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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 22621 | . . 3 ⊢ (𝐴 ∈ (𝐽 fClus 𝐹) → 𝐽 ∈ Top) | |
2 | istopon 21522 | . . . 4 ⊢ (𝐽 ∈ (TopOn‘𝑋) ↔ (𝐽 ∈ Top ∧ 𝑋 = ∪ 𝐽)) | |
3 | 2 | baib 538 | . . 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 2823 | . . . . 5 ⊢ ∪ 𝐽 = ∪ 𝐽 | |
6 | 5 | fclsfil 22620 | . . . 4 ⊢ (𝐴 ∈ (𝐽 fClus 𝐹) → 𝐹 ∈ (Fil‘∪ 𝐽)) |
7 | fveq2 6672 | . . . . 5 ⊢ (𝑋 = ∪ 𝐽 → (Fil‘𝑋) = (Fil‘∪ 𝐽)) | |
8 | 7 | eleq2d 2900 | . . . 4 ⊢ (𝑋 = ∪ 𝐽 → (𝐹 ∈ (Fil‘𝑋) ↔ 𝐹 ∈ (Fil‘∪ 𝐽))) |
9 | 6, 8 | syl5ibrcom 249 | . . 3 ⊢ (𝐴 ∈ (𝐽 fClus 𝐹) → (𝑋 = ∪ 𝐽 → 𝐹 ∈ (Fil‘𝑋))) |
10 | filunibas 22491 | . . . . 5 ⊢ (𝐹 ∈ (Fil‘∪ 𝐽) → ∪ 𝐹 = ∪ 𝐽) | |
11 | 6, 10 | syl 17 | . . . 4 ⊢ (𝐴 ∈ (𝐽 fClus 𝐹) → ∪ 𝐹 = ∪ 𝐽) |
12 | filunibas 22491 | . . . . 5 ⊢ (𝐹 ∈ (Fil‘𝑋) → ∪ 𝐹 = 𝑋) | |
13 | 12 | eqeq1d 2825 | . . . 4 ⊢ (𝐹 ∈ (Fil‘𝑋) → (∪ 𝐹 = ∪ 𝐽 ↔ 𝑋 = ∪ 𝐽)) |
14 | 11, 13 | syl5ibcom 247 | . . 3 ⊢ (𝐴 ∈ (𝐽 fClus 𝐹) → (𝐹 ∈ (Fil‘𝑋) → 𝑋 = ∪ 𝐽)) |
15 | 9, 14 | impbid 214 | . 2 ⊢ (𝐴 ∈ (𝐽 fClus 𝐹) → (𝑋 = ∪ 𝐽 ↔ 𝐹 ∈ (Fil‘𝑋))) |
16 | 4, 15 | bitrd 281 | 1 ⊢ (𝐴 ∈ (𝐽 fClus 𝐹) → (𝐽 ∈ (TopOn‘𝑋) ↔ 𝐹 ∈ (Fil‘𝑋))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 = wceq 1537 ∈ wcel 2114 ∪ cuni 4840 ‘cfv 6357 (class class class)co 7158 Topctop 21503 TopOnctopon 21520 Filcfil 22455 fClus cfcls 22546 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-int 4879 df-iin 4924 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-iota 6316 df-fun 6359 df-fn 6360 df-fv 6365 df-ov 7161 df-oprab 7162 df-mpo 7163 df-fbas 20544 df-topon 21521 df-fil 22456 df-fcls 22551 |
This theorem is referenced by: fclsopni 22625 fclselbas 22626 fclsss1 22632 fclsss2 22633 fclscf 22635 |
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