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Mirrors > Home > MPE Home > Th. List > mrcid | Structured version Visualization version GIF version |
Description: The closure of a closed set is itself. (Contributed by Stefan O'Rear, 31-Jan-2015.) |
Ref | Expression |
---|---|
mrcfval.f | ⊢ 𝐹 = (mrCls‘𝐶) |
Ref | Expression |
---|---|
mrcid | ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ∈ 𝐶) → (𝐹‘𝑈) = 𝑈) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mress 17300 | . . 3 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ∈ 𝐶) → 𝑈 ⊆ 𝑋) | |
2 | mrcfval.f | . . . 4 ⊢ 𝐹 = (mrCls‘𝐶) | |
3 | 2 | mrcval 17317 | . . 3 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝑋) → (𝐹‘𝑈) = ∩ {𝑠 ∈ 𝐶 ∣ 𝑈 ⊆ 𝑠}) |
4 | 1, 3 | syldan 591 | . 2 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ∈ 𝐶) → (𝐹‘𝑈) = ∩ {𝑠 ∈ 𝐶 ∣ 𝑈 ⊆ 𝑠}) |
5 | intmin 4905 | . . 3 ⊢ (𝑈 ∈ 𝐶 → ∩ {𝑠 ∈ 𝐶 ∣ 𝑈 ⊆ 𝑠} = 𝑈) | |
6 | 5 | adantl 482 | . 2 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ∈ 𝐶) → ∩ {𝑠 ∈ 𝐶 ∣ 𝑈 ⊆ 𝑠} = 𝑈) |
7 | 4, 6 | eqtrd 2780 | 1 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ∈ 𝐶) → (𝐹‘𝑈) = 𝑈) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1542 ∈ wcel 2110 {crab 3070 ⊆ wss 3892 ∩ cint 4885 ‘cfv 6432 Moorecmre 17289 mrClscmrc 17290 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2015 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2711 ax-sep 5227 ax-nul 5234 ax-pow 5292 ax-pr 5356 ax-un 7582 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2072 df-mo 2542 df-eu 2571 df-clab 2718 df-cleq 2732 df-clel 2818 df-nfc 2891 df-ne 2946 df-ral 3071 df-rex 3072 df-rab 3075 df-v 3433 df-sbc 3721 df-csb 3838 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-nul 4263 df-if 4466 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4846 df-int 4886 df-br 5080 df-opab 5142 df-mpt 5163 df-id 5490 df-xp 5596 df-rel 5597 df-cnv 5598 df-co 5599 df-dm 5600 df-rn 5601 df-res 5602 df-ima 5603 df-iota 6390 df-fun 6434 df-fn 6435 df-f 6436 df-fv 6440 df-mre 17293 df-mrc 17294 |
This theorem is referenced by: mrcidb 17322 mrcidm 17326 mrcsscl 17327 isacs4lem 18260 dprdsn 19637 isnacs3 40529 |
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