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Theorem lmrcl 14128
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 14127 . . 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 5140 . 2  |-  dom  ~~> t  C_  Top
3 df-br 4019 . . 3  |-  ( F ( ~~> t `  J
) P  <->  <. F ,  P >.  e.  ( ~~> t `  J ) )
41funmpt2 5271 . . . . 5  |-  Fun  ~~> t
5 funrel 5249 . . . . 5  |-  ( Fun  ~~> t  ->  Rel  ~~> t )
64, 5ax-mp 5 . . . 4  |-  Rel  ~~> t
7 relelfvdm 5563 . . . 4  |-  ( ( Rel  ~~> t  /\  <. F ,  P >.  e.  ( ~~> t `  J ) )  ->  J  e.  dom 
~~> t )
86, 7mpan 424 . . 3  |-  ( <. F ,  P >.  e.  ( ~~> t `  J
)  ->  J  e.  dom 
~~> t )
93, 8sylbi 121 . 2  |-  ( F ( ~~> t `  J
) P  ->  J  e.  dom  ~~> t )
102, 9sselid 3168 1  |-  ( F ( ~~> t `  J
) P  ->  J  e.  Top )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ w3a 980    e. wcel 2160   A.wral 2468   E.wrex 2469   <.cop 3610   U.cuni 3824   class class class wbr 4018   {copab 4078   dom cdm 4641   ran crn 4642    |` cres 4643   Rel wrel 4646   Fun wfun 5226   -->wf 5228   ` cfv 5232  (class class class)co 5892    ^pm cpm 6670   CCcc 7834   ZZ>=cuz 9553   Topctop 13934   ~~> tclm 14124
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4189  ax-pr 4224
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-rab 2477  df-v 2754  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-br 4019  df-opab 4080  df-mpt 4081  df-id 4308  df-xp 4647  df-rel 4648  df-cnv 4649  df-co 4650  df-dm 4651  df-rn 4652  df-res 4653  df-ima 4654  df-iota 5193  df-fun 5234  df-fv 5240  df-lm 14127
This theorem is referenced by:  lmcvg  14154  lmtopcnp  14187
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