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Theorem ismrcd1 25939
Description: Any function from the subsets of a set to itself, which is extensive (satisfies mrcssid 13391), isotone (satisfies mrcss 13390), and idempotent (satisfies mrcidm 13393) 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 25940 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 3296 . . . 4  |-  ( F  i^i  _I  )  C_  F
2 dmss 4785 . . . 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 5250 . . . 4  |-  ( F : ~P B --> ~P B  ->  dom  F  =  ~P B )
64, 5syl 17 . . 3  |-  ( ph  ->  dom  F  =  ~P B )
73, 6syl5sseq 3147 . 2  |-  ( ph  ->  dom  (  F  i^i  _I  )  C_  ~P B
)
8 ssid 3118 . . . . . . 7  |-  B  C_  B
9 ismrcd.b . . . . . . . 8  |-  ( ph  ->  B  e.  V )
10 elpwg 3537 . . . . . . . 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 5515 . . . . . 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 5391 . . . . . 6  |-  ( F `
 B )  e. 
_V
1615elpw 3536 . . . . 5  |-  ( ( F `  B )  e.  ~P B  <->  ( F `  B )  C_  B
)
1714, 16sylib 190 . . . 4  |-  ( ph  ->  ( F `  B
)  C_  B )
18 vex 2730 . . . . . . . 8  |-  x  e. 
_V
1918elpw 3536 . . . . . . 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 2588 . . . . 5  |-  ( ph  ->  A. x  e.  ~P  B x  C_  ( F `
 x ) )
23 id 21 . . . . . . 7  |-  ( x  =  B  ->  x  =  B )
24 fveq2 5377 . . . . . . 7  |-  ( x  =  B  ->  ( F `  x )  =  ( F `  B ) )
2523, 24sseq12d 3128 . . . . . 6  |-  ( x  =  B  ->  (
x  C_  ( F `  x )  <->  B  C_  ( F `  B )
) )
2625rcla4va 2819 . . . . 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 3117 . . 3  |-  ( ph  ->  ( F `  B
)  =  B )
29 ffn 5246 . . . . 5  |-  ( F : ~P B --> ~P B  ->  F  Fn  ~P B
)
304, 29syl 17 . . . 4  |-  ( ph  ->  F  Fn  ~P B
)
31 fnelfp 25921 . . . 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 3110 . . . . . . . . . . . 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 3785 . . . . . . . . . . . 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 4118 . . . . . . . . . . 11  |-  U. ~P B  =  B
4139, 40syl6sseq 3145 . . . . . . . . . 10  |-  ( (
ph  /\  z  C_  dom  (  F  i^i  _I  )  /\  z  =/=  (/) )  ->  |^| z  C_  B )
42 intex 4065 . . . . . . . . . . . 12  |-  ( z  =/=  (/)  <->  |^| z  e.  _V )
43 elpwg 3537 . . . . . . . . . . . 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 2012 . . . . . . . . . 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 3103 . . . . . . . . . 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 3775 . . . . . . . . . 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 3120 . . . . . . . . . . 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 5377 . . . . . . . . . . 11  |-  ( y  =  |^| z  -> 
( F `  y
)  =  ( F `
 |^| z ) )
6160sseq1d 3126 . . . . . . . . . 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 2805 . . . . . . . 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 3103 . . . . . . . 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 25921 . . . . . . . . 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 3135 . . . . . 6  |-  ( ( ( ph  /\  z  C_ 
dom  (  F  i^i  _I  )  /\  z  =/=  (/) )  /\  x  e.  z )  ->  ( F `  |^| z ) 
C_  x )
7271ralrimiva 2588 . . . . 5  |-  ( (
ph  /\  z  C_  dom  (  F  i^i  _I  )  /\  z  =/=  (/) )  ->  A. x  e.  z  ( F `  |^| z )  C_  x )
73 ssint 3776 . . . . 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 5377 . . . . . . 7  |-  ( x  =  |^| z  -> 
( F `  x
)  =  ( F `
 |^| z ) )
7876, 77sseq12d 3128 . . . . . 6  |-  ( x  =  |^| z  -> 
( x  C_  ( F `  x )  <->  |^| z  C_  ( F `  |^| z ) ) )
7978rcla4va 2819 . . . . 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 3117 . . 3  |-  ( (
ph  /\  z  C_  dom  (  F  i^i  _I  )  /\  z  =/=  (/) )  ->  ( F `
 |^| z )  = 
|^| z )
82 fnelfp 25921 . . . 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 13376 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 2412   A.wral 2509   _Vcvv 2727    i^i cin 3077    C_ wss 3078   (/)c0 3362   ~Pcpw 3530   U.cuni 3727   |^|cint 3760    _I cid 4197   dom cdm 4580    Fn wfn 4587   -->wf 4588   ` cfv 4592  Moorecmre 13358
This theorem is referenced by:  ismrcd2  25940  istopclsd  25941  ismrc  25942
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 1926  ax-ext 2234  ax-sep 4038  ax-nul 4046  ax-pow 4082  ax-pr 4108  ax-un 4403
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 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-ral 2513  df-rex 2514  df-rab 2516  df-v 2729  df-sbc 2922  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-nul 3363  df-if 3471  df-pw 3532  df-sn 3550  df-pr 3551  df-op 3553  df-uni 3728  df-int 3761  df-br 3921  df-opab 3975  df-mpt 3976  df-id 4202  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-fun 4602  df-fn 4603  df-f 4604  df-fv 4608  df-mre 13361
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