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Theorem lmrel 18675
Description: The topological space convergence relation is a relation. (Contributed by NM, 7-Dec-2006.) (Revised by Mario Carneiro, 14-Nov-2013.)
Assertion
Ref Expression
lmrel  |-  Rel  ( ~~> t `  J )

Proof of Theorem lmrel
Dummy variables  j 
f  x  y  u are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-lm 18674 . 2  |-  ~~> 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 ) ) } )
21relmptopab 6297 1  |-  Rel  ( ~~> t `  J )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ w3a 958    e. wcel 1755   A.wral 2705   E.wrex 2706   U.cuni 4079   ran crn 4828    |` cres 4829   Rel wrel 4832   -->wf 5402   ` cfv 5406  (class class class)co 6080    ^pm cpm 7203   CCcc 9267   ZZ>=cuz 10848   Topctop 18339   ~~> tclm 18671
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1594  ax-4 1605  ax-5 1669  ax-6 1707  ax-7 1727  ax-8 1757  ax-9 1759  ax-10 1774  ax-11 1779  ax-12 1791  ax-13 1942  ax-ext 2414  ax-sep 4401  ax-nul 4409  ax-pow 4458  ax-pr 4519
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 960  df-tru 1365  df-ex 1590  df-nf 1593  df-sb 1700  df-eu 2258  df-mo 2259  df-clab 2420  df-cleq 2426  df-clel 2429  df-nfc 2558  df-ne 2598  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2964  df-sbc 3176  df-csb 3277  df-dif 3319  df-un 3321  df-in 3323  df-ss 3330  df-nul 3626  df-if 3780  df-sn 3866  df-pr 3868  df-op 3872  df-uni 4080  df-br 4281  df-opab 4339  df-mpt 4340  df-id 4623  df-xp 4833  df-rel 4834  df-cnv 4835  df-co 4836  df-dm 4837  df-rn 4838  df-res 4839  df-ima 4840  df-iota 5369  df-fun 5408  df-fv 5414  df-lm 18674
This theorem is referenced by:  lmfun  18826  cmetcaulem  20640  lmle  20653  heibor1lem  28549  rrncmslem  28572
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