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Theorem mrisval 16005
Description: Value of the set of independent sets of a Moore system. (Contributed by David Moews, 1-May-2017.)
Hypotheses
Ref Expression
mrisval.1 𝑁 = (mrCls‘𝐴)
mrisval.2 𝐼 = (mrInd‘𝐴)
Assertion
Ref Expression
mrisval (𝐴 ∈ (Moore‘𝑋) → 𝐼 = {𝑠 ∈ 𝒫 𝑋 ∣ ∀𝑥𝑠 ¬ 𝑥 ∈ (𝑁‘(𝑠 ∖ {𝑥}))})
Distinct variable groups:   𝐴,𝑠,𝑥   𝑋,𝑠
Allowed substitution hints:   𝐼(𝑥,𝑠)   𝑁(𝑥,𝑠)   𝑋(𝑥)

Proof of Theorem mrisval
Dummy variable 𝑐 is distinct from all other variables.
StepHypRef Expression
1 mrisval.2 . . 3 𝐼 = (mrInd‘𝐴)
2 fvssunirn 6011 . . . . 5 (Moore‘𝑋) ⊆ ran Moore
32sseli 3468 . . . 4 (𝐴 ∈ (Moore‘𝑋) → 𝐴 ran Moore)
4 unieq 4278 . . . . . . 7 (𝑐 = 𝐴 𝑐 = 𝐴)
54pweqd 4016 . . . . . 6 (𝑐 = 𝐴 → 𝒫 𝑐 = 𝒫 𝐴)
6 fveq2 5987 . . . . . . . . . . 11 (𝑐 = 𝐴 → (mrCls‘𝑐) = (mrCls‘𝐴))
7 mrisval.1 . . . . . . . . . . 11 𝑁 = (mrCls‘𝐴)
86, 7syl6eqr 2566 . . . . . . . . . 10 (𝑐 = 𝐴 → (mrCls‘𝑐) = 𝑁)
98fveq1d 5989 . . . . . . . . 9 (𝑐 = 𝐴 → ((mrCls‘𝑐)‘(𝑠 ∖ {𝑥})) = (𝑁‘(𝑠 ∖ {𝑥})))
109eleq2d 2577 . . . . . . . 8 (𝑐 = 𝐴 → (𝑥 ∈ ((mrCls‘𝑐)‘(𝑠 ∖ {𝑥})) ↔ 𝑥 ∈ (𝑁‘(𝑠 ∖ {𝑥}))))
1110notbid 306 . . . . . . 7 (𝑐 = 𝐴 → (¬ 𝑥 ∈ ((mrCls‘𝑐)‘(𝑠 ∖ {𝑥})) ↔ ¬ 𝑥 ∈ (𝑁‘(𝑠 ∖ {𝑥}))))
1211ralbidv 2873 . . . . . 6 (𝑐 = 𝐴 → (∀𝑥𝑠 ¬ 𝑥 ∈ ((mrCls‘𝑐)‘(𝑠 ∖ {𝑥})) ↔ ∀𝑥𝑠 ¬ 𝑥 ∈ (𝑁‘(𝑠 ∖ {𝑥}))))
135, 12rabeqbidv 3072 . . . . 5 (𝑐 = 𝐴 → {𝑠 ∈ 𝒫 𝑐 ∣ ∀𝑥𝑠 ¬ 𝑥 ∈ ((mrCls‘𝑐)‘(𝑠 ∖ {𝑥}))} = {𝑠 ∈ 𝒫 𝐴 ∣ ∀𝑥𝑠 ¬ 𝑥 ∈ (𝑁‘(𝑠 ∖ {𝑥}))})
14 df-mri 15963 . . . . 5 mrInd = (𝑐 ran Moore ↦ {𝑠 ∈ 𝒫 𝑐 ∣ ∀𝑥𝑠 ¬ 𝑥 ∈ ((mrCls‘𝑐)‘(𝑠 ∖ {𝑥}))})
15 vuniex 6728 . . . . . . 7 𝑐 ∈ V
1615pwex 4673 . . . . . 6 𝒫 𝑐 ∈ V
1716rabex 4639 . . . . 5 {𝑠 ∈ 𝒫 𝑐 ∣ ∀𝑥𝑠 ¬ 𝑥 ∈ ((mrCls‘𝑐)‘(𝑠 ∖ {𝑥}))} ∈ V
1813, 14, 17fvmpt3i 6080 . . . 4 (𝐴 ran Moore → (mrInd‘𝐴) = {𝑠 ∈ 𝒫 𝐴 ∣ ∀𝑥𝑠 ¬ 𝑥 ∈ (𝑁‘(𝑠 ∖ {𝑥}))})
193, 18syl 17 . . 3 (𝐴 ∈ (Moore‘𝑋) → (mrInd‘𝐴) = {𝑠 ∈ 𝒫 𝐴 ∣ ∀𝑥𝑠 ¬ 𝑥 ∈ (𝑁‘(𝑠 ∖ {𝑥}))})
201, 19syl5eq 2560 . 2 (𝐴 ∈ (Moore‘𝑋) → 𝐼 = {𝑠 ∈ 𝒫 𝐴 ∣ ∀𝑥𝑠 ¬ 𝑥 ∈ (𝑁‘(𝑠 ∖ {𝑥}))})
21 mreuni 15975 . . . 4 (𝐴 ∈ (Moore‘𝑋) → 𝐴 = 𝑋)
2221pweqd 4016 . . 3 (𝐴 ∈ (Moore‘𝑋) → 𝒫 𝐴 = 𝒫 𝑋)
2322rabeqdv 3071 . 2 (𝐴 ∈ (Moore‘𝑋) → {𝑠 ∈ 𝒫 𝐴 ∣ ∀𝑥𝑠 ¬ 𝑥 ∈ (𝑁‘(𝑠 ∖ {𝑥}))} = {𝑠 ∈ 𝒫 𝑋 ∣ ∀𝑥𝑠 ¬ 𝑥 ∈ (𝑁‘(𝑠 ∖ {𝑥}))})
2420, 23eqtrd 2548 1 (𝐴 ∈ (Moore‘𝑋) → 𝐼 = {𝑠 ∈ 𝒫 𝑋 ∣ ∀𝑥𝑠 ¬ 𝑥 ∈ (𝑁‘(𝑠 ∖ {𝑥}))})
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1474  wcel 1938  wral 2800  {crab 2804  cdif 3441  𝒫 cpw 4011  {csn 4028   cuni 4270  ran crn 4933  cfv 5689  Moorecmre 15957  mrClscmrc 15958  mrIndcmri 15959
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1700  ax-4 1713  ax-5 1793  ax-6 1838  ax-7 1885  ax-8 1940  ax-9 1947  ax-10 1966  ax-11 1971  ax-12 1983  ax-13 2137  ax-ext 2494  ax-sep 4607  ax-nul 4616  ax-pow 4668  ax-pr 4732  ax-un 6723
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1699  df-sb 1831  df-eu 2366  df-mo 2367  df-clab 2501  df-cleq 2507  df-clel 2510  df-nfc 2644  df-ne 2686  df-ral 2805  df-rex 2806  df-rab 2809  df-v 3079  df-sbc 3307  df-dif 3447  df-un 3449  df-in 3451  df-ss 3458  df-nul 3778  df-if 3940  df-pw 4013  df-sn 4029  df-pr 4031  df-op 4035  df-uni 4271  df-br 4482  df-opab 4542  df-mpt 4543  df-id 4847  df-xp 4938  df-rel 4939  df-cnv 4940  df-co 4941  df-dm 4942  df-rn 4943  df-iota 5653  df-fun 5691  df-fv 5697  df-mre 15961  df-mri 15963
This theorem is referenced by:  ismri  16006
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