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Mirrors > Home > MPE Home > Th. List > mrcval | Structured version Visualization version GIF version |
Description: Evaluation of the Moore closure of a set. (Contributed by Stefan O'Rear, 31-Jan-2015.) (Proof shortened by Fan Zheng, 6-Jun-2016.) |
Ref | Expression |
---|---|
mrcfval.f | ⊢ 𝐹 = (mrCls‘𝐶) |
Ref | Expression |
---|---|
mrcval | ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝑋) → (𝐹‘𝑈) = ∩ {𝑠 ∈ 𝐶 ∣ 𝑈 ⊆ 𝑠}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mrcfval.f | . . . 4 ⊢ 𝐹 = (mrCls‘𝐶) | |
2 | 1 | mrcfval 16268 | . . 3 ⊢ (𝐶 ∈ (Moore‘𝑋) → 𝐹 = (𝑥 ∈ 𝒫 𝑋 ↦ ∩ {𝑠 ∈ 𝐶 ∣ 𝑥 ⊆ 𝑠})) |
3 | 2 | adantr 481 | . 2 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝑋) → 𝐹 = (𝑥 ∈ 𝒫 𝑋 ↦ ∩ {𝑠 ∈ 𝐶 ∣ 𝑥 ⊆ 𝑠})) |
4 | sseq1 3626 | . . . . 5 ⊢ (𝑥 = 𝑈 → (𝑥 ⊆ 𝑠 ↔ 𝑈 ⊆ 𝑠)) | |
5 | 4 | rabbidv 3189 | . . . 4 ⊢ (𝑥 = 𝑈 → {𝑠 ∈ 𝐶 ∣ 𝑥 ⊆ 𝑠} = {𝑠 ∈ 𝐶 ∣ 𝑈 ⊆ 𝑠}) |
6 | 5 | inteqd 4480 | . . 3 ⊢ (𝑥 = 𝑈 → ∩ {𝑠 ∈ 𝐶 ∣ 𝑥 ⊆ 𝑠} = ∩ {𝑠 ∈ 𝐶 ∣ 𝑈 ⊆ 𝑠}) |
7 | 6 | adantl 482 | . 2 ⊢ (((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝑋) ∧ 𝑥 = 𝑈) → ∩ {𝑠 ∈ 𝐶 ∣ 𝑥 ⊆ 𝑠} = ∩ {𝑠 ∈ 𝐶 ∣ 𝑈 ⊆ 𝑠}) |
8 | mre1cl 16254 | . . . 4 ⊢ (𝐶 ∈ (Moore‘𝑋) → 𝑋 ∈ 𝐶) | |
9 | elpw2g 4827 | . . . 4 ⊢ (𝑋 ∈ 𝐶 → (𝑈 ∈ 𝒫 𝑋 ↔ 𝑈 ⊆ 𝑋)) | |
10 | 8, 9 | syl 17 | . . 3 ⊢ (𝐶 ∈ (Moore‘𝑋) → (𝑈 ∈ 𝒫 𝑋 ↔ 𝑈 ⊆ 𝑋)) |
11 | 10 | biimpar 502 | . 2 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝑋) → 𝑈 ∈ 𝒫 𝑋) |
12 | 8 | adantr 481 | . . . . 5 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝑋) → 𝑋 ∈ 𝐶) |
13 | simpr 477 | . . . . 5 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝑋) → 𝑈 ⊆ 𝑋) | |
14 | sseq2 3627 | . . . . . 6 ⊢ (𝑠 = 𝑋 → (𝑈 ⊆ 𝑠 ↔ 𝑈 ⊆ 𝑋)) | |
15 | 14 | elrab 3363 | . . . . 5 ⊢ (𝑋 ∈ {𝑠 ∈ 𝐶 ∣ 𝑈 ⊆ 𝑠} ↔ (𝑋 ∈ 𝐶 ∧ 𝑈 ⊆ 𝑋)) |
16 | 12, 13, 15 | sylanbrc 698 | . . . 4 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝑋) → 𝑋 ∈ {𝑠 ∈ 𝐶 ∣ 𝑈 ⊆ 𝑠}) |
17 | ne0i 3921 | . . . 4 ⊢ (𝑋 ∈ {𝑠 ∈ 𝐶 ∣ 𝑈 ⊆ 𝑠} → {𝑠 ∈ 𝐶 ∣ 𝑈 ⊆ 𝑠} ≠ ∅) | |
18 | 16, 17 | syl 17 | . . 3 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝑋) → {𝑠 ∈ 𝐶 ∣ 𝑈 ⊆ 𝑠} ≠ ∅) |
19 | intex 4820 | . . 3 ⊢ ({𝑠 ∈ 𝐶 ∣ 𝑈 ⊆ 𝑠} ≠ ∅ ↔ ∩ {𝑠 ∈ 𝐶 ∣ 𝑈 ⊆ 𝑠} ∈ V) | |
20 | 18, 19 | sylib 208 | . 2 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝑋) → ∩ {𝑠 ∈ 𝐶 ∣ 𝑈 ⊆ 𝑠} ∈ V) |
21 | 3, 7, 11, 20 | fvmptd 6288 | 1 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝑋) → (𝐹‘𝑈) = ∩ {𝑠 ∈ 𝐶 ∣ 𝑈 ⊆ 𝑠}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ≠ wne 2794 {crab 2916 Vcvv 3200 ⊆ wss 3574 ∅c0 3915 𝒫 cpw 4158 ∩ cint 4475 ↦ cmpt 4729 ‘cfv 5888 Moorecmre 16242 mrClscmrc 16243 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-int 4476 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-fv 5896 df-mre 16246 df-mrc 16247 |
This theorem is referenced by: mrcid 16273 mrcss 16276 mrcssid 16277 cycsubg2 17631 aspval2 19347 |
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