Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 4547 | . . 3 ⊢ (𝑥 ∈ 𝒫 𝑋 → 𝑥 ⊆ 𝑋) | |
2 | clscld.1 | . . . . 5 ⊢ 𝑋 = ∪ 𝐽 | |
3 | 2 | clsval 21573 | . . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑥 ⊆ 𝑋) → ((cls‘𝐽)‘𝑥) = ∩ {𝑦 ∈ (Clsd‘𝐽) ∣ 𝑥 ⊆ 𝑦}) |
4 | fvex 6676 | . . . 4 ⊢ ((cls‘𝐽)‘𝑥) ∈ V | |
5 | 3, 4 | syl6eqelr 2919 | . . 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 21561 | . 2 ⊢ (𝐽 ∈ Top → (cls‘𝐽) = (𝑥 ∈ 𝒫 𝑋 ↦ ∩ {𝑦 ∈ (Clsd‘𝐽) ∣ 𝑥 ⊆ 𝑦})) |
8 | elpwi 4547 | . . 3 ⊢ (𝑦 ∈ 𝒫 𝑋 → 𝑦 ⊆ 𝑋) | |
9 | 2 | clscld 21583 | . . 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 6879 | 1 ⊢ (𝐽 ∈ Top → (cls‘𝐽):𝒫 𝑋⟶(Clsd‘𝐽)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1528 ∈ wcel 2105 {crab 3139 Vcvv 3492 ⊆ wss 3933 𝒫 cpw 4535 ∪ cuni 4830 ∩ cint 4867 ⟶wf 6344 ‘cfv 6348 Topctop 21429 Clsdccld 21552 clsccl 21554 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-reu 3142 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-int 4868 df-iun 4912 df-iin 4913 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 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 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-top 21430 df-cld 21555 df-cls 21557 |
This theorem is referenced by: clsf2 40354 |
Copyright terms: Public domain | W3C validator |