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Theorem ismrcd2 26774
Description: Second half of ismrcd1 26773. (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
ismrcd2  |-  ( ph  ->  F  =  (mrCls `  dom  ( F  i^i  _I  ) ) )
Distinct variable groups:    ph, x, y   
x, B, y    x, F, y    x, V, y

Proof of Theorem ismrcd2
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 ismrcd.f . . 3  |-  ( ph  ->  F : ~P B --> ~P B )
2 ffn 5389 . . 3  |-  ( F : ~P B --> ~P B  ->  F  Fn  ~P B
)
31, 2syl 15 . 2  |-  ( ph  ->  F  Fn  ~P B
)
4 ismrcd.b . . . 4  |-  ( ph  ->  B  e.  V )
5 ismrcd.e . . . 4  |-  ( (
ph  /\  x  C_  B
)  ->  x  C_  ( F `  x )
)
6 ismrcd.m . . . 4  |-  ( (
ph  /\  x  C_  B  /\  y  C_  x )  ->  ( F `  y )  C_  ( F `  x )
)
7 ismrcd.i . . . 4  |-  ( (
ph  /\  x  C_  B
)  ->  ( F `  ( F `  x
) )  =  ( F `  x ) )
84, 1, 5, 6, 7ismrcd1 26773 . . 3  |-  ( ph  ->  dom  ( F  i^i  _I  )  e.  (Moore `  B ) )
9 eqid 2283 . . . 4  |-  (mrCls `  dom  ( F  i^i  _I  ) )  =  (mrCls `  dom  ( F  i^i  _I  ) )
109mrcf 13511 . . 3  |-  ( dom  ( F  i^i  _I  )  e.  (Moore `  B
)  ->  (mrCls `  dom  ( F  i^i  _I  )
) : ~P B --> dom  ( F  i^i  _I  ) )
11 ffn 5389 . . 3  |-  ( (mrCls `  dom  ( F  i^i  _I  ) ) : ~P B
--> dom  ( F  i^i  _I  )  ->  (mrCls `  dom  ( F  i^i  _I  )
)  Fn  ~P B
)
128, 10, 113syl 18 . 2  |-  ( ph  ->  (mrCls `  dom  ( F  i^i  _I  ) )  Fn  ~P B )
139mrcssv 13516 . . . . . . 7  |-  ( dom  ( F  i^i  _I  )  e.  (Moore `  B
)  ->  ( (mrCls ` 
dom  ( F  i^i  _I  ) ) `  z
)  C_  B )
148, 13syl 15 . . . . . 6  |-  ( ph  ->  ( (mrCls `  dom  ( F  i^i  _I  )
) `  z )  C_  B )
1514adantr 451 . . . . 5  |-  ( (
ph  /\  z  e.  ~P B )  ->  (
(mrCls `  dom  ( F  i^i  _I  ) ) `
 z )  C_  B )
16 elpwi 3633 . . . . . 6  |-  ( z  e.  ~P B  -> 
z  C_  B )
179mrcssid 13519 . . . . . 6  |-  ( ( dom  ( F  i^i  _I  )  e.  (Moore `  B )  /\  z  C_  B )  ->  z  C_  ( (mrCls `  dom  ( F  i^i  _I  )
) `  z )
)
188, 16, 17syl2an 463 . . . . 5  |-  ( (
ph  /\  z  e.  ~P B )  ->  z  C_  ( (mrCls `  dom  ( F  i^i  _I  )
) `  z )
)
1963expib 1154 . . . . . . . 8  |-  ( ph  ->  ( ( x  C_  B  /\  y  C_  x
)  ->  ( F `  y )  C_  ( F `  x )
) )
2019alrimivv 1618 . . . . . . 7  |-  ( ph  ->  A. y A. x
( ( x  C_  B  /\  y  C_  x
)  ->  ( F `  y )  C_  ( F `  x )
) )
21 vex 2791 . . . . . . . 8  |-  z  e. 
_V
22 fvex 5539 . . . . . . . 8  |-  ( (mrCls `  dom  ( F  i^i  _I  ) ) `  z
)  e.  _V
23 sseq1 3199 . . . . . . . . . . . 12  |-  ( x  =  ( (mrCls `  dom  ( F  i^i  _I  ) ) `  z
)  ->  ( x  C_  B  <->  ( (mrCls `  dom  ( F  i^i  _I  ) ) `  z
)  C_  B )
)
2423adantl 452 . . . . . . . . . . 11  |-  ( ( y  =  z  /\  x  =  ( (mrCls ` 
dom  ( F  i^i  _I  ) ) `  z
) )  ->  (
x  C_  B  <->  ( (mrCls ` 
dom  ( F  i^i  _I  ) ) `  z
)  C_  B )
)
25 sseq12 3201 . . . . . . . . . . 11  |-  ( ( y  =  z  /\  x  =  ( (mrCls ` 
dom  ( F  i^i  _I  ) ) `  z
) )  ->  (
y  C_  x  <->  z  C_  ( (mrCls `  dom  ( F  i^i  _I  ) ) `
 z ) ) )
2624, 25anbi12d 691 . . . . . . . . . 10  |-  ( ( y  =  z  /\  x  =  ( (mrCls ` 
dom  ( F  i^i  _I  ) ) `  z
) )  ->  (
( x  C_  B  /\  y  C_  x )  <-> 
( ( (mrCls `  dom  ( F  i^i  _I  ) ) `  z
)  C_  B  /\  z  C_  ( (mrCls `  dom  ( F  i^i  _I  ) ) `  z
) ) ) )
27 fveq2 5525 . . . . . . . . . . 11  |-  ( y  =  z  ->  ( F `  y )  =  ( F `  z ) )
28 fveq2 5525 . . . . . . . . . . 11  |-  ( x  =  ( (mrCls `  dom  ( F  i^i  _I  ) ) `  z
)  ->  ( F `  x )  =  ( F `  ( (mrCls `  dom  ( F  i^i  _I  ) ) `  z
) ) )
29 sseq12 3201 . . . . . . . . . . 11  |-  ( ( ( F `  y
)  =  ( F `
 z )  /\  ( F `  x )  =  ( F `  ( (mrCls `  dom  ( F  i^i  _I  ) ) `
 z ) ) )  ->  ( ( F `  y )  C_  ( F `  x
)  <->  ( F `  z )  C_  ( F `  ( (mrCls ` 
dom  ( F  i^i  _I  ) ) `  z
) ) ) )
3027, 28, 29syl2an 463 . . . . . . . . . 10  |-  ( ( y  =  z  /\  x  =  ( (mrCls ` 
dom  ( F  i^i  _I  ) ) `  z
) )  ->  (
( F `  y
)  C_  ( F `  x )  <->  ( F `  z )  C_  ( F `  ( (mrCls ` 
dom  ( F  i^i  _I  ) ) `  z
) ) ) )
3126, 30imbi12d 311 . . . . . . . . 9  |-  ( ( y  =  z  /\  x  =  ( (mrCls ` 
dom  ( F  i^i  _I  ) ) `  z
) )  ->  (
( ( x  C_  B  /\  y  C_  x
)  ->  ( F `  y )  C_  ( F `  x )
)  <->  ( ( ( (mrCls `  dom  ( F  i^i  _I  ) ) `
 z )  C_  B  /\  z  C_  (
(mrCls `  dom  ( F  i^i  _I  ) ) `
 z ) )  ->  ( F `  z )  C_  ( F `  ( (mrCls ` 
dom  ( F  i^i  _I  ) ) `  z
) ) ) ) )
3231spc2gv 2871 . . . . . . . 8  |-  ( ( z  e.  _V  /\  ( (mrCls `  dom  ( F  i^i  _I  ) ) `
 z )  e. 
_V )  ->  ( A. y A. x ( ( x  C_  B  /\  y  C_  x )  ->  ( F `  y )  C_  ( F `  x )
)  ->  ( (
( (mrCls `  dom  ( F  i^i  _I  )
) `  z )  C_  B  /\  z  C_  ( (mrCls `  dom  ( F  i^i  _I  ) ) `
 z ) )  ->  ( F `  z )  C_  ( F `  ( (mrCls ` 
dom  ( F  i^i  _I  ) ) `  z
) ) ) ) )
3321, 22, 32mp2an 653 . . . . . . 7  |-  ( A. y A. x ( ( x  C_  B  /\  y  C_  x )  -> 
( F `  y
)  C_  ( F `  x ) )  -> 
( ( ( (mrCls `  dom  ( F  i^i  _I  ) ) `  z
)  C_  B  /\  z  C_  ( (mrCls `  dom  ( F  i^i  _I  ) ) `  z
) )  ->  ( F `  z )  C_  ( F `  (
(mrCls `  dom  ( F  i^i  _I  ) ) `
 z ) ) ) )
3420, 33syl 15 . . . . . 6  |-  ( ph  ->  ( ( ( (mrCls `  dom  ( F  i^i  _I  ) ) `  z
)  C_  B  /\  z  C_  ( (mrCls `  dom  ( F  i^i  _I  ) ) `  z
) )  ->  ( F `  z )  C_  ( F `  (
(mrCls `  dom  ( F  i^i  _I  ) ) `
 z ) ) ) )
3534adantr 451 . . . . 5  |-  ( (
ph  /\  z  e.  ~P B )  ->  (
( ( (mrCls `  dom  ( F  i^i  _I  ) ) `  z
)  C_  B  /\  z  C_  ( (mrCls `  dom  ( F  i^i  _I  ) ) `  z
) )  ->  ( F `  z )  C_  ( F `  (
(mrCls `  dom  ( F  i^i  _I  ) ) `
 z ) ) ) )
3615, 18, 35mp2and 660 . . . 4  |-  ( (
ph  /\  z  e.  ~P B )  ->  ( F `  z )  C_  ( F `  (
(mrCls `  dom  ( F  i^i  _I  ) ) `
 z ) ) )
379mrccl 13513 . . . . . 6  |-  ( ( dom  ( F  i^i  _I  )  e.  (Moore `  B )  /\  z  C_  B )  ->  (
(mrCls `  dom  ( F  i^i  _I  ) ) `
 z )  e. 
dom  ( F  i^i  _I  ) )
388, 16, 37syl2an 463 . . . . 5  |-  ( (
ph  /\  z  e.  ~P B )  ->  (
(mrCls `  dom  ( F  i^i  _I  ) ) `
 z )  e. 
dom  ( F  i^i  _I  ) )
393adantr 451 . . . . . 6  |-  ( (
ph  /\  z  e.  ~P B )  ->  F  Fn  ~P B )
4022elpw 3631 . . . . . . . 8  |-  ( ( (mrCls `  dom  ( F  i^i  _I  ) ) `
 z )  e. 
~P B  <->  ( (mrCls ` 
dom  ( F  i^i  _I  ) ) `  z
)  C_  B )
4114, 40sylibr 203 . . . . . . 7  |-  ( ph  ->  ( (mrCls `  dom  ( F  i^i  _I  )
) `  z )  e.  ~P B )
4241adantr 451 . . . . . 6  |-  ( (
ph  /\  z  e.  ~P B )  ->  (
(mrCls `  dom  ( F  i^i  _I  ) ) `
 z )  e. 
~P B )
43 fnelfp 26755 . . . . . 6  |-  ( ( F  Fn  ~P B  /\  ( (mrCls `  dom  ( F  i^i  _I  )
) `  z )  e.  ~P B )  -> 
( ( (mrCls `  dom  ( F  i^i  _I  ) ) `  z
)  e.  dom  ( F  i^i  _I  )  <->  ( F `  ( (mrCls `  dom  ( F  i^i  _I  )
) `  z )
)  =  ( (mrCls `  dom  ( F  i^i  _I  ) ) `  z
) ) )
4439, 42, 43syl2anc 642 . . . . 5  |-  ( (
ph  /\  z  e.  ~P B )  ->  (
( (mrCls `  dom  ( F  i^i  _I  )
) `  z )  e.  dom  ( F  i^i  _I  )  <->  ( F `  ( (mrCls `  dom  ( F  i^i  _I  ) ) `
 z ) )  =  ( (mrCls `  dom  ( F  i^i  _I  ) ) `  z
) ) )
4538, 44mpbid 201 . . . 4  |-  ( (
ph  /\  z  e.  ~P B )  ->  ( F `  ( (mrCls ` 
dom  ( F  i^i  _I  ) ) `  z
) )  =  ( (mrCls `  dom  ( F  i^i  _I  ) ) `
 z ) )
4636, 45sseqtrd 3214 . . 3  |-  ( (
ph  /\  z  e.  ~P B )  ->  ( F `  z )  C_  ( (mrCls `  dom  ( F  i^i  _I  )
) `  z )
)
478adantr 451 . . . 4  |-  ( (
ph  /\  z  e.  ~P B )  ->  dom  ( F  i^i  _I  )  e.  (Moore `  B )
)
48 sseq1 3199 . . . . . . . 8  |-  ( x  =  z  ->  (
x  C_  B  <->  z  C_  B ) )
4948anbi2d 684 . . . . . . 7  |-  ( x  =  z  ->  (
( ph  /\  x  C_  B )  <->  ( ph  /\  z  C_  B )
) )
50 id 19 . . . . . . . 8  |-  ( x  =  z  ->  x  =  z )
51 fveq2 5525 . . . . . . . 8  |-  ( x  =  z  ->  ( F `  x )  =  ( F `  z ) )
5250, 51sseq12d 3207 . . . . . . 7  |-  ( x  =  z  ->  (
x  C_  ( F `  x )  <->  z  C_  ( F `  z ) ) )
5349, 52imbi12d 311 . . . . . 6  |-  ( x  =  z  ->  (
( ( ph  /\  x  C_  B )  ->  x  C_  ( F `  x ) )  <->  ( ( ph  /\  z  C_  B
)  ->  z  C_  ( F `  z ) ) ) )
5453, 5chvarv 1953 . . . . 5  |-  ( (
ph  /\  z  C_  B )  ->  z  C_  ( F `  z
) )
5516, 54sylan2 460 . . . 4  |-  ( (
ph  /\  z  e.  ~P B )  ->  z  C_  ( F `  z
) )
5651fveq2d 5529 . . . . . . . . 9  |-  ( x  =  z  ->  ( F `  ( F `  x ) )  =  ( F `  ( F `  z )
) )
5756, 51eqeq12d 2297 . . . . . . . 8  |-  ( x  =  z  ->  (
( F `  ( F `  x )
)  =  ( F `
 x )  <->  ( F `  ( F `  z
) )  =  ( F `  z ) ) )
5849, 57imbi12d 311 . . . . . . 7  |-  ( x  =  z  ->  (
( ( ph  /\  x  C_  B )  -> 
( F `  ( F `  x )
)  =  ( F `
 x ) )  <-> 
( ( ph  /\  z  C_  B )  -> 
( F `  ( F `  z )
)  =  ( F `
 z ) ) ) )
5958, 7chvarv 1953 . . . . . 6  |-  ( (
ph  /\  z  C_  B )  ->  ( F `  ( F `  z ) )  =  ( F `  z
) )
6016, 59sylan2 460 . . . . 5  |-  ( (
ph  /\  z  e.  ~P B )  ->  ( F `  ( F `  z ) )  =  ( F `  z
) )
61 ffvelrn 5663 . . . . . . 7  |-  ( ( F : ~P B --> ~P B  /\  z  e.  ~P B )  -> 
( F `  z
)  e.  ~P B
)
621, 61sylan 457 . . . . . 6  |-  ( (
ph  /\  z  e.  ~P B )  ->  ( F `  z )  e.  ~P B )
63 fnelfp 26755 . . . . . 6  |-  ( ( F  Fn  ~P B  /\  ( F `  z
)  e.  ~P B
)  ->  ( ( F `  z )  e.  dom  ( F  i^i  _I  )  <->  ( F `  ( F `  z ) )  =  ( F `
 z ) ) )
6439, 62, 63syl2anc 642 . . . . 5  |-  ( (
ph  /\  z  e.  ~P B )  ->  (
( F `  z
)  e.  dom  ( F  i^i  _I  )  <->  ( F `  ( F `  z
) )  =  ( F `  z ) ) )
6560, 64mpbird 223 . . . 4  |-  ( (
ph  /\  z  e.  ~P B )  ->  ( F `  z )  e.  dom  ( F  i^i  _I  ) )
669mrcsscl 13522 . . . 4  |-  ( ( dom  ( F  i^i  _I  )  e.  (Moore `  B )  /\  z  C_  ( F `  z
)  /\  ( F `  z )  e.  dom  ( F  i^i  _I  )
)  ->  ( (mrCls ` 
dom  ( F  i^i  _I  ) ) `  z
)  C_  ( F `  z ) )
6747, 55, 65, 66syl3anc 1182 . . 3  |-  ( (
ph  /\  z  e.  ~P B )  ->  (
(mrCls `  dom  ( F  i^i  _I  ) ) `
 z )  C_  ( F `  z ) )
6846, 67eqssd 3196 . 2  |-  ( (
ph  /\  z  e.  ~P B )  ->  ( F `  z )  =  ( (mrCls `  dom  ( F  i^i  _I  ) ) `  z
) )
693, 12, 68eqfnfvd 5625 1  |-  ( ph  ->  F  =  (mrCls `  dom  ( F  i^i  _I  ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934   A.wal 1527    = wceq 1623    e. wcel 1684   _Vcvv 2788    i^i cin 3151    C_ wss 3152   ~Pcpw 3625    _I cid 4304   dom cdm 4689    Fn wfn 5250   -->wf 5251   ` cfv 5255  Moorecmre 13484  mrClscmrc 13485
This theorem is referenced by:  istopclsd  26775  ismrc  26776
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-fv 5263  df-mre 13488  df-mrc 13489
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