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Theorem lmrcl 22717
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 22715 . . 3 𝑡 = (𝑗 ∈ Top ↦ {⟨𝑓, 𝑥⟩ ∣ (𝑓 ∈ ( 𝑗pm ℂ) ∧ 𝑥 𝑗 ∧ ∀𝑢𝑗 (𝑥𝑢 → ∃𝑦 ∈ ran ℤ(𝑓𝑦):𝑦𝑢))})
21dmmptss 6237 . 2 dom ⇝𝑡 ⊆ Top
3 df-br 5148 . . 3 (𝐹(⇝𝑡𝐽)𝑃 ↔ ⟨𝐹, 𝑃⟩ ∈ (⇝𝑡𝐽))
4 elfvdm 6925 . . 3 (⟨𝐹, 𝑃⟩ ∈ (⇝𝑡𝐽) → 𝐽 ∈ dom ⇝𝑡)
53, 4sylbi 216 . 2 (𝐹(⇝𝑡𝐽)𝑃𝐽 ∈ dom ⇝𝑡)
62, 5sselid 3979 1 (𝐹(⇝𝑡𝐽)𝑃𝐽 ∈ Top)
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1088  wcel 2107  wral 3062  wrex 3071  cop 4633   cuni 4907   class class class wbr 5147  {copab 5209  dom cdm 5675  ran crn 5676  cres 5677  wf 6536  cfv 6540  (class class class)co 7404  pm cpm 8817  cc 11104  cuz 12818  Topctop 22377  𝑡clm 22712
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5298  ax-nul 5305  ax-pr 5426
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-mpt 5231  df-xp 5681  df-rel 5682  df-cnv 5683  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-iota 6492  df-fv 6548  df-lm 22715
This theorem is referenced by:  lmcvg  22748
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