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Theorem ismrcd1 30835
Description: Any function from the subsets of a set to itself, which is extensive (satisfies mrcssid 15034), isotone (satisfies mrcss 15033), and idempotent (satisfies mrcidm 15036) 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 30836 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 3714 . . . 4  |-  ( F  i^i  _I  )  C_  F
2 dmss 5212 . . . 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 5741 . . . 4  |-  ( F : ~P B --> ~P B  ->  dom  F  =  ~P B )
64, 5syl 16 . . 3  |-  ( ph  ->  dom  F  =  ~P B )
73, 6syl5sseq 3547 . 2  |-  ( ph  ->  dom  ( F  i^i  _I  )  C_  ~P B
)
8 ssid 3518 . . . . . . 7  |-  B  C_  B
9 ismrcd.b . . . . . . . 8  |-  ( ph  ->  B  e.  V )
10 elpwg 4023 . . . . . . . 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 6033 . . . . 5  |-  ( ph  ->  ( F `  B
)  e.  ~P B
)
1413elpwid 4025 . . . 4  |-  ( ph  ->  ( F `  B
)  C_  B )
15 selpw 4022 . . . . . . 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 2871 . . . . 5  |-  ( ph  ->  A. x  e.  ~P  B x  C_  ( F `
 x ) )
19 id 22 . . . . . . 7  |-  ( x  =  B  ->  x  =  B )
20 fveq2 5872 . . . . . . 7  |-  ( x  =  B  ->  ( F `  x )  =  ( F `  B ) )
2119, 20sseq12d 3528 . . . . . 6  |-  ( x  =  B  ->  (
x  C_  ( F `  x )  <->  B  C_  ( F `  B )
) )
2221rspcva 3208 . . . . 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 3516 . . 3  |-  ( ph  ->  ( F `  B
)  =  B )
25 ffn 5737 . . . . 5  |-  ( F : ~P B --> ~P B  ->  F  Fn  ~P B
)
264, 25syl 16 . . . 4  |-  ( ph  ->  F  Fn  ~P B
)
27 fnelfp 6100 . . . 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 997 . . . . . . . . . . . . 13  |-  ( (
ph  /\  z  C_  dom  ( F  i^i  _I  )  /\  z  =/=  (/) )  -> 
z  C_  dom  ( F  i^i  _I  ) )
3173ad2ant1 1017 . . . . . . . . . . . . 13  |-  ( (
ph  /\  z  C_  dom  ( F  i^i  _I  )  /\  z  =/=  (/) )  ->  dom  ( F  i^i  _I  )  C_  ~P B )
3230, 31sstrd 3509 . . . . . . . . . . . 12  |-  ( (
ph  /\  z  C_  dom  ( F  i^i  _I  )  /\  z  =/=  (/) )  -> 
z  C_  ~P B
)
33 simp3 998 . . . . . . . . . . . 12  |-  ( (
ph  /\  z  C_  dom  ( F  i^i  _I  )  /\  z  =/=  (/) )  -> 
z  =/=  (/) )
34 intssuni2 4314 . . . . . . . . . . . 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 4706 . . . . . . . . . . 11  |-  U. ~P B  =  B
3735, 36syl6sseq 3545 . . . . . . . . . 10  |-  ( (
ph  /\  z  C_  dom  ( F  i^i  _I  )  /\  z  =/=  (/) )  ->  |^| z  C_  B )
38 intex 4612 . . . . . . . . . . . 12  |-  ( z  =/=  (/)  <->  |^| z  e.  _V )
39 elpwg 4023 . . . . . . . . . . . 12  |-  ( |^| z  e.  _V  ->  (
|^| z  e.  ~P B 
<-> 
|^| z  C_  B
) )
4038, 39sylbi 195 . . . . . . . . . . 11  |-  ( z  =/=  (/)  ->  ( |^| z  e.  ~P B  <->  |^| z  C_  B )
)
41403ad2ant3 1019 . . . . . . . . . 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 1199 . . . . . . . . . . 11  |-  ( ph  ->  ( ( x  C_  B  /\  y  C_  x
)  ->  ( F `  y )  C_  ( F `  x )
) )
4645alrimiv 1720 . . . . . . . . . 10  |-  ( ph  ->  A. y ( ( x  C_  B  /\  y  C_  x )  -> 
( F `  y
)  C_  ( F `  x ) ) )
47463ad2ant1 1017 . . . . . . . . 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 3499 . . . . . . . . . 10  |-  ( ( ( ph  /\  z  C_ 
dom  ( F  i^i  _I  )  /\  z  =/=  (/) )  /\  x  e.  z )  ->  x  e.  ~P B )
5049elpwid 4025 . . . . . . . . 9  |-  ( ( ( ph  /\  z  C_ 
dom  ( F  i^i  _I  )  /\  z  =/=  (/) )  /\  x  e.  z )  ->  x  C_  B )
51 intss1 4303 . . . . . . . . . 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 3520 . . . . . . . . . . 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 5872 . . . . . . . . . . 11  |-  ( y  =  |^| z  -> 
( F `  y
)  =  ( F `
 |^| z ) )
5756sseq1d 3526 . . . . . . . . . 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 3194 . . . . . . . 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 3499 . . . . . . . 8  |-  ( ( ( ph  /\  z  C_ 
dom  ( F  i^i  _I  )  /\  z  =/=  (/) )  /\  x  e.  z )  ->  x  e.  dom  ( F  i^i  _I  ) )
62263ad2ant1 1017 . . . . . . . . . 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 6100 . . . . . . . . 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 3535 . . . . . 6  |-  ( ( ( ph  /\  z  C_ 
dom  ( F  i^i  _I  )  /\  z  =/=  (/) )  /\  x  e.  z )  ->  ( F `  |^| z ) 
C_  x )
6867ralrimiva 2871 . . . . 5  |-  ( (
ph  /\  z  C_  dom  ( F  i^i  _I  )  /\  z  =/=  (/) )  ->  A. x  e.  z 
( F `  |^| z )  C_  x
)
69 ssint 4304 . . . . 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 1017 . . . . 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 5872 . . . . . . 7  |-  ( x  =  |^| z  -> 
( F `  x
)  =  ( F `
 |^| z ) )
7472, 73sseq12d 3528 . . . . . 6  |-  ( x  =  |^| z  -> 
( x  C_  ( F `  x )  <->  |^| z  C_  ( F `  |^| z ) ) )
7574rspcva 3208 . . . . 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 3516 . . 3  |-  ( (
ph  /\  z  C_  dom  ( F  i^i  _I  )  /\  z  =/=  (/) )  -> 
( F `  |^| z )  =  |^| z )
78 fnelfp 6100 . . . 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 15019 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 973   A.wal 1393    = wceq 1395    e. wcel 1819    =/= wne 2652   A.wral 2807   _Vcvv 3109    i^i cin 3470    C_ wss 3471   (/)c0 3793   ~Pcpw 4015   U.cuni 4251   |^|cint 4288    _I cid 4799   dom cdm 5008    Fn wfn 5589   -->wf 5590   ` cfv 5594  Moorecmre 14999
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-int 4289  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-fv 5602  df-mre 15003
This theorem is referenced by:  ismrcd2  30836  istopclsd  30837  ismrc  30838
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