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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 22501 | . . 3 ⊢ (𝑎 ∈ (𝐽 fLim 𝐹) → 𝐽 ∈ Top) | |
2 | eqid 2818 | . . . 4 ⊢ ∪ 𝐽 = ∪ 𝐽 | |
3 | 2 | flimfil 22505 | . . 3 ⊢ (𝑎 ∈ (𝐽 fLim 𝐹) → 𝐹 ∈ (Fil‘∪ 𝐽)) |
4 | flimclsi 22514 | . . . . . 6 ⊢ (𝑥 ∈ 𝐹 → (𝐽 fLim 𝐹) ⊆ ((cls‘𝐽)‘𝑥)) | |
5 | 4 | sseld 3963 | . . . . 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 3178 | . . 3 ⊢ (𝑎 ∈ (𝐽 fLim 𝐹) → ∀𝑥 ∈ 𝐹 𝑎 ∈ ((cls‘𝐽)‘𝑥)) |
8 | 2 | isfcls 22545 | . . 3 ⊢ (𝑎 ∈ (𝐽 fClus 𝐹) ↔ (𝐽 ∈ Top ∧ 𝐹 ∈ (Fil‘∪ 𝐽) ∧ ∀𝑥 ∈ 𝐹 𝑎 ∈ ((cls‘𝐽)‘𝑥))) |
9 | 1, 3, 7, 8 | syl3anbrc 1335 | . 2 ⊢ (𝑎 ∈ (𝐽 fLim 𝐹) → 𝑎 ∈ (𝐽 fClus 𝐹)) |
10 | 9 | ssriv 3968 | 1 ⊢ (𝐽 fLim 𝐹) ⊆ (𝐽 fClus 𝐹) |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2105 ∀wral 3135 ⊆ wss 3933 ∪ cuni 4830 ‘cfv 6348 (class class class)co 7145 Topctop 21429 clsccl 21554 Filcfil 22381 fLim cflim 22470 fClus cfcls 22472 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-int 4868 df-iun 4912 df-iin 4913 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-ov 7148 df-oprab 7149 df-mpo 7150 df-fbas 20470 df-top 21430 df-cld 21555 df-ntr 21556 df-cls 21557 df-nei 21634 df-fil 22382 df-flim 22475 df-fcls 22477 |
This theorem is referenced by: fclsfnflim 22563 flimfnfcls 22564 uffclsflim 22567 flfssfcf 22574 cnpfcf 22577 cfilfcls 23804 |
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