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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 17571 | . . . 4 ⊢ (𝐴 ∈ (Moore‘𝑋) → 𝐼 = {𝑠 ∈ 𝒫 𝑋 ∣ ∀𝑥 ∈ 𝑠 ¬ 𝑥 ∈ (𝑁‘(𝑠 ∖ {𝑥}))}) |
4 | 3 | eleq2d 2820 | . . 3 ⊢ (𝐴 ∈ (Moore‘𝑋) → (𝑆 ∈ 𝐼 ↔ 𝑆 ∈ {𝑠 ∈ 𝒫 𝑋 ∣ ∀𝑥 ∈ 𝑠 ¬ 𝑥 ∈ (𝑁‘(𝑠 ∖ {𝑥}))})) |
5 | difeq1 4115 | . . . . . . . 8 ⊢ (𝑠 = 𝑆 → (𝑠 ∖ {𝑥}) = (𝑆 ∖ {𝑥})) | |
6 | 5 | fveq2d 6893 | . . . . . . 7 ⊢ (𝑠 = 𝑆 → (𝑁‘(𝑠 ∖ {𝑥})) = (𝑁‘(𝑆 ∖ {𝑥}))) |
7 | 6 | eleq2d 2820 | . . . . . 6 ⊢ (𝑠 = 𝑆 → (𝑥 ∈ (𝑁‘(𝑠 ∖ {𝑥})) ↔ 𝑥 ∈ (𝑁‘(𝑆 ∖ {𝑥})))) |
8 | 7 | notbid 318 | . . . . 5 ⊢ (𝑠 = 𝑆 → (¬ 𝑥 ∈ (𝑁‘(𝑠 ∖ {𝑥})) ↔ ¬ 𝑥 ∈ (𝑁‘(𝑆 ∖ {𝑥})))) |
9 | 8 | raleqbi1dv 3334 | . . . 4 ⊢ (𝑠 = 𝑆 → (∀𝑥 ∈ 𝑠 ¬ 𝑥 ∈ (𝑁‘(𝑠 ∖ {𝑥})) ↔ ∀𝑥 ∈ 𝑆 ¬ 𝑥 ∈ (𝑁‘(𝑆 ∖ {𝑥})))) |
10 | 9 | elrab 3683 | . . 3 ⊢ (𝑆 ∈ {𝑠 ∈ 𝒫 𝑋 ∣ ∀𝑥 ∈ 𝑠 ¬ 𝑥 ∈ (𝑁‘(𝑠 ∖ {𝑥}))} ↔ (𝑆 ∈ 𝒫 𝑋 ∧ ∀𝑥 ∈ 𝑆 ¬ 𝑥 ∈ (𝑁‘(𝑆 ∖ {𝑥})))) |
11 | 4, 10 | bitrdi 287 | . 2 ⊢ (𝐴 ∈ (Moore‘𝑋) → (𝑆 ∈ 𝐼 ↔ (𝑆 ∈ 𝒫 𝑋 ∧ ∀𝑥 ∈ 𝑆 ¬ 𝑥 ∈ (𝑁‘(𝑆 ∖ {𝑥}))))) |
12 | elfvex 6927 | . . . 4 ⊢ (𝐴 ∈ (Moore‘𝑋) → 𝑋 ∈ V) | |
13 | elpw2g 5344 | . . . 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 631 | . 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 279 | 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 205 ∧ wa 397 = wceq 1542 ∈ wcel 2107 ∀wral 3062 {crab 3433 Vcvv 3475 ∖ cdif 3945 ⊆ wss 3948 𝒫 cpw 4602 {csn 4628 ‘cfv 6541 Moorecmre 17523 mrClscmrc 17524 mrIndcmri 17525 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7722 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-ral 3063 df-rex 3072 df-rab 3434 df-v 3477 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-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-iota 6493 df-fun 6543 df-fv 6549 df-mre 17527 df-mri 17529 |
This theorem is referenced by: ismri2 17573 mriss 17576 lbsacsbs 20762 |
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