Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  lmrel Structured version   Visualization version   GIF version

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  ℂcc 9886  ℤ≥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
 Copyright terms: Public domain W3C validator