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Theorem lmrel 18961
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 18960 . 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 6413 1  |-  Rel  ( ~~> t `  J )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ w3a 965    e. wcel 1758   A.wral 2796   E.wrex 2797   U.cuni 4194   ran crn 4944    |` cres 4945   Rel wrel 4948   -->wf 5517   ` cfv 5521  (class class class)co 6195    ^pm cpm 7320   CCcc 9386   ZZ>=cuz 10967   Topctop 18625   ~~> tclm 18957
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1954  ax-ext 2431  ax-sep 4516  ax-nul 4524  ax-pow 4573  ax-pr 4634
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2265  df-mo 2266  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2602  df-ne 2647  df-ral 2801  df-rex 2802  df-rab 2805  df-v 3074  df-sbc 3289  df-csb 3391  df-dif 3434  df-un 3436  df-in 3438  df-ss 3445  df-nul 3741  df-if 3895  df-sn 3981  df-pr 3983  df-op 3987  df-uni 4195  df-br 4396  df-opab 4454  df-mpt 4455  df-id 4739  df-xp 4949  df-rel 4950  df-cnv 4951  df-co 4952  df-dm 4953  df-rn 4954  df-res 4955  df-ima 4956  df-iota 5484  df-fun 5523  df-fv 5529  df-lm 18960
This theorem is referenced by:  lmfun  19112  cmetcaulem  20926  lmle  20939  heibor1lem  28851  rrncmslem  28874
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