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Mirrors > Home > MPE Home > Th. List > mrccls | Structured version Visualization version GIF version |
Description: Moore closure generalizes closure in a topology. (Contributed by Stefan O'Rear, 31-Jan-2015.) |
Ref | Expression |
---|---|
mrccls.f | ⊢ 𝐹 = (mrCls‘(Clsd‘𝐽)) |
Ref | Expression |
---|---|
mrccls | ⊢ (𝐽 ∈ Top → (cls‘𝐽) = 𝐹) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2771 | . . 3 ⊢ ∪ 𝐽 = ∪ 𝐽 | |
2 | 1 | clsfval 21352 | . 2 ⊢ (𝐽 ∈ Top → (cls‘𝐽) = (𝑎 ∈ 𝒫 ∪ 𝐽 ↦ ∩ {𝑏 ∈ (Clsd‘𝐽) ∣ 𝑎 ⊆ 𝑏})) |
3 | 1 | cldmre 21405 | . . 3 ⊢ (𝐽 ∈ Top → (Clsd‘𝐽) ∈ (Moore‘∪ 𝐽)) |
4 | mrccls.f | . . . 4 ⊢ 𝐹 = (mrCls‘(Clsd‘𝐽)) | |
5 | 4 | mrcfval 16749 | . . 3 ⊢ ((Clsd‘𝐽) ∈ (Moore‘∪ 𝐽) → 𝐹 = (𝑎 ∈ 𝒫 ∪ 𝐽 ↦ ∩ {𝑏 ∈ (Clsd‘𝐽) ∣ 𝑎 ⊆ 𝑏})) |
6 | 3, 5 | syl 17 | . 2 ⊢ (𝐽 ∈ Top → 𝐹 = (𝑎 ∈ 𝒫 ∪ 𝐽 ↦ ∩ {𝑏 ∈ (Clsd‘𝐽) ∣ 𝑎 ⊆ 𝑏})) |
7 | 2, 6 | eqtr4d 2810 | 1 ⊢ (𝐽 ∈ Top → (cls‘𝐽) = 𝐹) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1508 ∈ wcel 2051 {crab 3085 ⊆ wss 3822 𝒫 cpw 4416 ∪ cuni 4708 ∩ cint 4745 ↦ cmpt 5004 ‘cfv 6185 Moorecmre 16723 mrClscmrc 16724 Topctop 21220 Clsdccld 21343 clsccl 21345 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1759 ax-4 1773 ax-5 1870 ax-6 1929 ax-7 1966 ax-8 2053 ax-9 2060 ax-10 2080 ax-11 2094 ax-12 2107 ax-13 2302 ax-ext 2743 ax-rep 5045 ax-sep 5056 ax-nul 5063 ax-pow 5115 ax-pr 5182 ax-un 7277 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 835 df-3an 1071 df-tru 1511 df-ex 1744 df-nf 1748 df-sb 2017 df-mo 2548 df-eu 2585 df-clab 2752 df-cleq 2764 df-clel 2839 df-nfc 2911 df-ne 2961 df-ral 3086 df-rex 3087 df-reu 3088 df-rab 3090 df-v 3410 df-sbc 3675 df-csb 3780 df-dif 3825 df-un 3827 df-in 3829 df-ss 3836 df-nul 4173 df-if 4345 df-pw 4418 df-sn 4436 df-pr 4438 df-op 4442 df-uni 4709 df-int 4746 df-iun 4790 df-iin 4791 df-br 4926 df-opab 4988 df-mpt 5005 df-id 5308 df-xp 5409 df-rel 5410 df-cnv 5411 df-co 5412 df-dm 5413 df-rn 5414 df-res 5415 df-ima 5416 df-iota 6149 df-fun 6187 df-fn 6188 df-f 6189 df-f1 6190 df-fo 6191 df-f1o 6192 df-fv 6193 df-mre 16727 df-mrc 16728 df-top 21221 df-cld 21346 df-cls 21348 |
This theorem is referenced by: istopclsd 38730 |
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