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Mirrors > Home > MPE Home > Th. List > lmcl | Structured version Visualization version GIF version |
Description: Closure of a limit. (Contributed by NM, 19-Dec-2006.) (Revised by Mario Carneiro, 23-Dec-2013.) |
Ref | Expression |
---|---|
lmcl | ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹(⇝𝑡‘𝐽)𝑃) → 𝑃 ∈ 𝑋) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | id 22 | . . . 4 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ (TopOn‘𝑋)) | |
2 | 1 | lmbr 21943 | . . 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 481 | . 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 1141 | 1 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹(⇝𝑡‘𝐽)𝑃) → 𝑃 ∈ 𝑋) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 400 ∧ w3a 1085 ∈ wcel 2112 ∀wral 3068 ∃wrex 3069 class class class wbr 5025 ran crn 5518 ↾ cres 5519 ⟶wf 6324 ‘cfv 6328 (class class class)co 7143 ↑pm cpm 8410 ℂcc 10558 ℤ≥cuz 12267 TopOnctopon 21595 ⇝𝑡clm 21911 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1912 ax-6 1971 ax-7 2016 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2159 ax-12 2176 ax-ext 2730 ax-sep 5162 ax-nul 5169 ax-pow 5227 ax-pr 5291 ax-un 7452 |
This theorem depends on definitions: df-bi 210 df-an 401 df-or 846 df-3an 1087 df-tru 1542 df-ex 1783 df-nf 1787 df-sb 2071 df-mo 2558 df-eu 2589 df-clab 2737 df-cleq 2751 df-clel 2831 df-nfc 2899 df-ne 2950 df-ral 3073 df-rex 3074 df-rab 3077 df-v 3409 df-sbc 3694 df-csb 3802 df-dif 3857 df-un 3859 df-in 3861 df-ss 3871 df-nul 4222 df-if 4414 df-pw 4489 df-sn 4516 df-pr 4518 df-op 4522 df-uni 4792 df-br 5026 df-opab 5088 df-mpt 5106 df-id 5423 df-xp 5523 df-rel 5524 df-cnv 5525 df-co 5526 df-dm 5527 df-rn 5528 df-res 5529 df-iota 6287 df-fun 6330 df-fn 6331 df-f 6332 df-fv 6336 df-ov 7146 df-top 21579 df-topon 21596 df-lm 21914 |
This theorem is referenced by: lmss 21983 lmff 21986 lmcls 21987 lmcn 21990 lmmo 22065 1stccn 22148 1stckgenlem 22238 1stckgen 22239 cmetcaulem 23973 iscmet3lem2 23977 nglmle 23987 minvecolem4b 28745 minvecolem4 28747 axhcompl-zf 28865 heiborlem9 35522 bfplem2 35526 climreeq 42606 xlimcl 42815 |
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