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Theorem ismrcd1 26126
Description: Any function from the subsets of a set to itself, which is extensive (satisfies mrcssid 13467), isotone (satisfies mrcss 13466), and idempotent (satisfies mrcidm 13469) has a collection of fixed points which is a Moore collection, and itself is the closure operator for that collection. This can be taken as an alternate definition for the closure operators. This is the first half, ismrcd2 26127 is the second. (Contributed by Stefan O'Rear, 1-Feb-2015.)
Hypotheses
Ref Expression
ismrcd.b  |-  ( ph  ->  B  e.  V )
ismrcd.f  |-  ( ph  ->  F : ~P B --> ~P B )
ismrcd.e  |-  ( (
ph  /\  x  C_  B
)  ->  x  C_  ( F `  x )
)
ismrcd.m  |-  ( (
ph  /\  x  C_  B  /\  y  C_  x )  ->  ( F `  y )  C_  ( F `  x )
)
ismrcd.i  |-  ( (
ph  /\  x  C_  B
)  ->  ( F `  ( F `  x
) )  =  ( F `  x ) )
Assertion
Ref Expression
ismrcd1  |-  ( ph  ->  dom  (  F  i^i  _I  )  e.  (Moore `  B ) )
Distinct variable groups:    ph, x, y   
x, B, y    x, F, y    x, V, y

Proof of Theorem ismrcd1
StepHypRef Expression
1 inss1 3350 . . . 4  |-  ( F  i^i  _I  )  C_  F
2 dmss 4852 . . . 4  |-  ( ( F  i^i  _I  )  C_  F  ->  dom  (  F  i^i  _I  )  C_  dom  F )
31, 2ax-mp 10 . . 3  |-  dom  (  F  i^i  _I  )  C_  dom  F
4 ismrcd.f . . . 4  |-  ( ph  ->  F : ~P B --> ~P B )
5 fdm 5317 . . . 4  |-  ( F : ~P B --> ~P B  ->  dom  F  =  ~P B )
64, 5syl 17 . . 3  |-  ( ph  ->  dom  F  =  ~P B )
73, 6syl5sseq 3187 . 2  |-  ( ph  ->  dom  (  F  i^i  _I  )  C_  ~P B
)
8 ssid 3158 . . . . . . 7  |-  B  C_  B
9 ismrcd.b . . . . . . . 8  |-  ( ph  ->  B  e.  V )
10 elpwg 3592 . . . . . . . 8  |-  ( B  e.  V  ->  ( B  e.  ~P B  <->  B 
C_  B ) )
119, 10syl 17 . . . . . . 7  |-  ( ph  ->  ( B  e.  ~P B 
<->  B  C_  B )
)
128, 11mpbiri 226 . . . . . 6  |-  ( ph  ->  B  e.  ~P B
)
13 ffvelrn 5583 . . . . . 6  |-  ( ( F : ~P B --> ~P B  /\  B  e. 
~P B )  -> 
( F `  B
)  e.  ~P B
)
144, 12, 13syl2anc 645 . . . . 5  |-  ( ph  ->  ( F `  B
)  e.  ~P B
)
15 fvex 5458 . . . . . 6  |-  ( F `
 B )  e. 
_V
1615elpw 3591 . . . . 5  |-  ( ( F `  B )  e.  ~P B  <->  ( F `  B )  C_  B
)
1714, 16sylib 190 . . . 4  |-  ( ph  ->  ( F `  B
)  C_  B )
18 vex 2760 . . . . . . . 8  |-  x  e. 
_V
1918elpw 3591 . . . . . . 7  |-  ( x  e.  ~P B  <->  x  C_  B
)
20 ismrcd.e . . . . . . 7  |-  ( (
ph  /\  x  C_  B
)  ->  x  C_  ( F `  x )
)
2119, 20sylan2b 463 . . . . . 6  |-  ( (
ph  /\  x  e.  ~P B )  ->  x  C_  ( F `  x
) )
2221ralrimiva 2599 . . . . 5  |-  ( ph  ->  A. x  e.  ~P  B x  C_  ( F `
 x ) )
23 id 21 . . . . . . 7  |-  ( x  =  B  ->  x  =  B )
24 fveq2 5444 . . . . . . 7  |-  ( x  =  B  ->  ( F `  x )  =  ( F `  B ) )
2523, 24sseq12d 3168 . . . . . 6  |-  ( x  =  B  ->  (
x  C_  ( F `  x )  <->  B  C_  ( F `  B )
) )
2625rcla4va 2850 . . . . 5  |-  ( ( B  e.  ~P B  /\  A. x  e.  ~P  B x  C_  ( F `
 x ) )  ->  B  C_  ( F `  B )
)
2712, 22, 26syl2anc 645 . . . 4  |-  ( ph  ->  B  C_  ( F `  B ) )
2817, 27eqssd 3157 . . 3  |-  ( ph  ->  ( F `  B
)  =  B )
29 ffn 5313 . . . . 5  |-  ( F : ~P B --> ~P B  ->  F  Fn  ~P B
)
304, 29syl 17 . . . 4  |-  ( ph  ->  F  Fn  ~P B
)
31 fnelfp 26108 . . . 4  |-  ( ( F  Fn  ~P B  /\  B  e.  ~P B )  ->  ( B  e.  dom  (  F  i^i  _I  )  <->  ( F `  B )  =  B ) )
3230, 12, 31syl2anc 645 . . 3  |-  ( ph  ->  ( B  e.  dom  (  F  i^i  _I  )  <->  ( F `  B )  =  B ) )
3328, 32mpbird 225 . 2  |-  ( ph  ->  B  e.  dom  (  F  i^i  _I  ) )
34 simp2 961 . . . . . . . . . . . . 13  |-  ( (
ph  /\  z  C_  dom  (  F  i^i  _I  )  /\  z  =/=  (/) )  ->  z  C_  dom  (  F  i^i  _I  ) )
3573ad2ant1 981 . . . . . . . . . . . . 13  |-  ( (
ph  /\  z  C_  dom  (  F  i^i  _I  )  /\  z  =/=  (/) )  ->  dom  (  F  i^i  _I  )  C_  ~P B )
3634, 35sstrd 3150 . . . . . . . . . . . 12  |-  ( (
ph  /\  z  C_  dom  (  F  i^i  _I  )  /\  z  =/=  (/) )  ->  z  C_  ~P B )
37 simp3 962 . . . . . . . . . . . 12  |-  ( (
ph  /\  z  C_  dom  (  F  i^i  _I  )  /\  z  =/=  (/) )  ->  z  =/=  (/) )
38 intssuni2 3847 . . . . . . . . . . . 12  |-  ( ( z  C_  ~P B  /\  z  =/=  (/) )  ->  |^| z  C_  U. ~P B )
3936, 37, 38syl2anc 645 . . . . . . . . . . 11  |-  ( (
ph  /\  z  C_  dom  (  F  i^i  _I  )  /\  z  =/=  (/) )  ->  |^| z  C_ 
U. ~P B )
40 unipw 4182 . . . . . . . . . . 11  |-  U. ~P B  =  B
4139, 40syl6sseq 3185 . . . . . . . . . 10  |-  ( (
ph  /\  z  C_  dom  (  F  i^i  _I  )  /\  z  =/=  (/) )  ->  |^| z  C_  B )
42 intex 4129 . . . . . . . . . . . 12  |-  ( z  =/=  (/)  <->  |^| z  e.  _V )
43 elpwg 3592 . . . . . . . . . . . 12  |-  ( |^| z  e.  _V  ->  (
|^| z  e.  ~P B 
<-> 
|^| z  C_  B
) )
4442, 43sylbi 189 . . . . . . . . . . 11  |-  ( z  =/=  (/)  ->  ( |^| z  e.  ~P B  <->  |^| z  C_  B )
)
45443ad2ant3 983 . . . . . . . . . 10  |-  ( (
ph  /\  z  C_  dom  (  F  i^i  _I  )  /\  z  =/=  (/) )  ->  ( |^| z  e.  ~P B  <->  |^| z  C_  B )
)
4641, 45mpbird 225 . . . . . . . . 9  |-  ( (
ph  /\  z  C_  dom  (  F  i^i  _I  )  /\  z  =/=  (/) )  ->  |^| z  e.  ~P B )
4746adantr 453 . . . . . . . 8  |-  ( ( ( ph  /\  z  C_ 
dom  (  F  i^i  _I  )  /\  z  =/=  (/) )  /\  x  e.  z )  ->  |^| z  e.  ~P B )
48 ismrcd.m . . . . . . . . . . . 12  |-  ( (
ph  /\  x  C_  B  /\  y  C_  x )  ->  ( F `  y )  C_  ( F `  x )
)
49483expib 1159 . . . . . . . . . . 11  |-  ( ph  ->  ( ( x  C_  B  /\  y  C_  x
)  ->  ( F `  y )  C_  ( F `  x )
) )
5049alrimiv 2013 . . . . . . . . . 10  |-  ( ph  ->  A. y ( ( x  C_  B  /\  y  C_  x )  -> 
( F `  y
)  C_  ( F `  x ) ) )
51503ad2ant1 981 . . . . . . . . 9  |-  ( (
ph  /\  z  C_  dom  (  F  i^i  _I  )  /\  z  =/=  (/) )  ->  A. y
( ( x  C_  B  /\  y  C_  x
)  ->  ( F `  y )  C_  ( F `  x )
) )
5251adantr 453 . . . . . . . 8  |-  ( ( ( ph  /\  z  C_ 
dom  (  F  i^i  _I  )  /\  z  =/=  (/) )  /\  x  e.  z )  ->  A. y
( ( x  C_  B  /\  y  C_  x
)  ->  ( F `  y )  C_  ( F `  x )
) )
5336sselda 3141 . . . . . . . . . 10  |-  ( ( ( ph  /\  z  C_ 
dom  (  F  i^i  _I  )  /\  z  =/=  (/) )  /\  x  e.  z )  ->  x  e.  ~P B )
5453, 19sylib 190 . . . . . . . . 9  |-  ( ( ( ph  /\  z  C_ 
dom  (  F  i^i  _I  )  /\  z  =/=  (/) )  /\  x  e.  z )  ->  x  C_  B )
55 intss1 3837 . . . . . . . . . 10  |-  ( x  e.  z  ->  |^| z  C_  x )
5655adantl 454 . . . . . . . . 9  |-  ( ( ( ph  /\  z  C_ 
dom  (  F  i^i  _I  )  /\  z  =/=  (/) )  /\  x  e.  z )  ->  |^| z  C_  x )
5754, 56jca 520 . . . . . . . 8  |-  ( ( ( ph  /\  z  C_ 
dom  (  F  i^i  _I  )  /\  z  =/=  (/) )  /\  x  e.  z )  ->  (
x  C_  B  /\  |^| z  C_  x )
)
58 sseq1 3160 . . . . . . . . . . 11  |-  ( y  =  |^| z  -> 
( y  C_  x  <->  |^| z  C_  x )
)
5958anbi2d 687 . . . . . . . . . 10  |-  ( y  =  |^| z  -> 
( ( x  C_  B  /\  y  C_  x
)  <->  ( x  C_  B  /\  |^| z  C_  x
) ) )
60 fveq2 5444 . . . . . . . . . . 11  |-  ( y  =  |^| z  -> 
( F `  y
)  =  ( F `
 |^| z ) )
6160sseq1d 3166 . . . . . . . . . 10  |-  ( y  =  |^| z  -> 
( ( F `  y )  C_  ( F `  x )  <->  ( F `  |^| z
)  C_  ( F `  x ) ) )
6259, 61imbi12d 313 . . . . . . . . 9  |-  ( y  =  |^| z  -> 
( ( ( x 
C_  B  /\  y  C_  x )  ->  ( F `  y )  C_  ( F `  x
) )  <->  ( (
x  C_  B  /\  |^| z  C_  x )  ->  ( F `  |^| z )  C_  ( F `  x )
) ) )
6362cla4gv 2836 . . . . . . . 8  |-  ( |^| z  e.  ~P B  ->  ( A. y ( ( x  C_  B  /\  y  C_  x )  ->  ( F `  y )  C_  ( F `  x )
)  ->  ( (
x  C_  B  /\  |^| z  C_  x )  ->  ( F `  |^| z )  C_  ( F `  x )
) ) )
6447, 52, 57, 63syl3c 59 . . . . . . 7  |-  ( ( ( ph  /\  z  C_ 
dom  (  F  i^i  _I  )  /\  z  =/=  (/) )  /\  x  e.  z )  ->  ( F `  |^| z ) 
C_  ( F `  x ) )
6534sselda 3141 . . . . . . . 8  |-  ( ( ( ph  /\  z  C_ 
dom  (  F  i^i  _I  )  /\  z  =/=  (/) )  /\  x  e.  z )  ->  x  e.  dom  (  F  i^i  _I  ) )
66303ad2ant1 981 . . . . . . . . . 10  |-  ( (
ph  /\  z  C_  dom  (  F  i^i  _I  )  /\  z  =/=  (/) )  ->  F  Fn  ~P B )
6766adantr 453 . . . . . . . . 9  |-  ( ( ( ph  /\  z  C_ 
dom  (  F  i^i  _I  )  /\  z  =/=  (/) )  /\  x  e.  z )  ->  F  Fn  ~P B )
68 fnelfp 26108 . . . . . . . . 9  |-  ( ( F  Fn  ~P B  /\  x  e.  ~P B )  ->  (
x  e.  dom  (  F  i^i  _I  )  <->  ( F `  x )  =  x ) )
6967, 53, 68syl2anc 645 . . . . . . . 8  |-  ( ( ( ph  /\  z  C_ 
dom  (  F  i^i  _I  )  /\  z  =/=  (/) )  /\  x  e.  z )  ->  (
x  e.  dom  (  F  i^i  _I  )  <->  ( F `  x )  =  x ) )
7065, 69mpbid 203 . . . . . . 7  |-  ( ( ( ph  /\  z  C_ 
dom  (  F  i^i  _I  )  /\  z  =/=  (/) )  /\  x  e.  z )  ->  ( F `  x )  =  x )
7164, 70sseqtrd 3175 . . . . . 6  |-  ( ( ( ph  /\  z  C_ 
dom  (  F  i^i  _I  )  /\  z  =/=  (/) )  /\  x  e.  z )  ->  ( F `  |^| z ) 
C_  x )
7271ralrimiva 2599 . . . . 5  |-  ( (
ph  /\  z  C_  dom  (  F  i^i  _I  )  /\  z  =/=  (/) )  ->  A. x  e.  z  ( F `  |^| z )  C_  x )
73 ssint 3838 . . . . 5  |-  ( ( F `  |^| z
)  C_  |^| z  <->  A. x  e.  z  ( F `  |^| z )  C_  x )
7472, 73sylibr 205 . . . 4  |-  ( (
ph  /\  z  C_  dom  (  F  i^i  _I  )  /\  z  =/=  (/) )  ->  ( F `
 |^| z )  C_  |^| z )
75223ad2ant1 981 . . . . 5  |-  ( (
ph  /\  z  C_  dom  (  F  i^i  _I  )  /\  z  =/=  (/) )  ->  A. x  e.  ~P  B x  C_  ( F `  x ) )
76 id 21 . . . . . . 7  |-  ( x  =  |^| z  ->  x  =  |^| z )
77 fveq2 5444 . . . . . . 7  |-  ( x  =  |^| z  -> 
( F `  x
)  =  ( F `
 |^| z ) )
7876, 77sseq12d 3168 . . . . . 6  |-  ( x  =  |^| z  -> 
( x  C_  ( F `  x )  <->  |^| z  C_  ( F `  |^| z ) ) )
7978rcla4va 2850 . . . . 5  |-  ( (
|^| z  e.  ~P B  /\  A. x  e. 
~P  B x  C_  ( F `  x ) )  ->  |^| z  C_  ( F `  |^| z
) )
8046, 75, 79syl2anc 645 . . . 4  |-  ( (
ph  /\  z  C_  dom  (  F  i^i  _I  )  /\  z  =/=  (/) )  ->  |^| z  C_  ( F `  |^| z ) )
8174, 80eqssd 3157 . . 3  |-  ( (
ph  /\  z  C_  dom  (  F  i^i  _I  )  /\  z  =/=  (/) )  ->  ( F `
 |^| z )  = 
|^| z )
82 fnelfp 26108 . . . 4  |-  ( ( F  Fn  ~P B  /\  |^| z  e.  ~P B )  ->  ( |^| z  e.  dom  (  F  i^i  _I  )  <->  ( F `  |^| z
)  =  |^| z
) )
8366, 46, 82syl2anc 645 . . 3  |-  ( (
ph  /\  z  C_  dom  (  F  i^i  _I  )  /\  z  =/=  (/) )  ->  ( |^| z  e.  dom  (  F  i^i  _I  )  <->  ( F `  |^| z )  = 
|^| z ) )
8481, 83mpbird 225 . 2  |-  ( (
ph  /\  z  C_  dom  (  F  i^i  _I  )  /\  z  =/=  (/) )  ->  |^| z  e.  dom  (  F  i^i  _I  ) )
857, 33, 84ismred 13452 1  |-  ( ph  ->  dom  (  F  i^i  _I  )  e.  (Moore `  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 6    <-> wb 178    /\ wa 360    /\ w3a 939   A.wal 1532    = wceq 1619    e. wcel 1621    =/= wne 2419   A.wral 2516   _Vcvv 2757    i^i cin 3112    C_ wss 3113   (/)c0 3416   ~Pcpw 3585   U.cuni 3787   |^|cint 3822    _I cid 4262   dom cdm 4647    Fn wfn 4654   -->wf 4655   ` cfv 4659  Moorecmre 13432
This theorem is referenced by:  ismrcd2  26127  istopclsd  26128  ismrc  26129
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2237  ax-sep 4101  ax-nul 4109  ax-pow 4146  ax-pr 4172  ax-un 4470
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2121  df-mo 2122  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ne 2421  df-ral 2521  df-rex 2522  df-rab 2525  df-v 2759  df-sbc 2953  df-dif 3116  df-un 3118  df-in 3120  df-ss 3127  df-nul 3417  df-if 3526  df-pw 3587  df-sn 3606  df-pr 3607  df-op 3609  df-uni 3788  df-int 3823  df-br 3984  df-opab 4038  df-mpt 4039  df-id 4267  df-xp 4661  df-rel 4662  df-cnv 4663  df-co 4664  df-dm 4665  df-rn 4666  df-res 4667  df-ima 4668  df-fun 4669  df-fn 4670  df-f 4671  df-fv 4675  df-mre 13436
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