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Mirrors > Home > MPE Home > Th. List > cldval | Structured version Visualization version GIF version |
Description: The set of closed sets of a topology. (Note that the set of open sets is just the topology itself, so we don't have a separate definition.) (Contributed by NM, 2-Oct-2006.) (Revised by Mario Carneiro, 11-Nov-2013.) |
Ref | Expression |
---|---|
cldval.1 | ⊢ 𝑋 = ∪ 𝐽 |
Ref | Expression |
---|---|
cldval | ⊢ (𝐽 ∈ Top → (Clsd‘𝐽) = {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑥) ∈ 𝐽}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cldval.1 | . . . 4 ⊢ 𝑋 = ∪ 𝐽 | |
2 | 1 | topopn 20536 | . . 3 ⊢ (𝐽 ∈ Top → 𝑋 ∈ 𝐽) |
3 | pwexg 4776 | . . 3 ⊢ (𝑋 ∈ 𝐽 → 𝒫 𝑋 ∈ V) | |
4 | rabexg 4739 | . . 3 ⊢ (𝒫 𝑋 ∈ V → {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑥) ∈ 𝐽} ∈ V) | |
5 | 2, 3, 4 | 3syl 18 | . 2 ⊢ (𝐽 ∈ Top → {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑥) ∈ 𝐽} ∈ V) |
6 | unieq 4380 | . . . . . 6 ⊢ (𝑗 = 𝐽 → ∪ 𝑗 = ∪ 𝐽) | |
7 | 6, 1 | syl6eqr 2662 | . . . . 5 ⊢ (𝑗 = 𝐽 → ∪ 𝑗 = 𝑋) |
8 | 7 | pweqd 4113 | . . . 4 ⊢ (𝑗 = 𝐽 → 𝒫 ∪ 𝑗 = 𝒫 𝑋) |
9 | 7 | difeq1d 3689 | . . . . 5 ⊢ (𝑗 = 𝐽 → (∪ 𝑗 ∖ 𝑥) = (𝑋 ∖ 𝑥)) |
10 | eleq12 2678 | . . . . 5 ⊢ (((∪ 𝑗 ∖ 𝑥) = (𝑋 ∖ 𝑥) ∧ 𝑗 = 𝐽) → ((∪ 𝑗 ∖ 𝑥) ∈ 𝑗 ↔ (𝑋 ∖ 𝑥) ∈ 𝐽)) | |
11 | 9, 10 | mpancom 700 | . . . 4 ⊢ (𝑗 = 𝐽 → ((∪ 𝑗 ∖ 𝑥) ∈ 𝑗 ↔ (𝑋 ∖ 𝑥) ∈ 𝐽)) |
12 | 8, 11 | rabeqbidv 3168 | . . 3 ⊢ (𝑗 = 𝐽 → {𝑥 ∈ 𝒫 ∪ 𝑗 ∣ (∪ 𝑗 ∖ 𝑥) ∈ 𝑗} = {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑥) ∈ 𝐽}) |
13 | df-cld 20633 | . . 3 ⊢ Clsd = (𝑗 ∈ Top ↦ {𝑥 ∈ 𝒫 ∪ 𝑗 ∣ (∪ 𝑗 ∖ 𝑥) ∈ 𝑗}) | |
14 | 12, 13 | fvmptg 6189 | . 2 ⊢ ((𝐽 ∈ Top ∧ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑥) ∈ 𝐽} ∈ V) → (Clsd‘𝐽) = {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑥) ∈ 𝐽}) |
15 | 5, 14 | mpdan 699 | 1 ⊢ (𝐽 ∈ Top → (Clsd‘𝐽) = {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑥) ∈ 𝐽}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 = wceq 1475 ∈ wcel 1977 {crab 2900 Vcvv 3173 ∖ cdif 3537 𝒫 cpw 4108 ∪ cuni 4372 ‘cfv 5804 Topctop 20517 Clsdccld 20630 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-iota 5768 df-fun 5806 df-fv 5812 df-top 20521 df-cld 20633 |
This theorem is referenced by: iscld 20641 mretopd 20706 |
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