Theorem lmcl 21902

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(⇝𝑡‘𝐽)𝑃) → 𝑃 ∈ 𝑋)

Proof of Theorem lmcl

Dummy variables 𝑦 𝑢 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		id 22	. . . 4 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ (TopOn‘𝑋))
2	1	lmbr 21863	. . 3 ⊢ (𝐽 ∈ (TopOn‘𝑋) → (𝐹(⇝𝑡‘𝐽)𝑃 ↔ (𝐹 ∈ (𝑋 ↑pm ℂ) ∧ 𝑃 ∈ 𝑋 ∧ ∀𝑢 ∈ 𝐽 (𝑃 ∈ 𝑢 → ∃𝑦 ∈ ran ℤ≥(𝐹 ↾ 𝑦):𝑦⟶𝑢))))
3	2	biimpa 480	. 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹(⇝𝑡‘𝐽)𝑃) → (𝐹 ∈ (𝑋 ↑pm ℂ) ∧ 𝑃 ∈ 𝑋 ∧ ∀𝑢 ∈ 𝐽 (𝑃 ∈ 𝑢 → ∃𝑦 ∈ ran ℤ≥(𝐹 ↾ 𝑦):𝑦⟶𝑢)))
4	3	simp2d 1140	1 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹(⇝𝑡‘𝐽)𝑃) → 𝑃 ∈ 𝑋)

Syntax hints:   → wi 4   ∧ wa 399   ∧ w3a 1084   ∈ wcel 2111  ∀wral 3106  ∃wrex 3107   class class class wbr 5030  ran crn 5520   ↾ cres 5521  ⟶wf 6320  ‘cfv 6324  (class class class)co 7135   ↑_pm cpm 8390  ℂcc 10524  ℤ_≥cuz 12231  TopOnctopon 21515  ⇝_𝑡clm 21831

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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441

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 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-fv 6332  df-ov 7138  df-top 21499  df-topon 21516  df-lm 21834

This theorem is referenced by:  lmss  21903  lmff  21906  lmcls  21907  lmcn  21910  lmmo  21985  1stccn  22068  1stckgenlem  22158  1stckgen  22159  cmetcaulem  23892  iscmet3lem2  23896  nglmle  23906  minvecolem4b  28661  minvecolem4  28663  axhcompl-zf  28781  heiborlem9  35257  bfplem2  35261  climreeq  42255  xlimcl  42464