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| Mirrors > Home > MPE Home > Th. List > fclstopon | Structured version Visualization version GIF version | ||
| Description: Reverse closure for the cluster point predicate. (Contributed by Mario Carneiro, 26-Aug-2015.) |
| Ref | Expression |
|---|---|
| fclstopon | ⊢ (𝐴 ∈ (𝐽 fClus 𝐹) → (𝐽 ∈ (TopOn‘𝑋) ↔ 𝐹 ∈ (Fil‘𝑋))) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | fclstop 24019 | . . 3 ⊢ (𝐴 ∈ (𝐽 fClus 𝐹) → 𝐽 ∈ Top) | |
| 2 | istopon 22918 | . . . 4 ⊢ (𝐽 ∈ (TopOn‘𝑋) ↔ (𝐽 ∈ Top ∧ 𝑋 = ∪ 𝐽)) | |
| 3 | 2 | baib 535 | . . 3 ⊢ (𝐽 ∈ Top → (𝐽 ∈ (TopOn‘𝑋) ↔ 𝑋 = ∪ 𝐽)) |
| 4 | 1, 3 | syl 17 | . 2 ⊢ (𝐴 ∈ (𝐽 fClus 𝐹) → (𝐽 ∈ (TopOn‘𝑋) ↔ 𝑋 = ∪ 𝐽)) |
| 5 | eqid 2737 | . . . . 5 ⊢ ∪ 𝐽 = ∪ 𝐽 | |
| 6 | 5 | fclsfil 24018 | . . . 4 ⊢ (𝐴 ∈ (𝐽 fClus 𝐹) → 𝐹 ∈ (Fil‘∪ 𝐽)) |
| 7 | fveq2 6906 | . . . . 5 ⊢ (𝑋 = ∪ 𝐽 → (Fil‘𝑋) = (Fil‘∪ 𝐽)) | |
| 8 | 7 | eleq2d 2827 | . . . 4 ⊢ (𝑋 = ∪ 𝐽 → (𝐹 ∈ (Fil‘𝑋) ↔ 𝐹 ∈ (Fil‘∪ 𝐽))) |
| 9 | 6, 8 | syl5ibrcom 247 | . . 3 ⊢ (𝐴 ∈ (𝐽 fClus 𝐹) → (𝑋 = ∪ 𝐽 → 𝐹 ∈ (Fil‘𝑋))) |
| 10 | filunibas 23889 | . . . . 5 ⊢ (𝐹 ∈ (Fil‘∪ 𝐽) → ∪ 𝐹 = ∪ 𝐽) | |
| 11 | 6, 10 | syl 17 | . . . 4 ⊢ (𝐴 ∈ (𝐽 fClus 𝐹) → ∪ 𝐹 = ∪ 𝐽) |
| 12 | filunibas 23889 | . . . . 5 ⊢ (𝐹 ∈ (Fil‘𝑋) → ∪ 𝐹 = 𝑋) | |
| 13 | 12 | eqeq1d 2739 | . . . 4 ⊢ (𝐹 ∈ (Fil‘𝑋) → (∪ 𝐹 = ∪ 𝐽 ↔ 𝑋 = ∪ 𝐽)) |
| 14 | 11, 13 | syl5ibcom 245 | . . 3 ⊢ (𝐴 ∈ (𝐽 fClus 𝐹) → (𝐹 ∈ (Fil‘𝑋) → 𝑋 = ∪ 𝐽)) |
| 15 | 9, 14 | impbid 212 | . 2 ⊢ (𝐴 ∈ (𝐽 fClus 𝐹) → (𝑋 = ∪ 𝐽 ↔ 𝐹 ∈ (Fil‘𝑋))) |
| 16 | 4, 15 | bitrd 279 | 1 ⊢ (𝐴 ∈ (𝐽 fClus 𝐹) → (𝐽 ∈ (TopOn‘𝑋) ↔ 𝐹 ∈ (Fil‘𝑋))) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 = wceq 1540 ∈ wcel 2108 ∪ cuni 4907 ‘cfv 6561 (class class class)co 7431 Topctop 22899 TopOnctopon 22916 Filcfil 23853 fClus cfcls 23944 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-int 4947 df-iin 4994 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-fbas 21361 df-topon 22917 df-fil 23854 df-fcls 23949 |
| This theorem is referenced by: fclsopni 24023 fclselbas 24024 fclsss1 24030 fclsss2 24031 fclscf 24033 |
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