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Theorem lmrcl 20975
 Description: Reverse closure for the convergence relation. (Contributed by Mario Carneiro, 7-Sep-2015.)
Assertion
Ref Expression
lmrcl (𝐹(⇝𝑡𝐽)𝑃𝐽 ∈ 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 20973 . . 3 𝑡 = (𝑗 ∈ Top ↦ {⟨𝑓, 𝑥⟩ ∣ (𝑓 ∈ ( 𝑗pm ℂ) ∧ 𝑥 𝑗 ∧ ∀𝑢𝑗 (𝑥𝑢 → ∃𝑦 ∈ ran ℤ(𝑓𝑦):𝑦𝑢))})
21dmmptss 5600 . 2 dom ⇝𝑡 ⊆ Top
3 df-br 4624 . . 3 (𝐹(⇝𝑡𝐽)𝑃 ↔ ⟨𝐹, 𝑃⟩ ∈ (⇝𝑡𝐽))
4 elfvdm 6187 . . 3 (⟨𝐹, 𝑃⟩ ∈ (⇝𝑡𝐽) → 𝐽 ∈ dom ⇝𝑡)
53, 4sylbi 207 . 2 (𝐹(⇝𝑡𝐽)𝑃𝐽 ∈ dom ⇝𝑡)
62, 5sseldi 3586 1 (𝐹(⇝𝑡𝐽)𝑃𝐽 ∈ Top)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ w3a 1036   ∈ wcel 1987  ∀wral 2908  ∃wrex 2909  ⟨cop 4161  ∪ cuni 4409   class class class wbr 4623  {copab 4682  dom cdm 5084  ran crn 5085   ↾ cres 5086  ⟶wf 5853  ‘cfv 5857  (class class class)co 6615   ↑pm cpm 7818  ℂcc 9894  ℤ≥cuz 11647  Topctop 20638  ⇝𝑡clm 20970 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 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877 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-ne 2791  df-ral 2913  df-rex 2914  df-rab 2917  df-v 3192  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-sn 4156  df-pr 4158  df-op 4162  df-uni 4410  df-br 4624  df-opab 4684  df-mpt 4685  df-xp 5090  df-rel 5091  df-cnv 5092  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-iota 5820  df-fv 5865  df-lm 20973 This theorem is referenced by:  lmcvg  21006
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