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Theorem lmrcl 21767
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 21765 . . 3 𝑡 = (𝑗 ∈ Top ↦ {⟨𝑓, 𝑥⟩ ∣ (𝑓 ∈ ( 𝑗pm ℂ) ∧ 𝑥 𝑗 ∧ ∀𝑢𝑗 (𝑥𝑢 → ∃𝑦 ∈ ran ℤ(𝑓𝑦):𝑦𝑢))})
21dmmptss 6088 . 2 dom ⇝𝑡 ⊆ Top
3 df-br 5058 . . 3 (𝐹(⇝𝑡𝐽)𝑃 ↔ ⟨𝐹, 𝑃⟩ ∈ (⇝𝑡𝐽))
4 elfvdm 6695 . . 3 (⟨𝐹, 𝑃⟩ ∈ (⇝𝑡𝐽) → 𝐽 ∈ dom ⇝𝑡)
53, 4sylbi 218 . 2 (𝐹(⇝𝑡𝐽)𝑃𝐽 ∈ dom ⇝𝑡)
62, 5sseldi 3962 1 (𝐹(⇝𝑡𝐽)𝑃𝐽 ∈ Top)
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1079  wcel 2105  wral 3135  wrex 3136  cop 4563   cuni 4830   class class class wbr 5057  {copab 5119  dom cdm 5548  ran crn 5549  cres 5550  wf 6344  cfv 6348  (class class class)co 7145  pm cpm 8396  cc 10523  cuz 12231  Topctop 21429  𝑡clm 21762
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-mpt 5138  df-xp 5554  df-rel 5555  df-cnv 5556  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fv 6356  df-lm 21765
This theorem is referenced by:  lmcvg  21798
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