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Theorem ismrcd2 35510
Description: Second half of ismrcd1 35509. (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 5746 . . 3  |-  ( F : ~P B --> ~P B  ->  F  Fn  ~P B
)
31, 2syl 17 . 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 35509 . . 3  |-  ( ph  ->  dom  ( F  i^i  _I  )  e.  (Moore `  B ) )
9 eqid 2422 . . . 4  |-  (mrCls `  dom  ( F  i^i  _I  ) )  =  (mrCls `  dom  ( F  i^i  _I  ) )
109mrcf 15514 . . 3  |-  ( dom  ( F  i^i  _I  )  e.  (Moore `  B
)  ->  (mrCls `  dom  ( F  i^i  _I  )
) : ~P B --> dom  ( F  i^i  _I  ) )
11 ffn 5746 . . 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 )
138, 9mrcssvd 15528 . . . . . 6  |-  ( ph  ->  ( (mrCls `  dom  ( F  i^i  _I  )
) `  z )  C_  B )
1413adantr 466 . . . . 5  |-  ( (
ph  /\  z  e.  ~P B )  ->  (
(mrCls `  dom  ( F  i^i  _I  ) ) `
 z )  C_  B )
15 elpwi 3990 . . . . . 6  |-  ( z  e.  ~P B  -> 
z  C_  B )
169mrcssid 15522 . . . . . 6  |-  ( ( dom  ( F  i^i  _I  )  e.  (Moore `  B )  /\  z  C_  B )  ->  z  C_  ( (mrCls `  dom  ( F  i^i  _I  )
) `  z )
)
178, 15, 16syl2an 479 . . . . 5  |-  ( (
ph  /\  z  e.  ~P B )  ->  z  C_  ( (mrCls `  dom  ( F  i^i  _I  )
) `  z )
)
1863expib 1208 . . . . . . . 8  |-  ( ph  ->  ( ( x  C_  B  /\  y  C_  x
)  ->  ( F `  y )  C_  ( F `  x )
) )
1918alrimivv 1768 . . . . . . 7  |-  ( ph  ->  A. y A. x
( ( x  C_  B  /\  y  C_  x
)  ->  ( F `  y )  C_  ( F `  x )
) )
20 vex 3083 . . . . . . . 8  |-  z  e. 
_V
21 fvex 5891 . . . . . . . 8  |-  ( (mrCls `  dom  ( F  i^i  _I  ) ) `  z
)  e.  _V
22 sseq1 3485 . . . . . . . . . . . 12  |-  ( x  =  ( (mrCls `  dom  ( F  i^i  _I  ) ) `  z
)  ->  ( x  C_  B  <->  ( (mrCls `  dom  ( F  i^i  _I  ) ) `  z
)  C_  B )
)
2322adantl 467 . . . . . . . . . . 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 3487 . . . . . . . . . . 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 715 . . . . . . . . . 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 5881 . . . . . . . . . . 11  |-  ( y  =  z  ->  ( F `  y )  =  ( F `  z ) )
27 fveq2 5881 . . . . . . . . . . 11  |-  ( x  =  ( (mrCls `  dom  ( F  i^i  _I  ) ) `  z
)  ->  ( F `  x )  =  ( F `  ( (mrCls `  dom  ( F  i^i  _I  ) ) `  z
) ) )
28 sseq12 3487 . . . . . . . . . . 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 479 . . . . . . . . . 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 321 . . . . . . . . 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 3169 . . . . . . . 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 676 . . . . . . 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 17 . . . . . 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 466 . . . . 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 683 . . . 4  |-  ( (
ph  /\  z  e.  ~P B )  ->  ( F `  z )  C_  ( F `  (
(mrCls `  dom  ( F  i^i  _I  ) ) `
 z ) ) )
369mrccl 15516 . . . . . 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 479 . . . . 5  |-  ( (
ph  /\  z  e.  ~P B )  ->  (
(mrCls `  dom  ( F  i^i  _I  ) ) `
 z )  e. 
dom  ( F  i^i  _I  ) )
383adantr 466 . . . . . 6  |-  ( (
ph  /\  z  e.  ~P B )  ->  F  Fn  ~P B )
3921elpw 3987 . . . . . . . 8  |-  ( ( (mrCls `  dom  ( F  i^i  _I  ) ) `
 z )  e. 
~P B  <->  ( (mrCls ` 
dom  ( F  i^i  _I  ) ) `  z
)  C_  B )
4013, 39sylibr 215 . . . . . . 7  |-  ( ph  ->  ( (mrCls `  dom  ( F  i^i  _I  )
) `  z )  e.  ~P B )
4140adantr 466 . . . . . 6  |-  ( (
ph  /\  z  e.  ~P B )  ->  (
(mrCls `  dom  ( F  i^i  _I  ) ) `
 z )  e. 
~P B )
42 fnelfp 6107 . . . . . 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 665 . . . . 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 213 . . . 4  |-  ( (
ph  /\  z  e.  ~P B )  ->  ( F `  ( (mrCls ` 
dom  ( F  i^i  _I  ) ) `  z
) )  =  ( (mrCls `  dom  ( F  i^i  _I  ) ) `
 z ) )
4535, 44sseqtrd 3500 . . 3  |-  ( (
ph  /\  z  e.  ~P B )  ->  ( F `  z )  C_  ( (mrCls `  dom  ( F  i^i  _I  )
) `  z )
)
468adantr 466 . . . 4  |-  ( (
ph  /\  z  e.  ~P B )  ->  dom  ( F  i^i  _I  )  e.  (Moore `  B )
)
47 sseq1 3485 . . . . . . . 8  |-  ( x  =  z  ->  (
x  C_  B  <->  z  C_  B ) )
4847anbi2d 708 . . . . . . 7  |-  ( x  =  z  ->  (
( ph  /\  x  C_  B )  <->  ( ph  /\  z  C_  B )
) )
49 id 22 . . . . . . . 8  |-  ( x  =  z  ->  x  =  z )
50 fveq2 5881 . . . . . . . 8  |-  ( x  =  z  ->  ( F `  x )  =  ( F `  z ) )
5149, 50sseq12d 3493 . . . . . . 7  |-  ( x  =  z  ->  (
x  C_  ( F `  x )  <->  z  C_  ( F `  z ) ) )
5248, 51imbi12d 321 . . . . . 6  |-  ( x  =  z  ->  (
( ( ph  /\  x  C_  B )  ->  x  C_  ( F `  x ) )  <->  ( ( ph  /\  z  C_  B
)  ->  z  C_  ( F `  z ) ) ) )
5352, 5chvarv 2072 . . . . 5  |-  ( (
ph  /\  z  C_  B )  ->  z  C_  ( F `  z
) )
5415, 53sylan2 476 . . . 4  |-  ( (
ph  /\  z  e.  ~P B )  ->  z  C_  ( F `  z
) )
5550fveq2d 5885 . . . . . . . . 9  |-  ( x  =  z  ->  ( F `  ( F `  x ) )  =  ( F `  ( F `  z )
) )
5655, 50eqeq12d 2444 . . . . . . . 8  |-  ( x  =  z  ->  (
( F `  ( F `  x )
)  =  ( F `
 x )  <->  ( F `  ( F `  z
) )  =  ( F `  z ) ) )
5748, 56imbi12d 321 . . . . . . 7  |-  ( x  =  z  ->  (
( ( ph  /\  x  C_  B )  -> 
( F `  ( F `  x )
)  =  ( F `
 x ) )  <-> 
( ( ph  /\  z  C_  B )  -> 
( F `  ( F `  z )
)  =  ( F `
 z ) ) ) )
5857, 7chvarv 2072 . . . . . 6  |-  ( (
ph  /\  z  C_  B )  ->  ( F `  ( F `  z ) )  =  ( F `  z
) )
5915, 58sylan2 476 . . . . 5  |-  ( (
ph  /\  z  e.  ~P B )  ->  ( F `  ( F `  z ) )  =  ( F `  z
) )
601ffvelrnda 6037 . . . . . 6  |-  ( (
ph  /\  z  e.  ~P B )  ->  ( F `  z )  e.  ~P B )
61 fnelfp 6107 . . . . . 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 665 . . . . 5  |-  ( (
ph  /\  z  e.  ~P B )  ->  (
( F `  z
)  e.  dom  ( F  i^i  _I  )  <->  ( F `  ( F `  z
) )  =  ( F `  z ) ) )
6359, 62mpbird 235 . . . 4  |-  ( (
ph  /\  z  e.  ~P B )  ->  ( F `  z )  e.  dom  ( F  i^i  _I  ) )
649mrcsscl 15525 . . . 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 1264 . . 3  |-  ( (
ph  /\  z  e.  ~P B )  ->  (
(mrCls `  dom  ( F  i^i  _I  ) ) `
 z )  C_  ( F `  z ) )
6645, 65eqssd 3481 . 2  |-  ( (
ph  /\  z  e.  ~P B )  ->  ( F `  z )  =  ( (mrCls `  dom  ( F  i^i  _I  ) ) `  z
) )
673, 12, 66eqfnfvd 5994 1  |-  ( ph  ->  F  =  (mrCls `  dom  ( F  i^i  _I  ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    /\ wa 370    /\ w3a 982   A.wal 1435    = wceq 1437    e. wcel 1872   _Vcvv 3080    i^i cin 3435    C_ wss 3436   ~Pcpw 3981    _I cid 4763   dom cdm 4853    Fn wfn 5596   -->wf 5597   ` cfv 5601  Moorecmre 15487  mrClscmrc 15488
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401  ax-sep 4546  ax-nul 4555  ax-pow 4602  ax-pr 4660  ax-un 6597
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2273  df-mo 2274  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-ne 2616  df-ral 2776  df-rex 2777  df-rab 2780  df-v 3082  df-sbc 3300  df-csb 3396  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-nul 3762  df-if 3912  df-pw 3983  df-sn 3999  df-pr 4001  df-op 4005  df-uni 4220  df-int 4256  df-br 4424  df-opab 4483  df-mpt 4484  df-id 4768  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-rn 4864  df-res 4865  df-ima 4866  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-fv 5609  df-mre 15491  df-mrc 15492
This theorem is referenced by:  istopclsd  35511  ismrc  35512
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