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Mirrors > Home > MPE Home > Th. List > fclssscls | Structured version Visualization version GIF version |
Description: The set of cluster points is a subset of the closure of any filter element. (Contributed by Mario Carneiro, 11-Apr-2015.) (Revised by Stefan O'Rear, 8-Aug-2015.) |
Ref | Expression |
---|---|
fclssscls | ⊢ (𝑆 ∈ 𝐹 → (𝐽 fClus 𝐹) ⊆ ((cls‘𝐽)‘𝑆)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2732 | . . . . 5 ⊢ ∪ 𝐽 = ∪ 𝐽 | |
2 | 1 | isfcls 23520 | . . . 4 ⊢ (𝑥 ∈ (𝐽 fClus 𝐹) ↔ (𝐽 ∈ Top ∧ 𝐹 ∈ (Fil‘∪ 𝐽) ∧ ∀𝑠 ∈ 𝐹 𝑥 ∈ ((cls‘𝐽)‘𝑠))) |
3 | 2 | simp3bi 1147 | . . 3 ⊢ (𝑥 ∈ (𝐽 fClus 𝐹) → ∀𝑠 ∈ 𝐹 𝑥 ∈ ((cls‘𝐽)‘𝑠)) |
4 | fveq2 6891 | . . . . 5 ⊢ (𝑠 = 𝑆 → ((cls‘𝐽)‘𝑠) = ((cls‘𝐽)‘𝑆)) | |
5 | 4 | eleq2d 2819 | . . . 4 ⊢ (𝑠 = 𝑆 → (𝑥 ∈ ((cls‘𝐽)‘𝑠) ↔ 𝑥 ∈ ((cls‘𝐽)‘𝑆))) |
6 | 5 | rspcv 3608 | . . 3 ⊢ (𝑆 ∈ 𝐹 → (∀𝑠 ∈ 𝐹 𝑥 ∈ ((cls‘𝐽)‘𝑠) → 𝑥 ∈ ((cls‘𝐽)‘𝑆))) |
7 | 3, 6 | syl5 34 | . 2 ⊢ (𝑆 ∈ 𝐹 → (𝑥 ∈ (𝐽 fClus 𝐹) → 𝑥 ∈ ((cls‘𝐽)‘𝑆))) |
8 | 7 | ssrdv 3988 | 1 ⊢ (𝑆 ∈ 𝐹 → (𝐽 fClus 𝐹) ⊆ ((cls‘𝐽)‘𝑆)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1541 ∈ wcel 2106 ∀wral 3061 ⊆ wss 3948 ∪ cuni 4908 ‘cfv 6543 (class class class)co 7411 Topctop 22402 clsccl 22529 Filcfil 23356 fClus cfcls 23447 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-int 4951 df-iin 5000 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5574 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-iota 6495 df-fun 6545 df-fn 6546 df-fv 6551 df-ov 7414 df-oprab 7415 df-mpo 7416 df-fbas 20947 df-fil 23357 df-fcls 23452 |
This theorem is referenced by: fclscmp 23541 |
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