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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 23266 | . . 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 476 | . 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 1144 | 1 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹(⇝𝑡‘𝐽)𝑃) → 𝑃 ∈ 𝑋) |
| 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 ℂcc 11153 ℤ≥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 |
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