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Theorem mrisval 16337
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 6255 . . . . 5 (Moore‘𝑋) ⊆ ran Moore
32sseli 3632 . . . 4 (𝐴 ∈ (Moore‘𝑋) → 𝐴 ran Moore)
4 unieq 4476 . . . . . . 7 (𝑐 = 𝐴 𝑐 = 𝐴)
54pweqd 4196 . . . . . 6 (𝑐 = 𝐴 → 𝒫 𝑐 = 𝒫 𝐴)
6 fveq2 6229 . . . . . . . . . . 11 (𝑐 = 𝐴 → (mrCls‘𝑐) = (mrCls‘𝐴))
7 mrisval.1 . . . . . . . . . . 11 𝑁 = (mrCls‘𝐴)
86, 7syl6eqr 2703 . . . . . . . . . 10 (𝑐 = 𝐴 → (mrCls‘𝑐) = 𝑁)
98fveq1d 6231 . . . . . . . . 9 (𝑐 = 𝐴 → ((mrCls‘𝑐)‘(𝑠 ∖ {𝑥})) = (𝑁‘(𝑠 ∖ {𝑥})))
109eleq2d 2716 . . . . . . . 8 (𝑐 = 𝐴 → (𝑥 ∈ ((mrCls‘𝑐)‘(𝑠 ∖ {𝑥})) ↔ 𝑥 ∈ (𝑁‘(𝑠 ∖ {𝑥}))))
1110notbid 307 . . . . . . 7 (𝑐 = 𝐴 → (¬ 𝑥 ∈ ((mrCls‘𝑐)‘(𝑠 ∖ {𝑥})) ↔ ¬ 𝑥 ∈ (𝑁‘(𝑠 ∖ {𝑥}))))
1211ralbidv 3015 . . . . . 6 (𝑐 = 𝐴 → (∀𝑥𝑠 ¬ 𝑥 ∈ ((mrCls‘𝑐)‘(𝑠 ∖ {𝑥})) ↔ ∀𝑥𝑠 ¬ 𝑥 ∈ (𝑁‘(𝑠 ∖ {𝑥}))))
135, 12rabeqbidv 3226 . . . . 5 (𝑐 = 𝐴 → {𝑠 ∈ 𝒫 𝑐 ∣ ∀𝑥𝑠 ¬ 𝑥 ∈ ((mrCls‘𝑐)‘(𝑠 ∖ {𝑥}))} = {𝑠 ∈ 𝒫 𝐴 ∣ ∀𝑥𝑠 ¬ 𝑥 ∈ (𝑁‘(𝑠 ∖ {𝑥}))})
14 df-mri 16295 . . . . 5 mrInd = (𝑐 ran Moore ↦ {𝑠 ∈ 𝒫 𝑐 ∣ ∀𝑥𝑠 ¬ 𝑥 ∈ ((mrCls‘𝑐)‘(𝑠 ∖ {𝑥}))})
15 vuniex 6996 . . . . . . 7 𝑐 ∈ V
1615pwex 4878 . . . . . 6 𝒫 𝑐 ∈ V
1716rabex 4845 . . . . 5 {𝑠 ∈ 𝒫 𝑐 ∣ ∀𝑥𝑠 ¬ 𝑥 ∈ ((mrCls‘𝑐)‘(𝑠 ∖ {𝑥}))} ∈ V
1813, 14, 17fvmpt3i 6326 . . . 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 2697 . 2 (𝐴 ∈ (Moore‘𝑋) → 𝐼 = {𝑠 ∈ 𝒫 𝐴 ∣ ∀𝑥𝑠 ¬ 𝑥 ∈ (𝑁‘(𝑠 ∖ {𝑥}))})
21 mreuni 16307 . . . 4 (𝐴 ∈ (Moore‘𝑋) → 𝐴 = 𝑋)
2221pweqd 4196 . . 3 (𝐴 ∈ (Moore‘𝑋) → 𝒫 𝐴 = 𝒫 𝑋)
2322rabeqdv 3225 . 2 (𝐴 ∈ (Moore‘𝑋) → {𝑠 ∈ 𝒫 𝐴 ∣ ∀𝑥𝑠 ¬ 𝑥 ∈ (𝑁‘(𝑠 ∖ {𝑥}))} = {𝑠 ∈ 𝒫 𝑋 ∣ ∀𝑥𝑠 ¬ 𝑥 ∈ (𝑁‘(𝑠 ∖ {𝑥}))})
2420, 23eqtrd 2685 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 1523  wcel 2030  wral 2941  {crab 2945  cdif 3604  𝒫 cpw 4191  {csn 4210   cuni 4468  ran crn 5144  cfv 5926  Moorecmre 16289  mrClscmrc 16290  mrIndcmri 16291
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-iota 5889  df-fun 5928  df-fv 5934  df-mre 16293  df-mri 16295
This theorem is referenced by:  ismri  16338
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