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Mirrors > Home > MPE Home > Th. List > flimfcls | Structured version Visualization version GIF version |
Description: A limit point is a cluster point. (Contributed by Jeff Hankins, 12-Nov-2009.) (Revised by Stefan O'Rear, 8-Aug-2015.) |
Ref | Expression |
---|---|
flimfcls | ⊢ (𝐽 fLim 𝐹) ⊆ (𝐽 fClus 𝐹) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | flimtop 23889 | . . 3 ⊢ (𝑎 ∈ (𝐽 fLim 𝐹) → 𝐽 ∈ Top) | |
2 | eqid 2728 | . . . 4 ⊢ ∪ 𝐽 = ∪ 𝐽 | |
3 | 2 | flimfil 23893 | . . 3 ⊢ (𝑎 ∈ (𝐽 fLim 𝐹) → 𝐹 ∈ (Fil‘∪ 𝐽)) |
4 | flimclsi 23902 | . . . . . 6 ⊢ (𝑥 ∈ 𝐹 → (𝐽 fLim 𝐹) ⊆ ((cls‘𝐽)‘𝑥)) | |
5 | 4 | sseld 3981 | . . . . 5 ⊢ (𝑥 ∈ 𝐹 → (𝑎 ∈ (𝐽 fLim 𝐹) → 𝑎 ∈ ((cls‘𝐽)‘𝑥))) |
6 | 5 | com12 32 | . . . 4 ⊢ (𝑎 ∈ (𝐽 fLim 𝐹) → (𝑥 ∈ 𝐹 → 𝑎 ∈ ((cls‘𝐽)‘𝑥))) |
7 | 6 | ralrimiv 3142 | . . 3 ⊢ (𝑎 ∈ (𝐽 fLim 𝐹) → ∀𝑥 ∈ 𝐹 𝑎 ∈ ((cls‘𝐽)‘𝑥)) |
8 | 2 | isfcls 23933 | . . 3 ⊢ (𝑎 ∈ (𝐽 fClus 𝐹) ↔ (𝐽 ∈ Top ∧ 𝐹 ∈ (Fil‘∪ 𝐽) ∧ ∀𝑥 ∈ 𝐹 𝑎 ∈ ((cls‘𝐽)‘𝑥))) |
9 | 1, 3, 7, 8 | syl3anbrc 1340 | . 2 ⊢ (𝑎 ∈ (𝐽 fLim 𝐹) → 𝑎 ∈ (𝐽 fClus 𝐹)) |
10 | 9 | ssriv 3986 | 1 ⊢ (𝐽 fLim 𝐹) ⊆ (𝐽 fClus 𝐹) |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2098 ∀wral 3058 ⊆ wss 3949 ∪ cuni 4912 ‘cfv 6553 (class class class)co 7426 Topctop 22815 clsccl 22942 Filcfil 23769 fLim cflim 23858 fClus cfcls 23860 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7746 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-int 4954 df-iun 5002 df-iin 5003 df-br 5153 df-opab 5215 df-mpt 5236 df-id 5580 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-ov 7429 df-oprab 7430 df-mpo 7431 df-fbas 21283 df-top 22816 df-cld 22943 df-ntr 22944 df-cls 22945 df-nei 23022 df-fil 23770 df-flim 23863 df-fcls 23865 |
This theorem is referenced by: fclsfnflim 23951 flimfnfcls 23952 uffclsflim 23955 flfssfcf 23962 cnpfcf 23965 cfilfcls 25222 |
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