![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > mrccl | Structured version Visualization version GIF version |
Description: The Moore closure of a set is a closed set. (Contributed by Stefan O'Rear, 31-Jan-2015.) |
Ref | Expression |
---|---|
mrcfval.f | ⊢ 𝐹 = (mrCls‘𝐶) |
Ref | Expression |
---|---|
mrccl | ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝑋) → (𝐹‘𝑈) ∈ 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mrcfval.f | . . . 4 ⊢ 𝐹 = (mrCls‘𝐶) | |
2 | 1 | mrcf 17560 | . . 3 ⊢ (𝐶 ∈ (Moore‘𝑋) → 𝐹:𝒫 𝑋⟶𝐶) |
3 | 2 | adantr 480 | . 2 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝑋) → 𝐹:𝒫 𝑋⟶𝐶) |
4 | mre1cl 17545 | . . . 4 ⊢ (𝐶 ∈ (Moore‘𝑋) → 𝑋 ∈ 𝐶) | |
5 | elpw2g 5344 | . . . 4 ⊢ (𝑋 ∈ 𝐶 → (𝑈 ∈ 𝒫 𝑋 ↔ 𝑈 ⊆ 𝑋)) | |
6 | 4, 5 | syl 17 | . . 3 ⊢ (𝐶 ∈ (Moore‘𝑋) → (𝑈 ∈ 𝒫 𝑋 ↔ 𝑈 ⊆ 𝑋)) |
7 | 6 | biimpar 477 | . 2 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝑋) → 𝑈 ∈ 𝒫 𝑋) |
8 | 3, 7 | ffvelcdmd 7087 | 1 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝑋) → (𝐹‘𝑈) ∈ 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 = wceq 1540 ∈ wcel 2105 ⊆ wss 3948 𝒫 cpw 4602 ⟶wf 6539 ‘cfv 6543 Moorecmre 17533 mrClscmrc 17534 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-ral 3061 df-rex 3070 df-rab 3432 df-v 3475 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-int 4951 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5574 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-fv 6551 df-mre 17537 df-mrc 17538 |
This theorem is referenced by: mrcsncl 17563 mrcidb 17566 mrcidm 17570 submrc 17579 isacs2 17604 mrelatlub 18525 mreclatBAD 18526 gsumwspan 18769 cycsubg2cl 19133 symggen 19386 odf1o1 19488 cntzspan 19760 gsumzsplit 19843 gsumzoppg 19860 gsumpt 19878 dmdprdd 19917 dprdfeq0 19940 dprdspan 19945 dprdres 19946 dprdz 19948 subgdmdprd 19952 subgdprd 19953 dprd2dlem1 19959 dprd2da 19960 dmdprdsplit2lem 19963 mrccss 21557 ismrcd2 41900 proot1mul 42404 mrelatlubALT 47782 mreclat 47784 |
Copyright terms: Public domain | W3C validator |