ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  lmrcl Unicode version

Theorem lmrcl 12985
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 12984 . . 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 5107 . 2  |-  dom  ~~> t  C_  Top
3 df-br 3990 . . 3  |-  ( F ( ~~> t `  J
) P  <->  <. F ,  P >.  e.  ( ~~> t `  J ) )
41funmpt2 5237 . . . . 5  |-  Fun  ~~> t
5 funrel 5215 . . . . 5  |-  ( Fun  ~~> t  ->  Rel  ~~> t )
64, 5ax-mp 5 . . . 4  |-  Rel  ~~> t
7 relelfvdm 5528 . . . 4  |-  ( ( Rel  ~~> t  /\  <. F ,  P >.  e.  ( ~~> t `  J ) )  ->  J  e.  dom 
~~> t )
86, 7mpan 422 . . 3  |-  ( <. F ,  P >.  e.  ( ~~> t `  J
)  ->  J  e.  dom 
~~> t )
93, 8sylbi 120 . 2  |-  ( F ( ~~> t `  J
) P  ->  J  e.  dom  ~~> t )
102, 9sselid 3145 1  |-  ( F ( ~~> t `  J
) P  ->  J  e.  Top )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ w3a 973    e. wcel 2141   A.wral 2448   E.wrex 2449   <.cop 3586   U.cuni 3796   class class class wbr 3989   {copab 4049   dom cdm 4611   ran crn 4612    |` cres 4613   Rel wrel 4616   Fun wfun 5192   -->wf 5194   ` cfv 5198  (class class class)co 5853    ^pm cpm 6627   CCcc 7772   ZZ>=cuz 9487   Topctop 12789   ~~> tclm 12981
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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-rab 2457  df-v 2732  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-br 3990  df-opab 4051  df-mpt 4052  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fv 5206  df-lm 12984
This theorem is referenced by:  lmcvg  13011  lmtopcnp  13044
  Copyright terms: Public domain W3C validator