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Theorem ismrcd2 26877
Description: Second half of ismrcd1 26876. (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 5405 . . 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 26876 . . 3  |-  ( ph  ->  dom  ( F  i^i  _I  )  e.  (Moore `  B ) )
9 eqid 2296 . . . 4  |-  (mrCls `  dom  ( F  i^i  _I  ) )  =  (mrCls `  dom  ( F  i^i  _I  ) )
109mrcf 13527 . . 3  |-  ( dom  ( F  i^i  _I  )  e.  (Moore `  B
)  ->  (mrCls `  dom  ( F  i^i  _I  )
) : ~P B --> dom  ( F  i^i  _I  ) )
11 ffn 5405 . . 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 13532 . . . . . . 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 3646 . . . . . 6  |-  ( z  e.  ~P B  -> 
z  C_  B )
179mrcssid 13535 . . . . . 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 1622 . . . . . . 7  |-  ( ph  ->  A. y A. x
( ( x  C_  B  /\  y  C_  x
)  ->  ( F `  y )  C_  ( F `  x )
) )
21 vex 2804 . . . . . . . 8  |-  z  e. 
_V
22 fvex 5555 . . . . . . . 8  |-  ( (mrCls `  dom  ( F  i^i  _I  ) ) `  z
)  e.  _V
23 sseq1 3212 . . . . . . . . . . . 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 3214 . . . . . . . . . . 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 5541 . . . . . . . . . . 11  |-  ( y  =  z  ->  ( F `  y )  =  ( F `  z ) )
28 fveq2 5541 . . . . . . . . . . 11  |-  ( x  =  ( (mrCls `  dom  ( F  i^i  _I  ) ) `  z
)  ->  ( F `  x )  =  ( F `  ( (mrCls `  dom  ( F  i^i  _I  ) ) `  z
) ) )
29 sseq12 3214 . . . . . . . . . . 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 2884 . . . . . . . 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 13529 . . . . . 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 3644 . . . . . . . 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 26858 . . . . . 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 3227 . . 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 3212 . . . . . . . 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 5541 . . . . . . . 8  |-  ( x  =  z  ->  ( F `  x )  =  ( F `  z ) )
5250, 51sseq12d 3220 . . . . . . 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 1966 . . . . 5  |-  ( (
ph  /\  z  C_  B )  ->  z  C_  ( F `  z
) )
5516, 54sylan2 460 . . . 4  |-  ( (
ph  /\  z  e.  ~P B )  ->  z  C_  ( F `  z
) )
5651fveq2d 5545 . . . . . . . . 9  |-  ( x  =  z  ->  ( F `  ( F `  x ) )  =  ( F `  ( F `  z )
) )
5756, 51eqeq12d 2310 . . . . . . . 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 1966 . . . . . 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 5679 . . . . . . 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 26858 . . . . . 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 13538 . . . 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 3209 . 2  |-  ( (
ph  /\  z  e.  ~P B )  ->  ( F `  z )  =  ( (mrCls `  dom  ( F  i^i  _I  ) ) `  z
) )
693, 12, 68eqfnfvd 5641 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 1530    = wceq 1632    e. wcel 1696   _Vcvv 2801    i^i cin 3164    C_ wss 3165   ~Pcpw 3638    _I cid 4320   dom cdm 4705    Fn wfn 5266   -->wf 5267   ` cfv 5271  Moorecmre 13500  mrClscmrc 13501
This theorem is referenced by:  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  ax-un 4528
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-csb 3095  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-iun 3923  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-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-fv 5279  df-mre 13504  df-mrc 13505
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