Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 2610 | . . 3 ⊢ ∪ 𝐽 = ∪ 𝐽 | |
2 | 1 | clsfval 20639 | . 2 ⊢ (𝐽 ∈ Top → (cls‘𝐽) = (𝑎 ∈ 𝒫 ∪ 𝐽 ↦ ∩ {𝑏 ∈ (Clsd‘𝐽) ∣ 𝑎 ⊆ 𝑏})) |
3 | 1 | cldmre 20692 | . . 3 ⊢ (𝐽 ∈ Top → (Clsd‘𝐽) ∈ (Moore‘∪ 𝐽)) |
4 | mrccls.f | . . . 4 ⊢ 𝐹 = (mrCls‘(Clsd‘𝐽)) | |
5 | 4 | mrcfval 16091 | . . 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 2647 | 1 ⊢ (𝐽 ∈ Top → (cls‘𝐽) = 𝐹) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1475 ∈ wcel 1977 {crab 2900 ⊆ wss 3540 𝒫 cpw 4108 ∪ cuni 4372 ∩ cint 4410 ↦ cmpt 4643 ‘cfv 5804 Moorecmre 16065 mrClscmrc 16066 Topctop 20517 Clsdccld 20630 clsccl 20632 |
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-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
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-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 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-int 4411 df-iun 4457 df-iin 4458 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-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-mre 16069 df-mrc 16070 df-top 20521 df-cld 20633 df-cls 20635 |
This theorem is referenced by: istopclsd 36281 |
Copyright terms: Public domain | W3C validator |