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Theorem ismrcd1 29034
Description: Any function from the subsets of a set to itself, which is extensive (satisfies mrcssid 14555), isotone (satisfies mrcss 14554), and idempotent (satisfies mrcidm 14557) 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 29035 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 3570 . . . 4  |-  ( F  i^i  _I  )  C_  F
2 dmss 5039 . . . 4  |-  ( ( F  i^i  _I  )  C_  F  ->  dom  ( F  i^i  _I  )  C_  dom  F )
31, 2ax-mp 5 . . 3  |-  dom  ( F  i^i  _I  )  C_  dom  F
4 ismrcd.f . . . 4  |-  ( ph  ->  F : ~P B --> ~P B )
5 fdm 5563 . . . 4  |-  ( F : ~P B --> ~P B  ->  dom  F  =  ~P B )
64, 5syl 16 . . 3  |-  ( ph  ->  dom  F  =  ~P B )
73, 6syl5sseq 3404 . 2  |-  ( ph  ->  dom  ( F  i^i  _I  )  C_  ~P B
)
8 ssid 3375 . . . . . . 7  |-  B  C_  B
9 ismrcd.b . . . . . . . 8  |-  ( ph  ->  B  e.  V )
10 elpwg 3868 . . . . . . . 8  |-  ( B  e.  V  ->  ( B  e.  ~P B  <->  B 
C_  B ) )
119, 10syl 16 . . . . . . 7  |-  ( ph  ->  ( B  e.  ~P B 
<->  B  C_  B )
)
128, 11mpbiri 233 . . . . . 6  |-  ( ph  ->  B  e.  ~P B
)
134, 12ffvelrnd 5844 . . . . 5  |-  ( ph  ->  ( F `  B
)  e.  ~P B
)
1413elpwid 3870 . . . 4  |-  ( ph  ->  ( F `  B
)  C_  B )
15 selpw 3867 . . . . . . 7  |-  ( x  e.  ~P B  <->  x  C_  B
)
16 ismrcd.e . . . . . . 7  |-  ( (
ph  /\  x  C_  B
)  ->  x  C_  ( F `  x )
)
1715, 16sylan2b 475 . . . . . 6  |-  ( (
ph  /\  x  e.  ~P B )  ->  x  C_  ( F `  x
) )
1817ralrimiva 2799 . . . . 5  |-  ( ph  ->  A. x  e.  ~P  B x  C_  ( F `
 x ) )
19 id 22 . . . . . . 7  |-  ( x  =  B  ->  x  =  B )
20 fveq2 5691 . . . . . . 7  |-  ( x  =  B  ->  ( F `  x )  =  ( F `  B ) )
2119, 20sseq12d 3385 . . . . . 6  |-  ( x  =  B  ->  (
x  C_  ( F `  x )  <->  B  C_  ( F `  B )
) )
2221rspcva 3071 . . . . 5  |-  ( ( B  e.  ~P B  /\  A. x  e.  ~P  B x  C_  ( F `
 x ) )  ->  B  C_  ( F `  B )
)
2312, 18, 22syl2anc 661 . . . 4  |-  ( ph  ->  B  C_  ( F `  B ) )
2414, 23eqssd 3373 . . 3  |-  ( ph  ->  ( F `  B
)  =  B )
25 ffn 5559 . . . . 5  |-  ( F : ~P B --> ~P B  ->  F  Fn  ~P B
)
264, 25syl 16 . . . 4  |-  ( ph  ->  F  Fn  ~P B
)
27 fnelfp 5906 . . . 4  |-  ( ( F  Fn  ~P B  /\  B  e.  ~P B )  ->  ( B  e.  dom  ( F  i^i  _I  )  <->  ( F `  B )  =  B ) )
2826, 12, 27syl2anc 661 . . 3  |-  ( ph  ->  ( B  e.  dom  ( F  i^i  _I  )  <->  ( F `  B )  =  B ) )
2924, 28mpbird 232 . 2  |-  ( ph  ->  B  e.  dom  ( F  i^i  _I  ) )
30 simp2 989 . . . . . . . . . . . . 13  |-  ( (
ph  /\  z  C_  dom  ( F  i^i  _I  )  /\  z  =/=  (/) )  -> 
z  C_  dom  ( F  i^i  _I  ) )
3173ad2ant1 1009 . . . . . . . . . . . . 13  |-  ( (
ph  /\  z  C_  dom  ( F  i^i  _I  )  /\  z  =/=  (/) )  ->  dom  ( F  i^i  _I  )  C_  ~P B )
3230, 31sstrd 3366 . . . . . . . . . . . 12  |-  ( (
ph  /\  z  C_  dom  ( F  i^i  _I  )  /\  z  =/=  (/) )  -> 
z  C_  ~P B
)
33 simp3 990 . . . . . . . . . . . 12  |-  ( (
ph  /\  z  C_  dom  ( F  i^i  _I  )  /\  z  =/=  (/) )  -> 
z  =/=  (/) )
34 intssuni2 4153 . . . . . . . . . . . 12  |-  ( ( z  C_  ~P B  /\  z  =/=  (/) )  ->  |^| z  C_  U. ~P B )
3532, 33, 34syl2anc 661 . . . . . . . . . . 11  |-  ( (
ph  /\  z  C_  dom  ( F  i^i  _I  )  /\  z  =/=  (/) )  ->  |^| z  C_  U. ~P B )
36 unipw 4542 . . . . . . . . . . 11  |-  U. ~P B  =  B
3735, 36syl6sseq 3402 . . . . . . . . . 10  |-  ( (
ph  /\  z  C_  dom  ( F  i^i  _I  )  /\  z  =/=  (/) )  ->  |^| z  C_  B )
38 intex 4448 . . . . . . . . . . . 12  |-  ( z  =/=  (/)  <->  |^| z  e.  _V )
39 elpwg 3868 . . . . . . . . . . . 12  |-  ( |^| z  e.  _V  ->  (
|^| z  e.  ~P B 
<-> 
|^| z  C_  B
) )
4038, 39sylbi 195 . . . . . . . . . . 11  |-  ( z  =/=  (/)  ->  ( |^| z  e.  ~P B  <->  |^| z  C_  B )
)
41403ad2ant3 1011 . . . . . . . . . 10  |-  ( (
ph  /\  z  C_  dom  ( F  i^i  _I  )  /\  z  =/=  (/) )  -> 
( |^| z  e.  ~P B 
<-> 
|^| z  C_  B
) )
4237, 41mpbird 232 . . . . . . . . 9  |-  ( (
ph  /\  z  C_  dom  ( F  i^i  _I  )  /\  z  =/=  (/) )  ->  |^| z  e.  ~P B )
4342adantr 465 . . . . . . . 8  |-  ( ( ( ph  /\  z  C_ 
dom  ( F  i^i  _I  )  /\  z  =/=  (/) )  /\  x  e.  z )  ->  |^| z  e.  ~P B )
44 ismrcd.m . . . . . . . . . . . 12  |-  ( (
ph  /\  x  C_  B  /\  y  C_  x )  ->  ( F `  y )  C_  ( F `  x )
)
45443expib 1190 . . . . . . . . . . 11  |-  ( ph  ->  ( ( x  C_  B  /\  y  C_  x
)  ->  ( F `  y )  C_  ( F `  x )
) )
4645alrimiv 1685 . . . . . . . . . 10  |-  ( ph  ->  A. y ( ( x  C_  B  /\  y  C_  x )  -> 
( F `  y
)  C_  ( F `  x ) ) )
47463ad2ant1 1009 . . . . . . . . 9  |-  ( (
ph  /\  z  C_  dom  ( F  i^i  _I  )  /\  z  =/=  (/) )  ->  A. y ( ( x 
C_  B  /\  y  C_  x )  ->  ( F `  y )  C_  ( F `  x
) ) )
4847adantr 465 . . . . . . . 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 )
) )
4932sselda 3356 . . . . . . . . . 10  |-  ( ( ( ph  /\  z  C_ 
dom  ( F  i^i  _I  )  /\  z  =/=  (/) )  /\  x  e.  z )  ->  x  e.  ~P B )
5049elpwid 3870 . . . . . . . . 9  |-  ( ( ( ph  /\  z  C_ 
dom  ( F  i^i  _I  )  /\  z  =/=  (/) )  /\  x  e.  z )  ->  x  C_  B )
51 intss1 4143 . . . . . . . . . 10  |-  ( x  e.  z  ->  |^| z  C_  x )
5251adantl 466 . . . . . . . . 9  |-  ( ( ( ph  /\  z  C_ 
dom  ( F  i^i  _I  )  /\  z  =/=  (/) )  /\  x  e.  z )  ->  |^| z  C_  x )
5350, 52jca 532 . . . . . . . 8  |-  ( ( ( ph  /\  z  C_ 
dom  ( F  i^i  _I  )  /\  z  =/=  (/) )  /\  x  e.  z )  ->  (
x  C_  B  /\  |^| z  C_  x )
)
54 sseq1 3377 . . . . . . . . . . 11  |-  ( y  =  |^| z  -> 
( y  C_  x  <->  |^| z  C_  x )
)
5554anbi2d 703 . . . . . . . . . 10  |-  ( y  =  |^| z  -> 
( ( x  C_  B  /\  y  C_  x
)  <->  ( x  C_  B  /\  |^| z  C_  x
) ) )
56 fveq2 5691 . . . . . . . . . . 11  |-  ( y  =  |^| z  -> 
( F `  y
)  =  ( F `
 |^| z ) )
5756sseq1d 3383 . . . . . . . . . 10  |-  ( y  =  |^| z  -> 
( ( F `  y )  C_  ( F `  x )  <->  ( F `  |^| z
)  C_  ( F `  x ) ) )
5855, 57imbi12d 320 . . . . . . . . 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 )
) ) )
5958spcgv 3057 . . . . . . . 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 )
) ) )
6043, 48, 53, 59syl3c 61 . . . . . . 7  |-  ( ( ( ph  /\  z  C_ 
dom  ( F  i^i  _I  )  /\  z  =/=  (/) )  /\  x  e.  z )  ->  ( F `  |^| z ) 
C_  ( F `  x ) )
6130sselda 3356 . . . . . . . 8  |-  ( ( ( ph  /\  z  C_ 
dom  ( F  i^i  _I  )  /\  z  =/=  (/) )  /\  x  e.  z )  ->  x  e.  dom  ( F  i^i  _I  ) )
62263ad2ant1 1009 . . . . . . . . . 10  |-  ( (
ph  /\  z  C_  dom  ( F  i^i  _I  )  /\  z  =/=  (/) )  ->  F  Fn  ~P B
)
6362adantr 465 . . . . . . . . 9  |-  ( ( ( ph  /\  z  C_ 
dom  ( F  i^i  _I  )  /\  z  =/=  (/) )  /\  x  e.  z )  ->  F  Fn  ~P B )
64 fnelfp 5906 . . . . . . . . 9  |-  ( ( F  Fn  ~P B  /\  x  e.  ~P B )  ->  (
x  e.  dom  ( F  i^i  _I  )  <->  ( F `  x )  =  x ) )
6563, 49, 64syl2anc 661 . . . . . . . 8  |-  ( ( ( ph  /\  z  C_ 
dom  ( F  i^i  _I  )  /\  z  =/=  (/) )  /\  x  e.  z )  ->  (
x  e.  dom  ( F  i^i  _I  )  <->  ( F `  x )  =  x ) )
6661, 65mpbid 210 . . . . . . 7  |-  ( ( ( ph  /\  z  C_ 
dom  ( F  i^i  _I  )  /\  z  =/=  (/) )  /\  x  e.  z )  ->  ( F `  x )  =  x )
6760, 66sseqtrd 3392 . . . . . 6  |-  ( ( ( ph  /\  z  C_ 
dom  ( F  i^i  _I  )  /\  z  =/=  (/) )  /\  x  e.  z )  ->  ( F `  |^| z ) 
C_  x )
6867ralrimiva 2799 . . . . 5  |-  ( (
ph  /\  z  C_  dom  ( F  i^i  _I  )  /\  z  =/=  (/) )  ->  A. x  e.  z 
( F `  |^| z )  C_  x
)
69 ssint 4144 . . . . 5  |-  ( ( F `  |^| z
)  C_  |^| z  <->  A. x  e.  z  ( F `  |^| z )  C_  x )
7068, 69sylibr 212 . . . 4  |-  ( (
ph  /\  z  C_  dom  ( F  i^i  _I  )  /\  z  =/=  (/) )  -> 
( F `  |^| z )  C_  |^| z
)
71183ad2ant1 1009 . . . . 5  |-  ( (
ph  /\  z  C_  dom  ( F  i^i  _I  )  /\  z  =/=  (/) )  ->  A. x  e.  ~P  B x  C_  ( F `
 x ) )
72 id 22 . . . . . . 7  |-  ( x  =  |^| z  ->  x  =  |^| z )
73 fveq2 5691 . . . . . . 7  |-  ( x  =  |^| z  -> 
( F `  x
)  =  ( F `
 |^| z ) )
7472, 73sseq12d 3385 . . . . . 6  |-  ( x  =  |^| z  -> 
( x  C_  ( F `  x )  <->  |^| z  C_  ( F `  |^| z ) ) )
7574rspcva 3071 . . . . 5  |-  ( (
|^| z  e.  ~P B  /\  A. x  e. 
~P  B x  C_  ( F `  x ) )  ->  |^| z  C_  ( F `  |^| z
) )
7642, 71, 75syl2anc 661 . . . 4  |-  ( (
ph  /\  z  C_  dom  ( F  i^i  _I  )  /\  z  =/=  (/) )  ->  |^| z  C_  ( F `
 |^| z ) )
7770, 76eqssd 3373 . . 3  |-  ( (
ph  /\  z  C_  dom  ( F  i^i  _I  )  /\  z  =/=  (/) )  -> 
( F `  |^| z )  =  |^| z )
78 fnelfp 5906 . . . 4  |-  ( ( F  Fn  ~P B  /\  |^| z  e.  ~P B )  ->  ( |^| z  e.  dom  ( F  i^i  _I  )  <->  ( F `  |^| z
)  =  |^| z
) )
7962, 42, 78syl2anc 661 . . 3  |-  ( (
ph  /\  z  C_  dom  ( F  i^i  _I  )  /\  z  =/=  (/) )  -> 
( |^| z  e.  dom  ( F  i^i  _I  )  <->  ( F `  |^| z
)  =  |^| z
) )
8077, 79mpbird 232 . 2  |-  ( (
ph  /\  z  C_  dom  ( F  i^i  _I  )  /\  z  =/=  (/) )  ->  |^| z  e.  dom  ( F  i^i  _I  )
)
817, 29, 80ismred 14540 1  |-  ( ph  ->  dom  ( F  i^i  _I  )  e.  (Moore `  B ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965   A.wal 1367    = wceq 1369    e. wcel 1756    =/= wne 2606   A.wral 2715   _Vcvv 2972    i^i cin 3327    C_ wss 3328   (/)c0 3637   ~Pcpw 3860   U.cuni 4091   |^|cint 4128    _I cid 4631   dom cdm 4840    Fn wfn 5413   -->wf 5414   ` cfv 5418  Moorecmre 14520
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-ral 2720  df-rex 2721  df-rab 2724  df-v 2974  df-sbc 3187  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-op 3884  df-uni 4092  df-int 4129  df-br 4293  df-opab 4351  df-mpt 4352  df-id 4636  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-fv 5426  df-mre 14524
This theorem is referenced by:  ismrcd2  29035  istopclsd  29036  ismrc  29037
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