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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 23250 | . . 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 475 | . 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) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 ∧ w3a 1084 ∈ wcel 2099 ∀wral 3051 ∃wrex 3060 class class class wbr 5145 ran crn 5675 ↾ cres 5676 ⟶wf 6542 ‘cfv 6546 (class class class)co 7416 ↑pm cpm 8848 ℂcc 11147 ℤ≥cuz 12868 TopOnctopon 22900 ⇝𝑡clm 23218 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-sep 5296 ax-nul 5303 ax-pow 5361 ax-pr 5425 ax-un 7738 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-ral 3052 df-rex 3061 df-rab 3420 df-v 3464 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-nul 4323 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4906 df-br 5146 df-opab 5208 df-mpt 5229 df-id 5572 df-xp 5680 df-rel 5681 df-cnv 5682 df-co 5683 df-dm 5684 df-rn 5685 df-res 5686 df-iota 6498 df-fun 6548 df-fn 6549 df-f 6550 df-fv 6554 df-ov 7419 df-top 22884 df-topon 22901 df-lm 23221 |
This theorem is referenced by: lmss 23290 lmff 23293 lmcls 23294 lmcn 23297 lmmo 23372 1stccn 23455 1stckgenlem 23545 1stckgen 23546 cmetcaulem 25304 iscmet3lem2 25308 nglmle 25318 minvecolem4b 30808 minvecolem4 30810 axhcompl-zf 30928 heiborlem9 37533 bfplem2 37537 climreeq 45270 xlimcl 45479 |
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