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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 2819 | . . 3 ⊢ ∪ 𝐽 = ∪ 𝐽 | |
2 | 1 | clsfval 21625 | . 2 ⊢ (𝐽 ∈ Top → (cls‘𝐽) = (𝑎 ∈ 𝒫 ∪ 𝐽 ↦ ∩ {𝑏 ∈ (Clsd‘𝐽) ∣ 𝑎 ⊆ 𝑏})) |
3 | 1 | cldmre 21678 | . . 3 ⊢ (𝐽 ∈ Top → (Clsd‘𝐽) ∈ (Moore‘∪ 𝐽)) |
4 | mrccls.f | . . . 4 ⊢ 𝐹 = (mrCls‘(Clsd‘𝐽)) | |
5 | 4 | mrcfval 16871 | . . 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 2857 | 1 ⊢ (𝐽 ∈ Top → (cls‘𝐽) = 𝐹) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1531 ∈ wcel 2108 {crab 3140 ⊆ wss 3934 𝒫 cpw 4537 ∪ cuni 4830 ∩ cint 4867 ↦ cmpt 5137 ‘cfv 6348 Moorecmre 16845 mrClscmrc 16846 Topctop 21493 Clsdccld 21616 clsccl 21618 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1905 ax-6 1964 ax-7 2009 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2154 ax-12 2170 ax-ext 2791 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7453 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1084 df-tru 1534 df-ex 1775 df-nf 1779 df-sb 2064 df-mo 2616 df-eu 2648 df-clab 2798 df-cleq 2812 df-clel 2891 df-nfc 2961 df-ne 3015 df-ral 3141 df-rex 3142 df-reu 3143 df-rab 3145 df-v 3495 df-sbc 3771 df-csb 3882 df-dif 3937 df-un 3939 df-in 3941 df-ss 3950 df-nul 4290 df-if 4466 df-pw 4539 df-sn 4560 df-pr 4562 df-op 4566 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-mre 16849 df-mrc 16850 df-top 21494 df-cld 21619 df-cls 21621 |
This theorem is referenced by: istopclsd 39288 |
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