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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 2734 | . . . . . . . 8 ⊢ ∪ 𝐽 = ∪ 𝐽 | |
2 | 1 | fclsfil 24033 | . . . . . . 7 ⊢ (𝐴 ∈ (𝐽 fClus 𝐹) → 𝐹 ∈ (Fil‘∪ 𝐽)) |
3 | fclstopon 24035 | . . . . . . 7 ⊢ (𝐴 ∈ (𝐽 fClus 𝐹) → (𝐽 ∈ (TopOn‘∪ 𝐽) ↔ 𝐹 ∈ (Fil‘∪ 𝐽))) | |
4 | 2, 3 | mpbird 257 | . . . . . 6 ⊢ (𝐴 ∈ (𝐽 fClus 𝐹) → 𝐽 ∈ (TopOn‘∪ 𝐽)) |
5 | fclsopn 24037 | . . . . . 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 584 | . . . . 5 ⊢ (𝐴 ∈ (𝐽 fClus 𝐹) → (𝐴 ∈ (𝐽 fClus 𝐹) ↔ (𝐴 ∈ ∪ 𝐽 ∧ ∀𝑜 ∈ 𝐽 (𝐴 ∈ 𝑜 → ∀𝑠 ∈ 𝐹 (𝑜 ∩ 𝑠) ≠ ∅)))) |
7 | 6 | ibi 267 | . . . 4 ⊢ (𝐴 ∈ (𝐽 fClus 𝐹) → (𝐴 ∈ ∪ 𝐽 ∧ ∀𝑜 ∈ 𝐽 (𝐴 ∈ 𝑜 → ∀𝑠 ∈ 𝐹 (𝑜 ∩ 𝑠) ≠ ∅))) |
8 | eleq2 2827 | . . . . . 6 ⊢ (𝑜 = 𝑈 → (𝐴 ∈ 𝑜 ↔ 𝐴 ∈ 𝑈)) | |
9 | ineq1 4220 | . . . . . . . 8 ⊢ (𝑜 = 𝑈 → (𝑜 ∩ 𝑠) = (𝑈 ∩ 𝑠)) | |
10 | 9 | neeq1d 2997 | . . . . . . 7 ⊢ (𝑜 = 𝑈 → ((𝑜 ∩ 𝑠) ≠ ∅ ↔ (𝑈 ∩ 𝑠) ≠ ∅)) |
11 | 10 | ralbidv 3175 | . . . . . 6 ⊢ (𝑜 = 𝑈 → (∀𝑠 ∈ 𝐹 (𝑜 ∩ 𝑠) ≠ ∅ ↔ ∀𝑠 ∈ 𝐹 (𝑈 ∩ 𝑠) ≠ ∅)) |
12 | 8, 11 | imbi12d 344 | . . . . 5 ⊢ (𝑜 = 𝑈 → ((𝐴 ∈ 𝑜 → ∀𝑠 ∈ 𝐹 (𝑜 ∩ 𝑠) ≠ ∅) ↔ (𝐴 ∈ 𝑈 → ∀𝑠 ∈ 𝐹 (𝑈 ∩ 𝑠) ≠ ∅))) |
13 | 12 | rspccv 3618 | . . . 4 ⊢ (∀𝑜 ∈ 𝐽 (𝐴 ∈ 𝑜 → ∀𝑠 ∈ 𝐹 (𝑜 ∩ 𝑠) ≠ ∅) → (𝑈 ∈ 𝐽 → (𝐴 ∈ 𝑈 → ∀𝑠 ∈ 𝐹 (𝑈 ∩ 𝑠) ≠ ∅))) |
14 | 7, 13 | simpl2im 503 | . . 3 ⊢ (𝐴 ∈ (𝐽 fClus 𝐹) → (𝑈 ∈ 𝐽 → (𝐴 ∈ 𝑈 → ∀𝑠 ∈ 𝐹 (𝑈 ∩ 𝑠) ≠ ∅))) |
15 | ineq2 4221 | . . . . 5 ⊢ (𝑠 = 𝑆 → (𝑈 ∩ 𝑠) = (𝑈 ∩ 𝑆)) | |
16 | 15 | neeq1d 2997 | . . . 4 ⊢ (𝑠 = 𝑆 → ((𝑈 ∩ 𝑠) ≠ ∅ ↔ (𝑈 ∩ 𝑆) ≠ ∅)) |
17 | 16 | rspccv 3618 | . . 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 1348 | 1 ⊢ ((𝐴 ∈ (𝐽 fClus 𝐹) ∧ (𝑈 ∈ 𝐽 ∧ 𝐴 ∈ 𝑈 ∧ 𝑆 ∈ 𝐹)) → (𝑈 ∩ 𝑆) ≠ ∅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1086 = wceq 1536 ∈ wcel 2105 ≠ wne 2937 ∀wral 3058 ∩ cin 3961 ∅c0 4338 ∪ cuni 4911 ‘cfv 6562 (class class class)co 7430 TopOnctopon 22931 Filcfil 23868 fClus cfcls 23959 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-rep 5284 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-int 4951 df-iun 4997 df-iin 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5582 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-ov 7433 df-oprab 7434 df-mpo 7435 df-fbas 21378 df-top 22915 df-topon 22932 df-cld 23042 df-ntr 23043 df-cls 23044 df-fil 23869 df-fcls 23964 |
This theorem is referenced by: fclsneii 24040 supnfcls 24043 flimfnfcls 24051 cfilfcls 25321 |
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