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Theorem lmrcl 12203
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 12202 . . 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 4993 . 2  |-  dom  ~~> t  C_  Top
3 df-br 3896 . . 3  |-  ( F ( ~~> t `  J
) P  <->  <. F ,  P >.  e.  ( ~~> t `  J ) )
41funmpt2 5120 . . . . 5  |-  Fun  ~~> t
5 funrel 5098 . . . . 5  |-  ( Fun  ~~> t  ->  Rel  ~~> t )
64, 5ax-mp 7 . . . 4  |-  Rel  ~~> t
7 relelfvdm 5407 . . . 4  |-  ( ( Rel  ~~> t  /\  <. F ,  P >.  e.  ( ~~> t `  J ) )  ->  J  e.  dom 
~~> t )
86, 7mpan 418 . . 3  |-  ( <. F ,  P >.  e.  ( ~~> t `  J
)  ->  J  e.  dom 
~~> t )
93, 8sylbi 120 . 2  |-  ( F ( ~~> t `  J
) P  ->  J  e.  dom  ~~> t )
102, 9sseldi 3061 1  |-  ( F ( ~~> t `  J
) P  ->  J  e.  Top )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ w3a 945    e. wcel 1463   A.wral 2390   E.wrex 2391   <.cop 3496   U.cuni 3702   class class class wbr 3895   {copab 3948   dom cdm 4499   ran crn 4500    |` cres 4501   Rel wrel 4504   Fun wfun 5075   -->wf 5077   ` cfv 5081  (class class class)co 5728    ^pm cpm 6497   CCcc 7545   ZZ>=cuz 9228   Topctop 12007   ~~> tclm 12199
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4006  ax-pow 4058  ax-pr 4091
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2244  df-ral 2395  df-rex 2396  df-rab 2399  df-v 2659  df-un 3041  df-in 3043  df-ss 3050  df-pw 3478  df-sn 3499  df-pr 3500  df-op 3502  df-uni 3703  df-br 3896  df-opab 3950  df-mpt 3951  df-id 4175  df-xp 4505  df-rel 4506  df-cnv 4507  df-co 4508  df-dm 4509  df-rn 4510  df-res 4511  df-ima 4512  df-iota 5046  df-fun 5083  df-fv 5089  df-lm 12202
This theorem is referenced by:  lmcvg  12228  lmtopcnp  12261
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