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Mirrors > Home > MPE Home > Th. List > fclsopni | Structured version Visualization version GIF version |
Description: An open neighborhood of a cluster point of a filter intersects any element of that filter. (Contributed by Mario Carneiro, 11-Apr-2015.) (Revised by Stefan O'Rear, 8-Aug-2015.) |
Ref | Expression |
---|---|
fclsopni | ⊢ ((𝐴 ∈ (𝐽 fClus 𝐹) ∧ (𝑈 ∈ 𝐽 ∧ 𝐴 ∈ 𝑈 ∧ 𝑆 ∈ 𝐹)) → (𝑈 ∩ 𝑆) ≠ ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2736 | . . . . . . . 8 ⊢ ∪ 𝐽 = ∪ 𝐽 | |
2 | 1 | fclsfil 22861 | . . . . . . 7 ⊢ (𝐴 ∈ (𝐽 fClus 𝐹) → 𝐹 ∈ (Fil‘∪ 𝐽)) |
3 | fclstopon 22863 | . . . . . . 7 ⊢ (𝐴 ∈ (𝐽 fClus 𝐹) → (𝐽 ∈ (TopOn‘∪ 𝐽) ↔ 𝐹 ∈ (Fil‘∪ 𝐽))) | |
4 | 2, 3 | mpbird 260 | . . . . . 6 ⊢ (𝐴 ∈ (𝐽 fClus 𝐹) → 𝐽 ∈ (TopOn‘∪ 𝐽)) |
5 | fclsopn 22865 | . . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘∪ 𝐽) ∧ 𝐹 ∈ (Fil‘∪ 𝐽)) → (𝐴 ∈ (𝐽 fClus 𝐹) ↔ (𝐴 ∈ ∪ 𝐽 ∧ ∀𝑜 ∈ 𝐽 (𝐴 ∈ 𝑜 → ∀𝑠 ∈ 𝐹 (𝑜 ∩ 𝑠) ≠ ∅)))) | |
6 | 4, 2, 5 | syl2anc 587 | . . . . 5 ⊢ (𝐴 ∈ (𝐽 fClus 𝐹) → (𝐴 ∈ (𝐽 fClus 𝐹) ↔ (𝐴 ∈ ∪ 𝐽 ∧ ∀𝑜 ∈ 𝐽 (𝐴 ∈ 𝑜 → ∀𝑠 ∈ 𝐹 (𝑜 ∩ 𝑠) ≠ ∅)))) |
7 | 6 | ibi 270 | . . . 4 ⊢ (𝐴 ∈ (𝐽 fClus 𝐹) → (𝐴 ∈ ∪ 𝐽 ∧ ∀𝑜 ∈ 𝐽 (𝐴 ∈ 𝑜 → ∀𝑠 ∈ 𝐹 (𝑜 ∩ 𝑠) ≠ ∅))) |
8 | eleq2 2819 | . . . . . 6 ⊢ (𝑜 = 𝑈 → (𝐴 ∈ 𝑜 ↔ 𝐴 ∈ 𝑈)) | |
9 | ineq1 4106 | . . . . . . . 8 ⊢ (𝑜 = 𝑈 → (𝑜 ∩ 𝑠) = (𝑈 ∩ 𝑠)) | |
10 | 9 | neeq1d 2991 | . . . . . . 7 ⊢ (𝑜 = 𝑈 → ((𝑜 ∩ 𝑠) ≠ ∅ ↔ (𝑈 ∩ 𝑠) ≠ ∅)) |
11 | 10 | ralbidv 3108 | . . . . . 6 ⊢ (𝑜 = 𝑈 → (∀𝑠 ∈ 𝐹 (𝑜 ∩ 𝑠) ≠ ∅ ↔ ∀𝑠 ∈ 𝐹 (𝑈 ∩ 𝑠) ≠ ∅)) |
12 | 8, 11 | imbi12d 348 | . . . . 5 ⊢ (𝑜 = 𝑈 → ((𝐴 ∈ 𝑜 → ∀𝑠 ∈ 𝐹 (𝑜 ∩ 𝑠) ≠ ∅) ↔ (𝐴 ∈ 𝑈 → ∀𝑠 ∈ 𝐹 (𝑈 ∩ 𝑠) ≠ ∅))) |
13 | 12 | rspccv 3524 | . . . 4 ⊢ (∀𝑜 ∈ 𝐽 (𝐴 ∈ 𝑜 → ∀𝑠 ∈ 𝐹 (𝑜 ∩ 𝑠) ≠ ∅) → (𝑈 ∈ 𝐽 → (𝐴 ∈ 𝑈 → ∀𝑠 ∈ 𝐹 (𝑈 ∩ 𝑠) ≠ ∅))) |
14 | 7, 13 | simpl2im 507 | . . 3 ⊢ (𝐴 ∈ (𝐽 fClus 𝐹) → (𝑈 ∈ 𝐽 → (𝐴 ∈ 𝑈 → ∀𝑠 ∈ 𝐹 (𝑈 ∩ 𝑠) ≠ ∅))) |
15 | ineq2 4107 | . . . . 5 ⊢ (𝑠 = 𝑆 → (𝑈 ∩ 𝑠) = (𝑈 ∩ 𝑆)) | |
16 | 15 | neeq1d 2991 | . . . 4 ⊢ (𝑠 = 𝑆 → ((𝑈 ∩ 𝑠) ≠ ∅ ↔ (𝑈 ∩ 𝑆) ≠ ∅)) |
17 | 16 | rspccv 3524 | . . 3 ⊢ (∀𝑠 ∈ 𝐹 (𝑈 ∩ 𝑠) ≠ ∅ → (𝑆 ∈ 𝐹 → (𝑈 ∩ 𝑆) ≠ ∅)) |
18 | 14, 17 | syl8 76 | . 2 ⊢ (𝐴 ∈ (𝐽 fClus 𝐹) → (𝑈 ∈ 𝐽 → (𝐴 ∈ 𝑈 → (𝑆 ∈ 𝐹 → (𝑈 ∩ 𝑆) ≠ ∅)))) |
19 | 18 | 3imp2 1351 | 1 ⊢ ((𝐴 ∈ (𝐽 fClus 𝐹) ∧ (𝑈 ∈ 𝐽 ∧ 𝐴 ∈ 𝑈 ∧ 𝑆 ∈ 𝐹)) → (𝑈 ∩ 𝑆) ≠ ∅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 ∧ w3a 1089 = wceq 1543 ∈ wcel 2112 ≠ wne 2932 ∀wral 3051 ∩ cin 3852 ∅c0 4223 ∪ cuni 4805 ‘cfv 6358 (class class class)co 7191 TopOnctopon 21761 Filcfil 22696 fClus cfcls 22787 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-rep 5164 ax-sep 5177 ax-nul 5184 ax-pow 5243 ax-pr 5307 ax-un 7501 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-nel 3037 df-ral 3056 df-rex 3057 df-reu 3058 df-rab 3060 df-v 3400 df-sbc 3684 df-csb 3799 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-op 4534 df-uni 4806 df-int 4846 df-iun 4892 df-iin 4893 df-br 5040 df-opab 5102 df-mpt 5121 df-id 5440 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-ov 7194 df-oprab 7195 df-mpo 7196 df-fbas 20314 df-top 21745 df-topon 21762 df-cld 21870 df-ntr 21871 df-cls 21872 df-fil 22697 df-fcls 22792 |
This theorem is referenced by: fclsneii 22868 supnfcls 22871 flimfnfcls 22879 cfilfcls 24125 |
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