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Mirrors > Home > MPE Home > Th. List > clsf | Structured version Visualization version GIF version |
Description: The closure function is a function from subsets of the base to closed sets. (Contributed by Mario Carneiro, 11-Apr-2015.) |
Ref | Expression |
---|---|
clscld.1 | ⊢ 𝑋 = ∪ 𝐽 |
Ref | Expression |
---|---|
clsf | ⊢ (𝐽 ∈ Top → (cls‘𝐽):𝒫 𝑋⟶(Clsd‘𝐽)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elpwi 4602 | . . 3 ⊢ (𝑥 ∈ 𝒫 𝑋 → 𝑥 ⊆ 𝑋) | |
2 | clscld.1 | . . . . 5 ⊢ 𝑋 = ∪ 𝐽 | |
3 | 2 | clsval 22865 | . . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑥 ⊆ 𝑋) → ((cls‘𝐽)‘𝑥) = ∩ {𝑦 ∈ (Clsd‘𝐽) ∣ 𝑥 ⊆ 𝑦}) |
4 | fvex 6895 | . . . 4 ⊢ ((cls‘𝐽)‘𝑥) ∈ V | |
5 | 3, 4 | eqeltrrdi 2834 | . . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑥 ⊆ 𝑋) → ∩ {𝑦 ∈ (Clsd‘𝐽) ∣ 𝑥 ⊆ 𝑦} ∈ V) |
6 | 1, 5 | sylan2 592 | . 2 ⊢ ((𝐽 ∈ Top ∧ 𝑥 ∈ 𝒫 𝑋) → ∩ {𝑦 ∈ (Clsd‘𝐽) ∣ 𝑥 ⊆ 𝑦} ∈ V) |
7 | 2 | clsfval 22853 | . 2 ⊢ (𝐽 ∈ Top → (cls‘𝐽) = (𝑥 ∈ 𝒫 𝑋 ↦ ∩ {𝑦 ∈ (Clsd‘𝐽) ∣ 𝑥 ⊆ 𝑦})) |
8 | elpwi 4602 | . . 3 ⊢ (𝑦 ∈ 𝒫 𝑋 → 𝑦 ⊆ 𝑋) | |
9 | 2 | clscld 22875 | . . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑦 ⊆ 𝑋) → ((cls‘𝐽)‘𝑦) ∈ (Clsd‘𝐽)) |
10 | 8, 9 | sylan2 592 | . 2 ⊢ ((𝐽 ∈ Top ∧ 𝑦 ∈ 𝒫 𝑋) → ((cls‘𝐽)‘𝑦) ∈ (Clsd‘𝐽)) |
11 | 6, 7, 10 | fmpt2d 7116 | 1 ⊢ (𝐽 ∈ Top → (cls‘𝐽):𝒫 𝑋⟶(Clsd‘𝐽)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1533 ∈ wcel 2098 {crab 3424 Vcvv 3466 ⊆ wss 3941 𝒫 cpw 4595 ∪ cuni 4900 ∩ cint 4941 ⟶wf 6530 ‘cfv 6534 Topctop 22719 Clsdccld 22844 clsccl 22846 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-rep 5276 ax-sep 5290 ax-nul 5297 ax-pow 5354 ax-pr 5418 ax-un 7719 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-ral 3054 df-rex 3063 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-nul 4316 df-if 4522 df-pw 4597 df-sn 4622 df-pr 4624 df-op 4628 df-uni 4901 df-int 4942 df-iun 4990 df-iin 4991 df-br 5140 df-opab 5202 df-mpt 5223 df-id 5565 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-dm 5677 df-rn 5678 df-res 5679 df-ima 5680 df-iota 6486 df-fun 6536 df-fn 6537 df-f 6538 df-f1 6539 df-fo 6540 df-f1o 6541 df-fv 6542 df-top 22720 df-cld 22847 df-cls 22849 |
This theorem is referenced by: clsf2 43391 |
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