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Theorem ismrcd1 26876
Description: Any function from the subsets of a set to itself, which is extensive (satisfies mrcssid 13535), isotone (satisfies mrcss 13534), and idempotent (satisfies mrcidm 13537) 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 26877 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
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 inss1 3402 . . . 4  |-  ( F  i^i  _I  )  C_  F
2 dmss 4894 . . . 4  |-  ( ( F  i^i  _I  )  C_  F  ->  dom  ( F  i^i  _I  )  C_  dom  F )
31, 2ax-mp 8 . . 3  |-  dom  ( F  i^i  _I  )  C_  dom  F
4 ismrcd.f . . . 4  |-  ( ph  ->  F : ~P B --> ~P B )
5 fdm 5409 . . . 4  |-  ( F : ~P B --> ~P B  ->  dom  F  =  ~P B )
64, 5syl 15 . . 3  |-  ( ph  ->  dom  F  =  ~P B )
73, 6syl5sseq 3239 . 2  |-  ( ph  ->  dom  ( F  i^i  _I  )  C_  ~P B
)
8 ssid 3210 . . . . . . 7  |-  B  C_  B
9 ismrcd.b . . . . . . . 8  |-  ( ph  ->  B  e.  V )
10 elpwg 3645 . . . . . . . 8  |-  ( B  e.  V  ->  ( B  e.  ~P B  <->  B 
C_  B ) )
119, 10syl 15 . . . . . . 7  |-  ( ph  ->  ( B  e.  ~P B 
<->  B  C_  B )
)
128, 11mpbiri 224 . . . . . 6  |-  ( ph  ->  B  e.  ~P B
)
13 ffvelrn 5679 . . . . . 6  |-  ( ( F : ~P B --> ~P B  /\  B  e. 
~P B )  -> 
( F `  B
)  e.  ~P B
)
144, 12, 13syl2anc 642 . . . . 5  |-  ( ph  ->  ( F `  B
)  e.  ~P B
)
15 fvex 5555 . . . . . 6  |-  ( F `
 B )  e. 
_V
1615elpw 3644 . . . . 5  |-  ( ( F `  B )  e.  ~P B  <->  ( F `  B )  C_  B
)
1714, 16sylib 188 . . . 4  |-  ( ph  ->  ( F `  B
)  C_  B )
18 vex 2804 . . . . . . . 8  |-  x  e. 
_V
1918elpw 3644 . . . . . . 7  |-  ( x  e.  ~P B  <->  x  C_  B
)
20 ismrcd.e . . . . . . 7  |-  ( (
ph  /\  x  C_  B
)  ->  x  C_  ( F `  x )
)
2119, 20sylan2b 461 . . . . . 6  |-  ( (
ph  /\  x  e.  ~P B )  ->  x  C_  ( F `  x
) )
2221ralrimiva 2639 . . . . 5  |-  ( ph  ->  A. x  e.  ~P  B x  C_  ( F `
 x ) )
23 id 19 . . . . . . 7  |-  ( x  =  B  ->  x  =  B )
24 fveq2 5541 . . . . . . 7  |-  ( x  =  B  ->  ( F `  x )  =  ( F `  B ) )
2523, 24sseq12d 3220 . . . . . 6  |-  ( x  =  B  ->  (
x  C_  ( F `  x )  <->  B  C_  ( F `  B )
) )
2625rspcva 2895 . . . . 5  |-  ( ( B  e.  ~P B  /\  A. x  e.  ~P  B x  C_  ( F `
 x ) )  ->  B  C_  ( F `  B )
)
2712, 22, 26syl2anc 642 . . . 4  |-  ( ph  ->  B  C_  ( F `  B ) )
2817, 27eqssd 3209 . . 3  |-  ( ph  ->  ( F `  B
)  =  B )
29 ffn 5405 . . . . 5  |-  ( F : ~P B --> ~P B  ->  F  Fn  ~P B
)
304, 29syl 15 . . . 4  |-  ( ph  ->  F  Fn  ~P B
)
31 fnelfp 26858 . . . 4  |-  ( ( F  Fn  ~P B  /\  B  e.  ~P B )  ->  ( B  e.  dom  ( F  i^i  _I  )  <->  ( F `  B )  =  B ) )
3230, 12, 31syl2anc 642 . . 3  |-  ( ph  ->  ( B  e.  dom  ( F  i^i  _I  )  <->  ( F `  B )  =  B ) )
3328, 32mpbird 223 . 2  |-  ( ph  ->  B  e.  dom  ( F  i^i  _I  ) )
34 simp2 956 . . . . . . . . . . . . 13  |-  ( (
ph  /\  z  C_  dom  ( F  i^i  _I  )  /\  z  =/=  (/) )  -> 
z  C_  dom  ( F  i^i  _I  ) )
3573ad2ant1 976 . . . . . . . . . . . . 13  |-  ( (
ph  /\  z  C_  dom  ( F  i^i  _I  )  /\  z  =/=  (/) )  ->  dom  ( F  i^i  _I  )  C_  ~P B )
3634, 35sstrd 3202 . . . . . . . . . . . 12  |-  ( (
ph  /\  z  C_  dom  ( F  i^i  _I  )  /\  z  =/=  (/) )  -> 
z  C_  ~P B
)
37 simp3 957 . . . . . . . . . . . 12  |-  ( (
ph  /\  z  C_  dom  ( F  i^i  _I  )  /\  z  =/=  (/) )  -> 
z  =/=  (/) )
38 intssuni2 3903 . . . . . . . . . . . 12  |-  ( ( z  C_  ~P B  /\  z  =/=  (/) )  ->  |^| z  C_  U. ~P B )
3936, 37, 38syl2anc 642 . . . . . . . . . . 11  |-  ( (
ph  /\  z  C_  dom  ( F  i^i  _I  )  /\  z  =/=  (/) )  ->  |^| z  C_  U. ~P B )
40 unipw 4240 . . . . . . . . . . 11  |-  U. ~P B  =  B
4139, 40syl6sseq 3237 . . . . . . . . . 10  |-  ( (
ph  /\  z  C_  dom  ( F  i^i  _I  )  /\  z  =/=  (/) )  ->  |^| z  C_  B )
42 intex 4183 . . . . . . . . . . . 12  |-  ( z  =/=  (/)  <->  |^| z  e.  _V )
43 elpwg 3645 . . . . . . . . . . . 12  |-  ( |^| z  e.  _V  ->  (
|^| z  e.  ~P B 
<-> 
|^| z  C_  B
) )
4442, 43sylbi 187 . . . . . . . . . . 11  |-  ( z  =/=  (/)  ->  ( |^| z  e.  ~P B  <->  |^| z  C_  B )
)
45443ad2ant3 978 . . . . . . . . . 10  |-  ( (
ph  /\  z  C_  dom  ( F  i^i  _I  )  /\  z  =/=  (/) )  -> 
( |^| z  e.  ~P B 
<-> 
|^| z  C_  B
) )
4641, 45mpbird 223 . . . . . . . . 9  |-  ( (
ph  /\  z  C_  dom  ( F  i^i  _I  )  /\  z  =/=  (/) )  ->  |^| z  e.  ~P B )
4746adantr 451 . . . . . . . 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 1154 . . . . . . . . . . 11  |-  ( ph  ->  ( ( x  C_  B  /\  y  C_  x
)  ->  ( F `  y )  C_  ( F `  x )
) )
5049alrimiv 1621 . . . . . . . . . 10  |-  ( ph  ->  A. y ( ( x  C_  B  /\  y  C_  x )  -> 
( F `  y
)  C_  ( F `  x ) ) )
51503ad2ant1 976 . . . . . . . . 9  |-  ( (
ph  /\  z  C_  dom  ( F  i^i  _I  )  /\  z  =/=  (/) )  ->  A. y ( ( x 
C_  B  /\  y  C_  x )  ->  ( F `  y )  C_  ( F `  x
) ) )
5251adantr 451 . . . . . . . 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 3193 . . . . . . . . . 10  |-  ( ( ( ph  /\  z  C_ 
dom  ( F  i^i  _I  )  /\  z  =/=  (/) )  /\  x  e.  z )  ->  x  e.  ~P B )
5453, 19sylib 188 . . . . . . . . 9  |-  ( ( ( ph  /\  z  C_ 
dom  ( F  i^i  _I  )  /\  z  =/=  (/) )  /\  x  e.  z )  ->  x  C_  B )
55 intss1 3893 . . . . . . . . . 10  |-  ( x  e.  z  ->  |^| z  C_  x )
5655adantl 452 . . . . . . . . 9  |-  ( ( ( ph  /\  z  C_ 
dom  ( F  i^i  _I  )  /\  z  =/=  (/) )  /\  x  e.  z )  ->  |^| z  C_  x )
5754, 56jca 518 . . . . . . . 8  |-  ( ( ( ph  /\  z  C_ 
dom  ( F  i^i  _I  )  /\  z  =/=  (/) )  /\  x  e.  z )  ->  (
x  C_  B  /\  |^| z  C_  x )
)
58 sseq1 3212 . . . . . . . . . . 11  |-  ( y  =  |^| z  -> 
( y  C_  x  <->  |^| z  C_  x )
)
5958anbi2d 684 . . . . . . . . . 10  |-  ( y  =  |^| z  -> 
( ( x  C_  B  /\  y  C_  x
)  <->  ( x  C_  B  /\  |^| z  C_  x
) ) )
60 fveq2 5541 . . . . . . . . . . 11  |-  ( y  =  |^| z  -> 
( F `  y
)  =  ( F `
 |^| z ) )
6160sseq1d 3218 . . . . . . . . . 10  |-  ( y  =  |^| z  -> 
( ( F `  y )  C_  ( F `  x )  <->  ( F `  |^| z
)  C_  ( F `  x ) ) )
6259, 61imbi12d 311 . . . . . . . . 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 )
) ) )
6362spcgv 2881 . . . . . . . 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 57 . . . . . . 7  |-  ( ( ( ph  /\  z  C_ 
dom  ( F  i^i  _I  )  /\  z  =/=  (/) )  /\  x  e.  z )  ->  ( F `  |^| z ) 
C_  ( F `  x ) )
6534sselda 3193 . . . . . . . 8  |-  ( ( ( ph  /\  z  C_ 
dom  ( F  i^i  _I  )  /\  z  =/=  (/) )  /\  x  e.  z )  ->  x  e.  dom  ( F  i^i  _I  ) )
66303ad2ant1 976 . . . . . . . . . 10  |-  ( (
ph  /\  z  C_  dom  ( F  i^i  _I  )  /\  z  =/=  (/) )  ->  F  Fn  ~P B
)
6766adantr 451 . . . . . . . . 9  |-  ( ( ( ph  /\  z  C_ 
dom  ( F  i^i  _I  )  /\  z  =/=  (/) )  /\  x  e.  z )  ->  F  Fn  ~P B )
68 fnelfp 26858 . . . . . . . . 9  |-  ( ( F  Fn  ~P B  /\  x  e.  ~P B )  ->  (
x  e.  dom  ( F  i^i  _I  )  <->  ( F `  x )  =  x ) )
6967, 53, 68syl2anc 642 . . . . . . . 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 201 . . . . . . 7  |-  ( ( ( ph  /\  z  C_ 
dom  ( F  i^i  _I  )  /\  z  =/=  (/) )  /\  x  e.  z )  ->  ( F `  x )  =  x )
7164, 70sseqtrd 3227 . . . . . 6  |-  ( ( ( ph  /\  z  C_ 
dom  ( F  i^i  _I  )  /\  z  =/=  (/) )  /\  x  e.  z )  ->  ( F `  |^| z ) 
C_  x )
7271ralrimiva 2639 . . . . 5  |-  ( (
ph  /\  z  C_  dom  ( F  i^i  _I  )  /\  z  =/=  (/) )  ->  A. x  e.  z 
( F `  |^| z )  C_  x
)
73 ssint 3894 . . . . 5  |-  ( ( F `  |^| z
)  C_  |^| z  <->  A. x  e.  z  ( F `  |^| z )  C_  x )
7472, 73sylibr 203 . . . 4  |-  ( (
ph  /\  z  C_  dom  ( F  i^i  _I  )  /\  z  =/=  (/) )  -> 
( F `  |^| z )  C_  |^| z
)
75223ad2ant1 976 . . . . 5  |-  ( (
ph  /\  z  C_  dom  ( F  i^i  _I  )  /\  z  =/=  (/) )  ->  A. x  e.  ~P  B x  C_  ( F `
 x ) )
76 id 19 . . . . . . 7  |-  ( x  =  |^| z  ->  x  =  |^| z )
77 fveq2 5541 . . . . . . 7  |-  ( x  =  |^| z  -> 
( F `  x
)  =  ( F `
 |^| z ) )
7876, 77sseq12d 3220 . . . . . 6  |-  ( x  =  |^| z  -> 
( x  C_  ( F `  x )  <->  |^| z  C_  ( F `  |^| z ) ) )
7978rspcva 2895 . . . . 5  |-  ( (
|^| z  e.  ~P B  /\  A. x  e. 
~P  B x  C_  ( F `  x ) )  ->  |^| z  C_  ( F `  |^| z
) )
8046, 75, 79syl2anc 642 . . . 4  |-  ( (
ph  /\  z  C_  dom  ( F  i^i  _I  )  /\  z  =/=  (/) )  ->  |^| z  C_  ( F `
 |^| z ) )
8174, 80eqssd 3209 . . 3  |-  ( (
ph  /\  z  C_  dom  ( F  i^i  _I  )  /\  z  =/=  (/) )  -> 
( F `  |^| z )  =  |^| z )
82 fnelfp 26858 . . . 4  |-  ( ( F  Fn  ~P B  /\  |^| z  e.  ~P B )  ->  ( |^| z  e.  dom  ( F  i^i  _I  )  <->  ( F `  |^| z
)  =  |^| z
) )
8366, 46, 82syl2anc 642 . . 3  |-  ( (
ph  /\  z  C_  dom  ( F  i^i  _I  )  /\  z  =/=  (/) )  -> 
( |^| z  e.  dom  ( F  i^i  _I  )  <->  ( F `  |^| z
)  =  |^| z
) )
8481, 83mpbird 223 . 2  |-  ( (
ph  /\  z  C_  dom  ( F  i^i  _I  )  /\  z  =/=  (/) )  ->  |^| z  e.  dom  ( F  i^i  _I  )
)
857, 33, 84ismred 13520 1  |-  ( ph  ->  dom  ( F  i^i  _I  )  e.  (Moore `  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934   A.wal 1530    = wceq 1632    e. wcel 1696    =/= wne 2459   A.wral 2556   _Vcvv 2801    i^i cin 3164    C_ wss 3165   (/)c0 3468   ~Pcpw 3638   U.cuni 3843   |^|cint 3878    _I cid 4320   dom cdm 4705    Fn wfn 5266   -->wf 5267   ` cfv 5271  Moorecmre 13500
This theorem is referenced by:  ismrcd2  26877  istopclsd  26878  ismrc  26879
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-int 3879  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-fv 5279  df-mre 13504
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