =  ( X  i^i  |^| t ) )
4947, 48sseq12d 3207 . . . . 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
) ) )
5049elrab 2923 . . . 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
) ) )
516, 43, 50sylanbrc 645 . . 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 } )
522, 51ismred2 13505 . 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 )
)
53 fssxp 5400 . . . 4  |-  ( F : ~P X --> ~P X  ->  F  C_  ( ~P X  X.  ~P X ) )
54 pwexg 4194 . . . . 5  |-  ( X  e.  V  ->  ~P X  e.  _V )
55 xpexg 4800 . . . . 5  |-  ( ( ~P X  e.  _V  /\ 
~P X  e.  _V )  ->  ( ~P X  X.  ~P X )  e. 
_V )
5654, 54, 55syl2anc 642 . . . 4  |-  ( X  e.  V  ->  ( ~P X  X.  ~P X
)  e.  _V )
57 ssexg 4160 . . . 4  |-  ( ( F  C_  ( ~P X  X.  ~P X )  /\  ( ~P X  X.  ~P X )  e. 
_V )  ->  F  e.  _V )
5853, 56, 57syl2anr 464 . . 3  |-  ( ( X  e.  V  /\  F : ~P X --> ~P X
)  ->  F  e.  _V )
59 simpr 447 . . . 4  |-  ( ( X  e.  V  /\  F : ~P X --> ~P X
)  ->  F : ~P X --> ~P X )
60 pweq 3628 . . . . . . . . . 10  |-  ( s  =  t  ->  ~P s  =  ~P t
)
6160ineq1d 3369 . . . . . . . . 9  |-  ( s  =  t  ->  ( ~P s  i^i  Fin )  =  ( ~P t  i^i  Fin ) )
6261imaeq2d 5012 . . . . . . . 8  |-  ( s  =  t  ->  ( F " ( ~P s  i^i  Fin ) )  =  ( F " ( ~P t  i^i  Fin )
) )
6362unieqd 3838 . . . . . . 7  |-  ( s  =  t  ->  U. ( F " ( ~P s  i^i  Fin ) )  = 
U. ( F "
( ~P t  i^i 
Fin ) ) )
64 id 19 . . . . . . 7  |-  ( s  =  t  ->  s  =  t )
6563, 64sseq12d 3207 . . . . . 6  |-  ( s  =  t  ->  ( U. ( F " ( ~P s  i^i  Fin )
)  C_  s  <->  U. ( F " ( ~P t  i^i  Fin ) )  C_  t ) )
6665elrab3 2924 . . . . 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 ) )
6766rgen 2608 . . . 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 )
6859, 67jctir 524 . . 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 ) ) )
69 feq1 5375 . . . . 5  |-  ( f  =  F  ->  (
f : ~P X --> ~P X  <->  F : ~P X --> ~P X ) )
70 imaeq1 5007 . . . . . . . . 9  |-  ( f  =  F  ->  (
f " ( ~P t  i^i  Fin )
)  =  ( F
" ( ~P t  i^i  Fin ) ) )
7170unieqd 3838 . . . . . . . 8  |-  ( f  =  F  ->  U. (
f " ( ~P t  i^i  Fin )
)  =  U. ( F " ( ~P t  i^i  Fin ) ) )
7271sseq1d 3205 . . . . . . 7  |-  ( f  =  F  ->  ( U. ( f " ( ~P t  i^i  Fin )
)  C_  t  <->  U. ( F " ( ~P t  i^i  Fin ) )  C_  t ) )
7372bibi2d 309 . . . . . 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 ) ) )
7473ralbidv 2563 . . . . 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 )
) )
7569, 74anbi12d 691 . . . 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 ) ) ) )
7675spcegv 2869 . . 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 ) ) ) )
7758, 68, 76sylc 56 . 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 ) ) )
78 isacs 13553 . 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 ) ) ) )
7952, 77, 78sylanbrc 645 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 176    /\ wa 358   E.wex 1528    = wceq 1623    e. wcel 1684   A.wral 2543   {crab 2547   _Vcvv 2788    i^i cin 3151    C_ wss 3152   ~Pcpw 3625   U.cuni 3827   |^|cint 3862    X. cxp 4687   ran crn 4690   "cima 4692   -->wf 5251   ` cfv 5255   Fincfn 6863  Moorecmre 13484  ACScacs 13487
This theorem is referenced by:  acsfn  13561
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-int 3863  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-fv 5263  df-mre 13488  df-acs 13491
  Copyright terms: Public domain W3C validator