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Theorem lmrcl 17218
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 17216 . . 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 5307 . 2  |-  dom  ~~> t  C_  Top
3 df-br 4155 . . 3  |-  ( F ( ~~> t `  J
) P  <->  <. F ,  P >.  e.  ( ~~> t `  J ) )
4 elfvdm 5698 . . 3  |-  ( <. F ,  P >.  e.  ( ~~> t `  J
)  ->  J  e.  dom 
~~> t )
53, 4sylbi 188 . 2  |-  ( F ( ~~> t `  J
) P  ->  J  e.  dom  ~~> t )
62, 5sseldi 3290 1  |-  ( F ( ~~> t `  J
) P  ->  J  e.  Top )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ w3a 936    e. wcel 1717   A.wral 2650   E.wrex 2651   <.cop 3761   U.cuni 3958   class class class wbr 4154   {copab 4207   dom cdm 4819   ran crn 4820    |` cres 4821   -->wf 5391   ` cfv 5395  (class class class)co 6021    ^pm cpm 6956   CCcc 8922   ZZ>=cuz 10421   Topctop 16882   ~~> tclm 17213
This theorem is referenced by:  lmcvg  17249
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-sep 4272  ax-nul 4280  ax-pow 4319  ax-pr 4345
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-ral 2655  df-rex 2656  df-rab 2659  df-v 2902  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-nul 3573  df-if 3684  df-sn 3764  df-pr 3765  df-op 3767  df-uni 3959  df-br 4155  df-opab 4209  df-mpt 4210  df-xp 4825  df-rel 4826  df-cnv 4827  df-dm 4829  df-rn 4830  df-res 4831  df-ima 4832  df-iota 5359  df-fv 5403  df-lm 17216
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