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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 22908 | . . 3 ⊢ (𝐴 ∈ (𝐽 fClus 𝐹) → 𝐽 ∈ Top) | |
2 | istopon 21809 | . . . 4 ⊢ (𝐽 ∈ (TopOn‘𝑋) ↔ (𝐽 ∈ Top ∧ 𝑋 = ∪ 𝐽)) | |
3 | 2 | baib 539 | . . 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 22907 | . . . 4 ⊢ (𝐴 ∈ (𝐽 fClus 𝐹) → 𝐹 ∈ (Fil‘∪ 𝐽)) |
7 | fveq2 6717 | . . . . 5 ⊢ (𝑋 = ∪ 𝐽 → (Fil‘𝑋) = (Fil‘∪ 𝐽)) | |
8 | 7 | eleq2d 2823 | . . . 4 ⊢ (𝑋 = ∪ 𝐽 → (𝐹 ∈ (Fil‘𝑋) ↔ 𝐹 ∈ (Fil‘∪ 𝐽))) |
9 | 6, 8 | syl5ibrcom 250 | . . 3 ⊢ (𝐴 ∈ (𝐽 fClus 𝐹) → (𝑋 = ∪ 𝐽 → 𝐹 ∈ (Fil‘𝑋))) |
10 | filunibas 22778 | . . . . 5 ⊢ (𝐹 ∈ (Fil‘∪ 𝐽) → ∪ 𝐹 = ∪ 𝐽) | |
11 | 6, 10 | syl 17 | . . . 4 ⊢ (𝐴 ∈ (𝐽 fClus 𝐹) → ∪ 𝐹 = ∪ 𝐽) |
12 | filunibas 22778 | . . . . 5 ⊢ (𝐹 ∈ (Fil‘𝑋) → ∪ 𝐹 = 𝑋) | |
13 | 12 | eqeq1d 2739 | . . . 4 ⊢ (𝐹 ∈ (Fil‘𝑋) → (∪ 𝐹 = ∪ 𝐽 ↔ 𝑋 = ∪ 𝐽)) |
14 | 11, 13 | syl5ibcom 248 | . . 3 ⊢ (𝐴 ∈ (𝐽 fClus 𝐹) → (𝐹 ∈ (Fil‘𝑋) → 𝑋 = ∪ 𝐽)) |
15 | 9, 14 | impbid 215 | . 2 ⊢ (𝐴 ∈ (𝐽 fClus 𝐹) → (𝑋 = ∪ 𝐽 ↔ 𝐹 ∈ (Fil‘𝑋))) |
16 | 4, 15 | bitrd 282 | 1 ⊢ (𝐴 ∈ (𝐽 fClus 𝐹) → (𝐽 ∈ (TopOn‘𝑋) ↔ 𝐹 ∈ (Fil‘𝑋))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 = wceq 1543 ∈ wcel 2110 ∪ cuni 4819 ‘cfv 6380 (class class class)co 7213 Topctop 21790 TopOnctopon 21807 Filcfil 22742 fClus cfcls 22833 |
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 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-sep 5192 ax-nul 5199 ax-pow 5258 ax-pr 5322 ax-un 7523 |
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 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3066 df-rex 3067 df-rab 3070 df-v 3410 df-sbc 3695 df-csb 3812 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-nul 4238 df-if 4440 df-pw 4515 df-sn 4542 df-pr 4544 df-op 4548 df-uni 4820 df-int 4860 df-iin 4907 df-br 5054 df-opab 5116 df-mpt 5136 df-id 5455 df-xp 5557 df-rel 5558 df-cnv 5559 df-co 5560 df-dm 5561 df-rn 5562 df-res 5563 df-ima 5564 df-iota 6338 df-fun 6382 df-fn 6383 df-fv 6388 df-ov 7216 df-oprab 7217 df-mpo 7218 df-fbas 20360 df-topon 21808 df-fil 22743 df-fcls 22838 |
This theorem is referenced by: fclsopni 22912 fclselbas 22913 fclsss1 22919 fclsss2 22920 fclscf 22922 |
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