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Theorem lmcl 21011
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 20972 . . 3 (𝐽 ∈ (TopOn‘𝑋) → (𝐹(⇝𝑡𝐽)𝑃 ↔ (𝐹 ∈ (𝑋pm ℂ) ∧ 𝑃𝑋 ∧ ∀𝑢𝐽 (𝑃𝑢 → ∃𝑦 ∈ ran ℤ(𝐹𝑦):𝑦𝑢))))
32biimpa 501 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹(⇝𝑡𝐽)𝑃) → (𝐹 ∈ (𝑋pm ℂ) ∧ 𝑃𝑋 ∧ ∀𝑢𝐽 (𝑃𝑢 → ∃𝑦 ∈ ran ℤ(𝐹𝑦):𝑦𝑢)))
43simp2d 1072 1 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹(⇝𝑡𝐽)𝑃) → 𝑃𝑋)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  w3a 1036  wcel 1987  wral 2907  wrex 2908   class class class wbr 4613  ran crn 5075  cres 5076  wf 5843  cfv 5847  (class class class)co 6604  pm cpm 7803  cc 9878  cuz 11631  TopOnctopon 20618  𝑡clm 20940
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-fv 5855  df-ov 6607  df-top 20621  df-topon 20623  df-lm 20943
This theorem is referenced by:  lmss  21012  lmff  21015  lmcls  21016  lmcn  21019  lmmo  21094  1stccn  21176  1stckgenlem  21266  1stckgen  21267  cmetcaulem  22994  iscmet3lem2  22998  nglmle  23008  minvecolem4b  27580  minvecolem4  27582  axhcompl-zf  27701  heiborlem9  33247  bfplem2  33251  climreeq  39246
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