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

Theorem lmrcl 22955
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 22953 . . 3 𝑡 = (𝑗 ∈ Top ↦ {⟨𝑓, 𝑥⟩ ∣ (𝑓 ∈ ( 𝑗pm ℂ) ∧ 𝑥 𝑗 ∧ ∀𝑢𝑗 (𝑥𝑢 → ∃𝑦 ∈ ran ℤ(𝑓𝑦):𝑦𝑢))})
21dmmptss 6240 . 2 dom ⇝𝑡 ⊆ Top
3 df-br 5149 . . 3 (𝐹(⇝𝑡𝐽)𝑃 ↔ ⟨𝐹, 𝑃⟩ ∈ (⇝𝑡𝐽))
4 elfvdm 6928 . . 3 (⟨𝐹, 𝑃⟩ ∈ (⇝𝑡𝐽) → 𝐽 ∈ dom ⇝𝑡)
53, 4sylbi 216 . 2 (𝐹(⇝𝑡𝐽)𝑃𝐽 ∈ dom ⇝𝑡)
62, 5sselid 3980 1 (𝐹(⇝𝑡𝐽)𝑃𝐽 ∈ Top)
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1087  wcel 2106  wral 3061  wrex 3070  cop 4634   cuni 4908   class class class wbr 5148  {copab 5210  dom cdm 5676  ran crn 5677  cres 5678  wf 6539  cfv 6543  (class class class)co 7411  pm cpm 8823  cc 11110  cuz 12826  Topctop 22615  𝑡clm 22950
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pr 5427
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-xp 5682  df-rel 5683  df-cnv 5684  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fv 6551  df-lm 22953
This theorem is referenced by:  lmcvg  22986
  Copyright terms: Public domain W3C validator