Users' Mathboxes Mathbox for Mario Carneiro < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  mclsrcl Structured version   Unicode version

Theorem mclsrcl 29205
Description: Reverse closure for the closure function. (Contributed by Mario Carneiro, 18-Jul-2016.)
Hypotheses
Ref Expression
mclsval.d  |-  D  =  (mDV `  T )
mclsval.e  |-  E  =  (mEx `  T )
mclsval.c  |-  C  =  (mCls `  T )
Assertion
Ref Expression
mclsrcl  |-  ( A  e.  ( K C B )  ->  ( T  e.  _V  /\  K  C_  D  /\  B  C_  E ) )

Proof of Theorem mclsrcl
Dummy variables  h  d  t  c  m  o  p  s  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 n0i 3798 . . 3  |-  ( A  e.  ( K C B )  ->  -.  ( K C B )  =  (/) )
2 mclsval.c . . . . . 6  |-  C  =  (mCls `  T )
3 fvprc 5866 . . . . . 6  |-  ( -.  T  e.  _V  ->  (mCls `  T )  =  (/) )
42, 3syl5eq 2510 . . . . 5  |-  ( -.  T  e.  _V  ->  C  =  (/) )
54oveqd 6313 . . . 4  |-  ( -.  T  e.  _V  ->  ( K C B )  =  ( K (/) B ) )
6 df-ov 6299 . . . . 5  |-  ( K
(/) B )  =  ( (/) `  <. K ,  B >. )
7 0fv 5905 . . . . 5  |-  ( (/) ` 
<. K ,  B >. )  =  (/)
86, 7eqtri 2486 . . . 4  |-  ( K
(/) B )  =  (/)
95, 8syl6eq 2514 . . 3  |-  ( -.  T  e.  _V  ->  ( K C B )  =  (/) )
101, 9nsyl2 127 . 2  |-  ( A  e.  ( K C B )  ->  T  e.  _V )
11 fveq2 5872 . . . . . . . . 9  |-  ( t  =  T  ->  (mCls `  t )  =  (mCls `  T ) )
1211, 2syl6eqr 2516 . . . . . . . 8  |-  ( t  =  T  ->  (mCls `  t )  =  C )
1312oveqd 6313 . . . . . . 7  |-  ( t  =  T  ->  ( K (mCls `  t ) B )  =  ( K C B ) )
1413eleq2d 2527 . . . . . 6  |-  ( t  =  T  ->  ( A  e.  ( K
(mCls `  t ) B )  <->  A  e.  ( K C B ) ) )
15 fvex 5882 . . . . . . . . 9  |-  (mDV `  t )  e.  _V
1615elpw2 4620 . . . . . . . 8  |-  ( K  e.  ~P (mDV `  t )  <->  K  C_  (mDV `  t ) )
17 fveq2 5872 . . . . . . . . . 10  |-  ( t  =  T  ->  (mDV `  t )  =  (mDV
`  T ) )
18 mclsval.d . . . . . . . . . 10  |-  D  =  (mDV `  T )
1917, 18syl6eqr 2516 . . . . . . . . 9  |-  ( t  =  T  ->  (mDV `  t )  =  D )
2019sseq2d 3527 . . . . . . . 8  |-  ( t  =  T  ->  ( K  C_  (mDV `  t
)  <->  K  C_  D ) )
2116, 20syl5bb 257 . . . . . . 7  |-  ( t  =  T  ->  ( K  e.  ~P (mDV `  t )  <->  K  C_  D
) )
22 fvex 5882 . . . . . . . . 9  |-  (mEx `  t )  e.  _V
2322elpw2 4620 . . . . . . . 8  |-  ( B  e.  ~P (mEx `  t )  <->  B  C_  (mEx `  t ) )
24 fveq2 5872 . . . . . . . . . 10  |-  ( t  =  T  ->  (mEx `  t )  =  (mEx
`  T ) )
25 mclsval.e . . . . . . . . . 10  |-  E  =  (mEx `  T )
2624, 25syl6eqr 2516 . . . . . . . . 9  |-  ( t  =  T  ->  (mEx `  t )  =  E )
2726sseq2d 3527 . . . . . . . 8  |-  ( t  =  T  ->  ( B  C_  (mEx `  t
)  <->  B  C_  E ) )
2823, 27syl5bb 257 . . . . . . 7  |-  ( t  =  T  ->  ( B  e.  ~P (mEx `  t )  <->  B  C_  E
) )
2921, 28anbi12d 710 . . . . . 6  |-  ( t  =  T  ->  (
( K  e.  ~P (mDV `  t )  /\  B  e.  ~P (mEx `  t ) )  <->  ( K  C_  D  /\  B  C_  E ) ) )
3014, 29imbi12d 320 . . . . 5  |-  ( t  =  T  ->  (
( A  e.  ( K (mCls `  t
) B )  -> 
( K  e.  ~P (mDV `  t )  /\  B  e.  ~P (mEx `  t ) ) )  <-> 
( A  e.  ( K C B )  ->  ( K  C_  D  /\  B  C_  E
) ) ) )
31 vex 3112 . . . . . . 7  |-  t  e. 
_V
3215pwex 4639 . . . . . . . 8  |-  ~P (mDV `  t )  e.  _V
3322pwex 4639 . . . . . . . 8  |-  ~P (mEx `  t )  e.  _V
3432, 33mpt2ex 6876 . . . . . . 7  |-  ( d  e.  ~P (mDV `  t ) ,  h  e.  ~P (mEx `  t
)  |->  |^| { c  |  ( ( h  u. 
ran  (mVH `  t )
)  C_  c  /\  A. m A. o A. p ( <. m ,  o ,  p >.  e.  (mAx `  t
)  ->  A. s  e.  ran  (mSubst `  t
) ( ( ( s " ( o  u.  ran  (mVH `  t ) ) ) 
C_  c  /\  A. x A. y ( x m y  ->  (
( (mVars `  t
) `  ( s `  ( (mVH `  t
) `  x )
) )  X.  (
(mVars `  t ) `  ( s `  (
(mVH `  t ) `  y ) ) ) )  C_  d )
)  ->  ( s `  p )  e.  c ) ) ) } )  e.  _V
35 df-mcls 29141 . . . . . . . 8  |- mCls  =  ( t  e.  _V  |->  ( d  e.  ~P (mDV `  t ) ,  h  e.  ~P (mEx `  t
)  |->  |^| { c  |  ( ( h  u. 
ran  (mVH `  t )
)  C_  c  /\  A. m A. o A. p ( <. m ,  o ,  p >.  e.  (mAx `  t
)  ->  A. s  e.  ran  (mSubst `  t
) ( ( ( s " ( o  u.  ran  (mVH `  t ) ) ) 
C_  c  /\  A. x A. y ( x m y  ->  (
( (mVars `  t
) `  ( s `  ( (mVH `  t
) `  x )
) )  X.  (
(mVars `  t ) `  ( s `  (
(mVH `  t ) `  y ) ) ) )  C_  d )
)  ->  ( s `  p )  e.  c ) ) ) } ) )
3635fvmpt2 5964 . . . . . . 7  |-  ( ( t  e.  _V  /\  ( d  e.  ~P (mDV `  t ) ,  h  e.  ~P (mEx `  t )  |->  |^| { c  |  ( ( h  u.  ran  (mVH `  t ) )  C_  c  /\  A. m A. o A. p ( <.
m ,  o ,  p >.  e.  (mAx `  t )  ->  A. s  e.  ran  (mSubst `  t
) ( ( ( s " ( o  u.  ran  (mVH `  t ) ) ) 
C_  c  /\  A. x A. y ( x m y  ->  (
( (mVars `  t
) `  ( s `  ( (mVH `  t
) `  x )
) )  X.  (
(mVars `  t ) `  ( s `  (
(mVH `  t ) `  y ) ) ) )  C_  d )
)  ->  ( s `  p )  e.  c ) ) ) } )  e.  _V )  ->  (mCls `  t )  =  ( d  e. 
~P (mDV `  t
) ,  h  e. 
~P (mEx `  t
)  |->  |^| { c  |  ( ( h  u. 
ran  (mVH `  t )
)  C_  c  /\  A. m A. o A. p ( <. m ,  o ,  p >.  e.  (mAx `  t
)  ->  A. s  e.  ran  (mSubst `  t
) ( ( ( s " ( o  u.  ran  (mVH `  t ) ) ) 
C_  c  /\  A. x A. y ( x m y  ->  (
( (mVars `  t
) `  ( s `  ( (mVH `  t
) `  x )
) )  X.  (
(mVars `  t ) `  ( s `  (
(mVH `  t ) `  y ) ) ) )  C_  d )
)  ->  ( s `  p )  e.  c ) ) ) } ) )
3731, 34, 36mp2an 672 . . . . . 6  |-  (mCls `  t )  =  ( d  e.  ~P (mDV `  t ) ,  h  e.  ~P (mEx `  t
)  |->  |^| { c  |  ( ( h  u. 
ran  (mVH `  t )
)  C_  c  /\  A. m A. o A. p ( <. m ,  o ,  p >.  e.  (mAx `  t
)  ->  A. s  e.  ran  (mSubst `  t
) ( ( ( s " ( o  u.  ran  (mVH `  t ) ) ) 
C_  c  /\  A. x A. y ( x m y  ->  (
( (mVars `  t
) `  ( s `  ( (mVH `  t
) `  x )
) )  X.  (
(mVars `  t ) `  ( s `  (
(mVH `  t ) `  y ) ) ) )  C_  d )
)  ->  ( s `  p )  e.  c ) ) ) } )
3837elmpt2cl 6516 . . . . 5  |-  ( A  e.  ( K (mCls `  t ) B )  ->  ( K  e. 
~P (mDV `  t
)  /\  B  e.  ~P (mEx `  t )
) )
3930, 38vtoclg 3167 . . . 4  |-  ( T  e.  _V  ->  ( A  e.  ( K C B )  ->  ( K  C_  D  /\  B  C_  E ) ) )
4010, 39mpcom 36 . . 3  |-  ( A  e.  ( K C B )  ->  ( K  C_  D  /\  B  C_  E ) )
4140simpld 459 . 2  |-  ( A  e.  ( K C B )  ->  K  C_  D )
4240simprd 463 . 2  |-  ( A  e.  ( K C B )  ->  B  C_  E )
4310, 41, 423jca 1176 1  |-  ( A  e.  ( K C B )  ->  ( T  e.  _V  /\  K  C_  D  /\  B  C_  E ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    /\ w3a 973   A.wal 1393    = wceq 1395    e. wcel 1819   {cab 2442   A.wral 2807   _Vcvv 3109    u. cun 3469    C_ wss 3471   (/)c0 3793   ~Pcpw 4015   <.cop 4038   <.cotp 4040   |^|cint 4288   class class class wbr 4456    X. cxp 5006   ran crn 5009   "cima 5011   ` cfv 5594  (class class class)co 6296    |-> cmpt2 6298  mAxcmax 29109  mExcmex 29111  mDVcmdv 29112  mVarscmvrs 29113  mSubstcmsub 29115  mVHcmvh 29116  mClscmcls 29121
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-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591
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-reu 2814  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  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-iun 4334  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-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-1st 6799  df-2nd 6800  df-mcls 29141
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator