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Theorem ismrcd2 26755
Description: Second half of ismrcd1 26754. (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 5593 . . 3  |-  ( F : ~P B --> ~P B  ->  F  Fn  ~P B
)
31, 2syl 16 . 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 26754 . . 3  |-  ( ph  ->  dom  ( F  i^i  _I  )  e.  (Moore `  B ) )
9 eqid 2438 . . . 4  |-  (mrCls `  dom  ( F  i^i  _I  ) )  =  (mrCls `  dom  ( F  i^i  _I  ) )
109mrcf 13836 . . 3  |-  ( dom  ( F  i^i  _I  )  e.  (Moore `  B
)  ->  (mrCls `  dom  ( F  i^i  _I  )
) : ~P B --> dom  ( F  i^i  _I  ) )
11 ffn 5593 . . 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 19 . 2  |-  ( ph  ->  (mrCls `  dom  ( F  i^i  _I  ) )  Fn  ~P B )
138, 9mrcssvd 13850 . . . . . 6  |-  ( ph  ->  ( (mrCls `  dom  ( F  i^i  _I  )
) `  z )  C_  B )
1413adantr 453 . . . . 5  |-  ( (
ph  /\  z  e.  ~P B )  ->  (
(mrCls `  dom  ( F  i^i  _I  ) ) `
 z )  C_  B )
15 elpwi 3809 . . . . . 6  |-  ( z  e.  ~P B  -> 
z  C_  B )
169mrcssid 13844 . . . . . 6  |-  ( ( dom  ( F  i^i  _I  )  e.  (Moore `  B )  /\  z  C_  B )  ->  z  C_  ( (mrCls `  dom  ( F  i^i  _I  )
) `  z )
)
178, 15, 16syl2an 465 . . . . 5  |-  ( (
ph  /\  z  e.  ~P B )  ->  z  C_  ( (mrCls `  dom  ( F  i^i  _I  )
) `  z )
)
1863expib 1157 . . . . . . . 8  |-  ( ph  ->  ( ( x  C_  B  /\  y  C_  x
)  ->  ( F `  y )  C_  ( F `  x )
) )
1918alrimivv 1643 . . . . . . 7  |-  ( ph  ->  A. y A. x
( ( x  C_  B  /\  y  C_  x
)  ->  ( F `  y )  C_  ( F `  x )
) )
20 vex 2961 . . . . . . . 8  |-  z  e. 
_V
21 fvex 5744 . . . . . . . 8  |-  ( (mrCls `  dom  ( F  i^i  _I  ) ) `  z
)  e.  _V
22 sseq1 3371 . . . . . . . . . . . 12  |-  ( x  =  ( (mrCls `  dom  ( F  i^i  _I  ) ) `  z
)  ->  ( x  C_  B  <->  ( (mrCls `  dom  ( F  i^i  _I  ) ) `  z
)  C_  B )
)
2322adantl 454 . . . . . . . . . . 11  |-  ( ( y  =  z  /\  x  =  ( (mrCls ` 
dom  ( F  i^i  _I  ) ) `  z
) )  ->  (
x  C_  B  <->  ( (mrCls ` 
dom  ( F  i^i  _I  ) ) `  z
)  C_  B )
)
24 sseq12 3373 . . . . . . . . . . 11  |-  ( ( y  =  z  /\  x  =  ( (mrCls ` 
dom  ( F  i^i  _I  ) ) `  z
) )  ->  (
y  C_  x  <->  z  C_  ( (mrCls `  dom  ( F  i^i  _I  ) ) `
 z ) ) )
2523, 24anbi12d 693 . . . . . . . . . 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
) ) ) )
26 fveq2 5730 . . . . . . . . . . 11  |-  ( y  =  z  ->  ( F `  y )  =  ( F `  z ) )
27 fveq2 5730 . . . . . . . . . . 11  |-  ( x  =  ( (mrCls `  dom  ( F  i^i  _I  ) ) `  z
)  ->  ( F `  x )  =  ( F `  ( (mrCls `  dom  ( F  i^i  _I  ) ) `  z
) ) )
28 sseq12 3373 . . . . . . . . . . 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
) ) ) )
2926, 27, 28syl2an 465 . . . . . . . . . 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
) ) ) )
3025, 29imbi12d 313 . . . . . . . . 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
) ) ) ) )
3130spc2gv 3041 . . . . . . . 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
) ) ) ) )
3220, 21, 31mp2an 655 . . . . . . 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 ) ) ) )
3319, 32syl 16 . . . . . 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 ) ) ) )
3433adantr 453 . . . . 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 ) ) ) )
3514, 17, 34mp2and 662 . . . 4  |-  ( (
ph  /\  z  e.  ~P B )  ->  ( F `  z )  C_  ( F `  (
(mrCls `  dom  ( F  i^i  _I  ) ) `
 z ) ) )
369mrccl 13838 . . . . . 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  ) )
378, 15, 36syl2an 465 . . . . 5  |-  ( (
ph  /\  z  e.  ~P B )  ->  (
(mrCls `  dom  ( F  i^i  _I  ) ) `
 z )  e. 
dom  ( F  i^i  _I  ) )
383adantr 453 . . . . . 6  |-  ( (
ph  /\  z  e.  ~P B )  ->  F  Fn  ~P B )
3921elpw 3807 . . . . . . . 8  |-  ( ( (mrCls `  dom  ( F  i^i  _I  ) ) `
 z )  e. 
~P B  <->  ( (mrCls ` 
dom  ( F  i^i  _I  ) ) `  z
)  C_  B )
4013, 39sylibr 205 . . . . . . 7  |-  ( ph  ->  ( (mrCls `  dom  ( F  i^i  _I  )
) `  z )  e.  ~P B )
4140adantr 453 . . . . . 6  |-  ( (
ph  /\  z  e.  ~P B )  ->  (
(mrCls `  dom  ( F  i^i  _I  ) ) `
 z )  e. 
~P B )
42 fnelfp 26738 . . . . . 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
) ) )
4338, 41, 42syl2anc 644 . . . . 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
) ) )
4437, 43mpbid 203 . . . 4  |-  ( (
ph  /\  z  e.  ~P B )  ->  ( F `  ( (mrCls ` 
dom  ( F  i^i  _I  ) ) `  z
) )  =  ( (mrCls `  dom  ( F  i^i  _I  ) ) `
 z ) )
4535, 44sseqtrd 3386 . . 3  |-  ( (
ph  /\  z  e.  ~P B )  ->  ( F `  z )  C_  ( (mrCls `  dom  ( F  i^i  _I  )
) `  z )
)
468adantr 453 . . . 4  |-  ( (
ph  /\  z  e.  ~P B )  ->  dom  ( F  i^i  _I  )  e.  (Moore `  B )
)
47 sseq1 3371 . . . . . . . 8  |-  ( x  =  z  ->  (
x  C_  B  <->  z  C_  B ) )
4847anbi2d 686 . . . . . . 7  |-  ( x  =  z  ->  (
( ph  /\  x  C_  B )  <->  ( ph  /\  z  C_  B )
) )
49 id 21 . . . . . . . 8  |-  ( x  =  z  ->  x  =  z )
50 fveq2 5730 . . . . . . . 8  |-  ( x  =  z  ->  ( F `  x )  =  ( F `  z ) )
5149, 50sseq12d 3379 . . . . . . 7  |-  ( x  =  z  ->  (
x  C_  ( F `  x )  <->  z  C_  ( F `  z ) ) )
5248, 51imbi12d 313 . . . . . 6  |-  ( x  =  z  ->  (
( ( ph  /\  x  C_  B )  ->  x  C_  ( F `  x ) )  <->  ( ( ph  /\  z  C_  B
)  ->  z  C_  ( F `  z ) ) ) )
5352, 5chvarv 1970 . . . . 5  |-  ( (
ph  /\  z  C_  B )  ->  z  C_  ( F `  z
) )
5415, 53sylan2 462 . . . 4  |-  ( (
ph  /\  z  e.  ~P B )  ->  z  C_  ( F `  z
) )
5550fveq2d 5734 . . . . . . . . 9  |-  ( x  =  z  ->  ( F `  ( F `  x ) )  =  ( F `  ( F `  z )
) )
5655, 50eqeq12d 2452 . . . . . . . 8  |-  ( x  =  z  ->  (
( F `  ( F `  x )
)  =  ( F `
 x )  <->  ( F `  ( F `  z
) )  =  ( F `  z ) ) )
5748, 56imbi12d 313 . . . . . . 7  |-  ( x  =  z  ->  (
( ( ph  /\  x  C_  B )  -> 
( F `  ( F `  x )
)  =  ( F `
 x ) )  <-> 
( ( ph  /\  z  C_  B )  -> 
( F `  ( F `  z )
)  =  ( F `
 z ) ) ) )
5857, 7chvarv 1970 . . . . . 6  |-  ( (
ph  /\  z  C_  B )  ->  ( F `  ( F `  z ) )  =  ( F `  z
) )
5915, 58sylan2 462 . . . . 5  |-  ( (
ph  /\  z  e.  ~P B )  ->  ( F `  ( F `  z ) )  =  ( F `  z
) )
601ffvelrnda 5872 . . . . . 6  |-  ( (
ph  /\  z  e.  ~P B )  ->  ( F `  z )  e.  ~P B )
61 fnelfp 26738 . . . . . 6  |-  ( ( F  Fn  ~P B  /\  ( F `  z
)  e.  ~P B
)  ->  ( ( F `  z )  e.  dom  ( F  i^i  _I  )  <->  ( F `  ( F `  z ) )  =  ( F `
 z ) ) )
6238, 60, 61syl2anc 644 . . . . 5  |-  ( (
ph  /\  z  e.  ~P B )  ->  (
( F `  z
)  e.  dom  ( F  i^i  _I  )  <->  ( F `  ( F `  z
) )  =  ( F `  z ) ) )
6359, 62mpbird 225 . . . 4  |-  ( (
ph  /\  z  e.  ~P B )  ->  ( F `  z )  e.  dom  ( F  i^i  _I  ) )
649mrcsscl 13847 . . . 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 ) )
6546, 54, 63, 64syl3anc 1185 . . 3  |-  ( (
ph  /\  z  e.  ~P B )  ->  (
(mrCls `  dom  ( F  i^i  _I  ) ) `
 z )  C_  ( F `  z ) )
6645, 65eqssd 3367 . 2  |-  ( (
ph  /\  z  e.  ~P B )  ->  ( F `  z )  =  ( (mrCls `  dom  ( F  i^i  _I  ) ) `  z
) )
673, 12, 66eqfnfvd 5832 1  |-  ( ph  ->  F  =  (mrCls `  dom  ( F  i^i  _I  ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    /\ w3a 937   A.wal 1550    = wceq 1653    e. wcel 1726   _Vcvv 2958    i^i cin 3321    C_ wss 3322   ~Pcpw 3801    _I cid 4495   dom cdm 4880    Fn wfn 5451   -->wf 5452   ` cfv 5456  Moorecmre 13809  mrClscmrc 13810
This theorem is referenced by:  istopclsd  26756  ismrc  26757
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-int 4053  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-fv 5464  df-mre 13813  df-mrc 13814
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