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

Theorem lmcl 23305
Description: Closure of a limit. (Contributed by NM, 19-Dec-2006.) (Revised by Mario Carneiro, 23-Dec-2013.)
Assertion
Ref Expression
lmcl ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹(⇝𝑡𝐽)𝑃) → 𝑃𝑋)

Proof of Theorem lmcl
Dummy variables 𝑦 𝑢 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 id 22 . . . 4 (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ (TopOn‘𝑋))
21lmbr 23266 . . 3 (𝐽 ∈ (TopOn‘𝑋) → (𝐹(⇝𝑡𝐽)𝑃 ↔ (𝐹 ∈ (𝑋pm ℂ) ∧ 𝑃𝑋 ∧ ∀𝑢𝐽 (𝑃𝑢 → ∃𝑦 ∈ ran ℤ(𝐹𝑦):𝑦𝑢))))
32biimpa 476 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹(⇝𝑡𝐽)𝑃) → (𝐹 ∈ (𝑋pm ℂ) ∧ 𝑃𝑋 ∧ ∀𝑢𝐽 (𝑃𝑢 → ∃𝑦 ∈ ran ℤ(𝐹𝑦):𝑦𝑢)))
43simp2d 1144 1 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹(⇝𝑡𝐽)𝑃) → 𝑃𝑋)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1087  wcel 2108  wral 3061  wrex 3070   class class class wbr 5143  ran crn 5686  cres 5687  wf 6557  cfv 6561  (class class class)co 7431  pm cpm 8867  cc 11153  cuz 12878  TopOnctopon 22916  𝑡clm 23234
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-fv 6569  df-ov 7434  df-top 22900  df-topon 22917  df-lm 23237
This theorem is referenced by:  lmss  23306  lmff  23309  lmcls  23310  lmcn  23313  lmmo  23388  1stccn  23471  1stckgenlem  23561  1stckgen  23562  cmetcaulem  25322  iscmet3lem2  25326  nglmle  25336  minvecolem4b  30897  minvecolem4  30899  axhcompl-zf  31017  heiborlem9  37826  bfplem2  37830  climreeq  45628  xlimcl  45837
  Copyright terms: Public domain W3C validator