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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 4519 | . . 3 ⊢ (𝑥 ∈ 𝒫 𝑋 → 𝑥 ⊆ 𝑋) | |
2 | clscld.1 | . . . . 5 ⊢ 𝑋 = ∪ 𝐽 | |
3 | 2 | clsval 21931 | . . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑥 ⊆ 𝑋) → ((cls‘𝐽)‘𝑥) = ∩ {𝑦 ∈ (Clsd‘𝐽) ∣ 𝑥 ⊆ 𝑦}) |
4 | fvex 6727 | . . . 4 ⊢ ((cls‘𝐽)‘𝑥) ∈ V | |
5 | 3, 4 | eqeltrrdi 2847 | . . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑥 ⊆ 𝑋) → ∩ {𝑦 ∈ (Clsd‘𝐽) ∣ 𝑥 ⊆ 𝑦} ∈ V) |
6 | 1, 5 | sylan2 596 | . 2 ⊢ ((𝐽 ∈ Top ∧ 𝑥 ∈ 𝒫 𝑋) → ∩ {𝑦 ∈ (Clsd‘𝐽) ∣ 𝑥 ⊆ 𝑦} ∈ V) |
7 | 2 | clsfval 21919 | . 2 ⊢ (𝐽 ∈ Top → (cls‘𝐽) = (𝑥 ∈ 𝒫 𝑋 ↦ ∩ {𝑦 ∈ (Clsd‘𝐽) ∣ 𝑥 ⊆ 𝑦})) |
8 | elpwi 4519 | . . 3 ⊢ (𝑦 ∈ 𝒫 𝑋 → 𝑦 ⊆ 𝑋) | |
9 | 2 | clscld 21941 | . . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑦 ⊆ 𝑋) → ((cls‘𝐽)‘𝑦) ∈ (Clsd‘𝐽)) |
10 | 8, 9 | sylan2 596 | . 2 ⊢ ((𝐽 ∈ Top ∧ 𝑦 ∈ 𝒫 𝑋) → ((cls‘𝐽)‘𝑦) ∈ (Clsd‘𝐽)) |
11 | 6, 7, 10 | fmpt2d 6937 | 1 ⊢ (𝐽 ∈ Top → (cls‘𝐽):𝒫 𝑋⟶(Clsd‘𝐽)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1543 ∈ wcel 2110 {crab 3062 Vcvv 3405 ⊆ wss 3863 𝒫 cpw 4510 ∪ cuni 4816 ∩ cint 4856 ⟶wf 6373 ‘cfv 6377 Topctop 21787 Clsdccld 21910 clsccl 21912 |
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-rep 5176 ax-sep 5189 ax-nul 5196 ax-pow 5255 ax-pr 5319 ax-un 7520 |
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 2940 df-ral 3063 df-rex 3064 df-reu 3065 df-rab 3067 df-v 3407 df-sbc 3692 df-csb 3809 df-dif 3866 df-un 3868 df-in 3870 df-ss 3880 df-nul 4235 df-if 4437 df-pw 4512 df-sn 4539 df-pr 4541 df-op 4545 df-uni 4817 df-int 4857 df-iun 4903 df-iin 4904 df-br 5051 df-opab 5113 df-mpt 5133 df-id 5452 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6335 df-fun 6379 df-fn 6380 df-f 6381 df-f1 6382 df-fo 6383 df-f1o 6384 df-fv 6385 df-top 21788 df-cld 21913 df-cls 21915 |
This theorem is referenced by: clsf2 41411 |
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