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Theorem lmcl 21891
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 21852 . . 3 (𝐽 ∈ (TopOn‘𝑋) → (𝐹(⇝𝑡𝐽)𝑃 ↔ (𝐹 ∈ (𝑋pm ℂ) ∧ 𝑃𝑋 ∧ ∀𝑢𝐽 (𝑃𝑢 → ∃𝑦 ∈ ran ℤ(𝐹𝑦):𝑦𝑢))))
32biimpa 480 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹(⇝𝑡𝐽)𝑃) → (𝐹 ∈ (𝑋pm ℂ) ∧ 𝑃𝑋 ∧ ∀𝑢𝐽 (𝑃𝑢 → ∃𝑦 ∈ ran ℤ(𝐹𝑦):𝑦𝑢)))
43simp2d 1140 1 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹(⇝𝑡𝐽)𝑃) → 𝑃𝑋)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  w3a 1084  wcel 2115  wral 3132  wrex 3133   class class class wbr 5047  ran crn 5537  cres 5538  wf 6332  cfv 6336  (class class class)co 7138  pm cpm 8390  cc 10520  cuz 12229  TopOnctopon 21504  𝑡clm 21820
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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5184  ax-nul 5191  ax-pow 5247  ax-pr 5311  ax-un 7444
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ral 3137  df-rex 3138  df-rab 3141  df-v 3481  df-sbc 3758  df-csb 3866  df-dif 3921  df-un 3923  df-in 3925  df-ss 3935  df-nul 4275  df-if 4449  df-pw 4522  df-sn 4549  df-pr 4551  df-op 4555  df-uni 4820  df-br 5048  df-opab 5110  df-mpt 5128  df-id 5441  df-xp 5542  df-rel 5543  df-cnv 5544  df-co 5545  df-dm 5546  df-rn 5547  df-res 5548  df-iota 6295  df-fun 6338  df-fn 6339  df-f 6340  df-fv 6344  df-ov 7141  df-top 21488  df-topon 21505  df-lm 21823
This theorem is referenced by:  lmss  21892  lmff  21895  lmcls  21896  lmcn  21899  lmmo  21974  1stccn  22057  1stckgenlem  22147  1stckgen  22148  cmetcaulem  23881  iscmet3lem2  23885  nglmle  23895  minvecolem4b  28650  minvecolem4  28652  axhcompl-zf  28770  heiborlem9  35157  bfplem2  35161  climreeq  42097  xlimcl  42306
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