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Theorem ismrcd1 26105
Description: Any function from the subsets of a set to itself, which is extensive (satisfies mrcssid 13446), isotone (satisfies mrcss 13445), and idempotent (satisfies mrcidm 13448) 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 26106 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 3331 . . . 4  |-  ( F  i^i  _I  )  C_  F
2 dmss 4831 . . . 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 5296 . . . 4  |-  ( F : ~P B --> ~P B  ->  dom  F  =  ~P B )
64, 5syl 17 . . 3  |-  ( ph  ->  dom  F  =  ~P B )
73, 6syl5sseq 3168 . 2  |-  ( ph  ->  dom  (  F  i^i  _I  )  C_  ~P B
)
8 ssid 3139 . . . . . . 7  |-  B  C_  B
9 ismrcd.b . . . . . . . 8  |-  ( ph  ->  B  e.  V )
10 elpwg 3573 . . . . . . . 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 5562 . . . . . 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 5437 . . . . . 6  |-  ( F `
 B )  e. 
_V
1615elpw 3572 . . . . 5  |-  ( ( F `  B )  e.  ~P B  <->  ( F `  B )  C_  B
)
1714, 16sylib 190 . . . 4  |-  ( ph  ->  ( F `  B
)  C_  B )
18 vex 2743 . . . . . . . 8  |-  x  e. 
_V
1918elpw 3572 . . . . . . 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 2597 . . . . 5  |-  ( ph  ->  A. x  e.  ~P  B x  C_  ( F `
 x ) )
23 id 21 . . . . . . 7  |-  ( x  =  B  ->  x  =  B )
24 fveq2 5423 . . . . . . 7  |-  ( x  =  B  ->  ( F `  x )  =  ( F `  B ) )
2523, 24sseq12d 3149 . . . . . 6  |-  ( x  =  B  ->  (
x  C_  ( F `  x )  <->  B  C_  ( F `  B )
) )
2625rcla4va 2833 . . . . 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 3138 . . 3  |-  ( ph  ->  ( F `  B
)  =  B )
29 ffn 5292 . . . . 5  |-  ( F : ~P B --> ~P B  ->  F  Fn  ~P B
)
304, 29syl 17 . . . 4  |-  ( ph  ->  F  Fn  ~P B
)
31 fnelfp 26087 . . . 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 3131 . . . . . . . . . . . 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 3828 . . . . . . . . . . . 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 4162 . . . . . . . . . . 11  |-  U. ~P B  =  B
4139, 40syl6sseq 3166 . . . . . . . . . 10  |-  ( (
ph  /\  z  C_  dom  (  F  i^i  _I  )  /\  z  =/=  (/) )  ->  |^| z  C_  B )
42 intex 4109 . . . . . . . . . . . 12  |-  ( z  =/=  (/)  <->  |^| z  e.  _V )
43 elpwg 3573 . . . . . . . . . . . 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 3122 . . . . . . . . . 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 3818 . . . . . . . . . 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 3141 . . . . . . . . . . 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 5423 . . . . . . . . . . 11  |-  ( y  =  |^| z  -> 
( F `  y
)  =  ( F `
 |^| z ) )
6160sseq1d 3147 . . . . . . . . . 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 2819 . . . . . . . 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 3122 . . . . . . . 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 26087 . . . . . . . . 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 3156 . . . . . 6  |-  ( ( ( ph  /\  z  C_ 
dom  (  F  i^i  _I  )  /\  z  =/=  (/) )  /\  x  e.  z )  ->  ( F `  |^| z ) 
C_  x )
7271ralrimiva 2597 . . . . 5  |-  ( (
ph  /\  z  C_  dom  (  F  i^i  _I  )  /\  z  =/=  (/) )  ->  A. x  e.  z  ( F `  |^| z )  C_  x )
73 ssint 3819 . . . . 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 5423 . . . . . . 7  |-  ( x  =  |^| z  -> 
( F `  x
)  =  ( F `
 |^| z ) )
7876, 77sseq12d 3149 . . . . . 6  |-  ( x  =  |^| z  -> 
( x  C_  ( F `  x )  <->  |^| z  C_  ( F `  |^| z ) ) )
7978rcla4va 2833 . . . . 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 3138 . . 3  |-  ( (
ph  /\  z  C_  dom  (  F  i^i  _I  )  /\  z  =/=  (/) )  ->  ( F `
 |^| z )  = 
|^| z )
82 fnelfp 26087 . . . 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 13431 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 2740    i^i cin 3093    C_ wss 3094   (/)c0 3397   ~Pcpw 3566   U.cuni 3768   |^|cint 3803    _I cid 4241   dom cdm 4626    Fn wfn 4633   -->wf 4634   ` cfv 4638  Moorecmre 13411
This theorem is referenced by:  ismrcd2  26106  istopclsd  26107  ismrc  26108
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 4081  ax-nul 4089  ax-pow 4126  ax-pr 4152  ax-un 4449
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 2520  df-rex 2521  df-rab 2523  df-v 2742  df-sbc 2936  df-dif 3097  df-un 3099  df-in 3101  df-ss 3108  df-nul 3398  df-if 3507  df-pw 3568  df-sn 3587  df-pr 3588  df-op 3590  df-uni 3769  df-int 3804  df-br 3964  df-opab 4018  df-mpt 4019  df-id 4246  df-xp 4640  df-rel 4641  df-cnv 4642  df-co 4643  df-dm 4644  df-rn 4645  df-res 4646  df-ima 4647  df-fun 4648  df-fn 4649  df-f 4650  df-fv 4654  df-mre 13415
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