MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  lmrcl Unicode version

Theorem lmrcl 16961
Description: Reverse closure for the convergence relation. (Contributed by Mario Carneiro, 7-Sep-2015.)
Assertion
Ref Expression
lmrcl  |-  ( F ( ~~> t `  J
) P  ->  J  e.  Top )

Proof of Theorem lmrcl
Dummy variables  j 
f  x  y  u are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-lm 16959 . . 3  |-  ~~> t  =  ( j  e.  Top  |->  { <. f ,  x >.  |  ( f  e.  ( U. j  ^pm  CC )  /\  x  e. 
U. j  /\  A. u  e.  j  (
x  e.  u  ->  E. y  e.  ran  ZZ>= ( f  |`  y
) : y --> u ) ) } )
21dmmptss 5169 . 2  |-  dom  ~~> t  C_  Top
3 df-br 4024 . . 3  |-  ( F ( ~~> t `  J
) P  <->  <. F ,  P >.  e.  ( ~~> t `  J ) )
4 elfvdm 5554 . . 3  |-  ( <. F ,  P >.  e.  ( ~~> t `  J
)  ->  J  e.  dom 
~~> t )
53, 4sylbi 187 . 2  |-  ( F ( ~~> t `  J
) P  ->  J  e.  dom  ~~> t )
62, 5sseldi 3178 1  |-  ( F ( ~~> t `  J
) P  ->  J  e.  Top )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ w3a 934    e. wcel 1684   A.wral 2543   E.wrex 2544   <.cop 3643   U.cuni 3827   class class class wbr 4023   {copab 4076   dom cdm 4689   ran crn 4690    |` cres 4691   -->wf 5251   ` cfv 5255  (class class class)co 5858    ^pm cpm 6773   CCcc 8735   ZZ>=cuz 10230   Topctop 16631   ~~> tclm 16956
This theorem is referenced by:  lmcvg  16992
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
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-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-xp 4695  df-rel 4696  df-cnv 4697  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fv 5263  df-lm 16959
  Copyright terms: Public domain W3C validator