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Mirrors > Home > MPE Home > Th. List > ismri | Structured version Visualization version GIF version |
Description: Criterion for a set to be an independent set of a Moore system. (Contributed by David Moews, 1-May-2017.) |
Ref | Expression |
---|---|
ismri.1 | ⊢ 𝑁 = (mrCls‘𝐴) |
ismri.2 | ⊢ 𝐼 = (mrInd‘𝐴) |
Ref | Expression |
---|---|
ismri | ⊢ (𝐴 ∈ (Moore‘𝑋) → (𝑆 ∈ 𝐼 ↔ (𝑆 ⊆ 𝑋 ∧ ∀𝑥 ∈ 𝑆 ¬ 𝑥 ∈ (𝑁‘(𝑆 ∖ {𝑥}))))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ismri.1 | . . . . 5 ⊢ 𝑁 = (mrCls‘𝐴) | |
2 | ismri.2 | . . . . 5 ⊢ 𝐼 = (mrInd‘𝐴) | |
3 | 1, 2 | mrisval 16893 | . . . 4 ⊢ (𝐴 ∈ (Moore‘𝑋) → 𝐼 = {𝑠 ∈ 𝒫 𝑋 ∣ ∀𝑥 ∈ 𝑠 ¬ 𝑥 ∈ (𝑁‘(𝑠 ∖ {𝑥}))}) |
4 | 3 | eleq2d 2875 | . . 3 ⊢ (𝐴 ∈ (Moore‘𝑋) → (𝑆 ∈ 𝐼 ↔ 𝑆 ∈ {𝑠 ∈ 𝒫 𝑋 ∣ ∀𝑥 ∈ 𝑠 ¬ 𝑥 ∈ (𝑁‘(𝑠 ∖ {𝑥}))})) |
5 | difeq1 4043 | . . . . . . . 8 ⊢ (𝑠 = 𝑆 → (𝑠 ∖ {𝑥}) = (𝑆 ∖ {𝑥})) | |
6 | 5 | fveq2d 6649 | . . . . . . 7 ⊢ (𝑠 = 𝑆 → (𝑁‘(𝑠 ∖ {𝑥})) = (𝑁‘(𝑆 ∖ {𝑥}))) |
7 | 6 | eleq2d 2875 | . . . . . 6 ⊢ (𝑠 = 𝑆 → (𝑥 ∈ (𝑁‘(𝑠 ∖ {𝑥})) ↔ 𝑥 ∈ (𝑁‘(𝑆 ∖ {𝑥})))) |
8 | 7 | notbid 321 | . . . . 5 ⊢ (𝑠 = 𝑆 → (¬ 𝑥 ∈ (𝑁‘(𝑠 ∖ {𝑥})) ↔ ¬ 𝑥 ∈ (𝑁‘(𝑆 ∖ {𝑥})))) |
9 | 8 | raleqbi1dv 3356 | . . . 4 ⊢ (𝑠 = 𝑆 → (∀𝑥 ∈ 𝑠 ¬ 𝑥 ∈ (𝑁‘(𝑠 ∖ {𝑥})) ↔ ∀𝑥 ∈ 𝑆 ¬ 𝑥 ∈ (𝑁‘(𝑆 ∖ {𝑥})))) |
10 | 9 | elrab 3628 | . . 3 ⊢ (𝑆 ∈ {𝑠 ∈ 𝒫 𝑋 ∣ ∀𝑥 ∈ 𝑠 ¬ 𝑥 ∈ (𝑁‘(𝑠 ∖ {𝑥}))} ↔ (𝑆 ∈ 𝒫 𝑋 ∧ ∀𝑥 ∈ 𝑆 ¬ 𝑥 ∈ (𝑁‘(𝑆 ∖ {𝑥})))) |
11 | 4, 10 | syl6bb 290 | . 2 ⊢ (𝐴 ∈ (Moore‘𝑋) → (𝑆 ∈ 𝐼 ↔ (𝑆 ∈ 𝒫 𝑋 ∧ ∀𝑥 ∈ 𝑆 ¬ 𝑥 ∈ (𝑁‘(𝑆 ∖ {𝑥}))))) |
12 | elfvex 6678 | . . . 4 ⊢ (𝐴 ∈ (Moore‘𝑋) → 𝑋 ∈ V) | |
13 | elpw2g 5211 | . . . 4 ⊢ (𝑋 ∈ V → (𝑆 ∈ 𝒫 𝑋 ↔ 𝑆 ⊆ 𝑋)) | |
14 | 12, 13 | syl 17 | . . 3 ⊢ (𝐴 ∈ (Moore‘𝑋) → (𝑆 ∈ 𝒫 𝑋 ↔ 𝑆 ⊆ 𝑋)) |
15 | 14 | anbi1d 632 | . 2 ⊢ (𝐴 ∈ (Moore‘𝑋) → ((𝑆 ∈ 𝒫 𝑋 ∧ ∀𝑥 ∈ 𝑆 ¬ 𝑥 ∈ (𝑁‘(𝑆 ∖ {𝑥}))) ↔ (𝑆 ⊆ 𝑋 ∧ ∀𝑥 ∈ 𝑆 ¬ 𝑥 ∈ (𝑁‘(𝑆 ∖ {𝑥}))))) |
16 | 11, 15 | bitrd 282 | 1 ⊢ (𝐴 ∈ (Moore‘𝑋) → (𝑆 ∈ 𝐼 ↔ (𝑆 ⊆ 𝑋 ∧ ∀𝑥 ∈ 𝑆 ¬ 𝑥 ∈ (𝑁‘(𝑆 ∖ {𝑥}))))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1538 ∈ wcel 2111 ∀wral 3106 {crab 3110 Vcvv 3441 ∖ cdif 3878 ⊆ wss 3881 𝒫 cpw 4497 {csn 4525 ‘cfv 6324 Moorecmre 16845 mrClscmrc 16846 mrIndcmri 16847 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-sbc 3721 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-iota 6283 df-fun 6326 df-fv 6332 df-mre 16849 df-mri 16851 |
This theorem is referenced by: ismri2 16895 mriss 16898 lbsacsbs 19921 |
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