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Theorem lmrcl 12287
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 12286 . . 3 𝑡 = (𝑗 ∈ Top ↦ {⟨𝑓, 𝑥⟩ ∣ (𝑓 ∈ ( 𝑗pm ℂ) ∧ 𝑥 𝑗 ∧ ∀𝑢𝑗 (𝑥𝑢 → ∃𝑦 ∈ ran ℤ(𝑓𝑦):𝑦𝑢))})
21dmmptss 5005 . 2 dom ⇝𝑡 ⊆ Top
3 df-br 3900 . . 3 (𝐹(⇝𝑡𝐽)𝑃 ↔ ⟨𝐹, 𝑃⟩ ∈ (⇝𝑡𝐽))
41funmpt2 5132 . . . . 5 Fun ⇝𝑡
5 funrel 5110 . . . . 5 (Fun ⇝𝑡 → Rel ⇝𝑡)
64, 5ax-mp 5 . . . 4 Rel ⇝𝑡
7 relelfvdm 5421 . . . 4 ((Rel ⇝𝑡 ∧ ⟨𝐹, 𝑃⟩ ∈ (⇝𝑡𝐽)) → 𝐽 ∈ dom ⇝𝑡)
86, 7mpan 420 . . 3 (⟨𝐹, 𝑃⟩ ∈ (⇝𝑡𝐽) → 𝐽 ∈ dom ⇝𝑡)
93, 8sylbi 120 . 2 (𝐹(⇝𝑡𝐽)𝑃𝐽 ∈ dom ⇝𝑡)
102, 9sseldi 3065 1 (𝐹(⇝𝑡𝐽)𝑃𝐽 ∈ Top)
Colors of variables: wff set class
Syntax hints:  wi 4  w3a 947  wcel 1465  wral 2393  wrex 2394  cop 3500   cuni 3706   class class class wbr 3899  {copab 3958  dom cdm 4509  ran crn 4510  cres 4511  Rel wrel 4514  Fun wfun 5087  wf 5089  cfv 5093  (class class class)co 5742  pm cpm 6511  cc 7586  cuz 9294  Topctop 12091  𝑡clm 12283
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-rab 2402  df-v 2662  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-br 3900  df-opab 3960  df-mpt 3961  df-id 4185  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-rn 4520  df-res 4521  df-ima 4522  df-iota 5058  df-fun 5095  df-fv 5101  df-lm 12286
This theorem is referenced by:  lmcvg  12313  lmtopcnp  12346
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