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Theorem lmrel 20957
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 (⇝𝑡𝐽)

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 20956 . 2 𝑡 = (𝑗 ∈ Top ↦ {⟨𝑓, 𝑥⟩ ∣ (𝑓 ∈ ( 𝑗pm ℂ) ∧ 𝑥 𝑗 ∧ ∀𝑢𝑗 (𝑥𝑢 → ∃𝑦 ∈ ran ℤ(𝑓𝑦):𝑦𝑢))})
21relmptopab 6843 1 Rel (⇝𝑡𝐽)
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1036  wcel 1987  wral 2907  wrex 2908   cuni 4407  ran crn 5080  cres 5081  Rel wrel 5084  wf 5848  cfv 5852  (class class class)co 6610  pm cpm 7810  cc 9886  cuz 11639  Topctop 20630  𝑡clm 20953
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-nul 3897  df-if 4064  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-br 4619  df-opab 4679  df-mpt 4680  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-iota 5815  df-fun 5854  df-fv 5860  df-lm 20956
This theorem is referenced by:  lmfun  21108  cmetcaulem  23009  lmle  23022  heibor1lem  33275  rrncmslem  33298
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