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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 23824 | . . 3 ⊢ (𝑎 ∈ (𝐽 fLim 𝐹) → 𝐽 ∈ Top) | |
2 | eqid 2726 | . . . 4 ⊢ ∪ 𝐽 = ∪ 𝐽 | |
3 | 2 | flimfil 23828 | . . 3 ⊢ (𝑎 ∈ (𝐽 fLim 𝐹) → 𝐹 ∈ (Fil‘∪ 𝐽)) |
4 | flimclsi 23837 | . . . . . 6 ⊢ (𝑥 ∈ 𝐹 → (𝐽 fLim 𝐹) ⊆ ((cls‘𝐽)‘𝑥)) | |
5 | 4 | sseld 3976 | . . . . 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 3139 | . . 3 ⊢ (𝑎 ∈ (𝐽 fLim 𝐹) → ∀𝑥 ∈ 𝐹 𝑎 ∈ ((cls‘𝐽)‘𝑥)) |
8 | 2 | isfcls 23868 | . . 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 3981 | 1 ⊢ (𝐽 fLim 𝐹) ⊆ (𝐽 fClus 𝐹) |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2098 ∀wral 3055 ⊆ wss 3943 ∪ cuni 4902 ‘cfv 6537 (class class class)co 7405 Topctop 22750 clsccl 22877 Filcfil 23704 fLim cflim 23793 fClus cfcls 23795 |
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 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-int 4944 df-iun 4992 df-iin 4993 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5567 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-ov 7408 df-oprab 7409 df-mpo 7410 df-fbas 21237 df-top 22751 df-cld 22878 df-ntr 22879 df-cls 22880 df-nei 22957 df-fil 23705 df-flim 23798 df-fcls 23800 |
This theorem is referenced by: fclsfnflim 23886 flimfnfcls 23887 uffclsflim 23890 flfssfcf 23897 cnpfcf 23900 cfilfcls 25157 |
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