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Theorem lmrcl 12287
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 12286 . . 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 5005 . 2  |-  dom  ~~> t  C_  Top
3 df-br 3900 . . 3  |-  ( F ( ~~> t `  J
) P  <->  <. F ,  P >.  e.  ( ~~> t `  J ) )
41funmpt2 5132 . . . . 5  |-  Fun  ~~> t
5 funrel 5110 . . . . 5  |-  ( Fun  ~~> t  ->  Rel  ~~> t )
64, 5ax-mp 5 . . . 4  |-  Rel  ~~> t
7 relelfvdm 5421 . . . 4  |-  ( ( Rel  ~~> t  /\  <. F ,  P >.  e.  ( ~~> t `  J ) )  ->  J  e.  dom 
~~> t )
86, 7mpan 420 . . 3  |-  ( <. F ,  P >.  e.  ( ~~> t `  J
)  ->  J  e.  dom 
~~> t )
93, 8sylbi 120 . 2  |-  ( F ( ~~> t `  J
) P  ->  J  e.  dom  ~~> t )
102, 9sseldi 3065 1  |-  ( F ( ~~> t `  J
) P  ->  J  e.  Top )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ w3a 947    e. wcel 1465   A.wral 2393   E.wrex 2394   <.cop 3500   U.cuni 3706   class class class wbr 3899   {copab 3958   dom cdm 4509   ran crn 4510    |` cres 4511   Rel wrel 4514   Fun wfun 5087   -->wf 5089   ` cfv 5093  (class class class)co 5742    ^pm cpm 6511   CCcc 7586   ZZ>=cuz 9294   Topctop 12091   ~~> tclm 12283
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-rab 2402  df-v 2662  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-br 3900  df-opab 3960  df-mpt 3961  df-id 4185  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-rn 4520  df-res 4521  df-ima 4522  df-iota 5058  df-fun 5095  df-fv 5101  df-lm 12286
This theorem is referenced by:  lmcvg  12313  lmtopcnp  12346
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