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Theorem mrisval 16897
 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 6678 . . . . 5 (Moore‘𝑋) ⊆ ran Moore
32sseli 3914 . . . 4 (𝐴 ∈ (Moore‘𝑋) → 𝐴 ran Moore)
4 unieq 4814 . . . . . . 7 (𝑐 = 𝐴 𝑐 = 𝐴)
54pweqd 4519 . . . . . 6 (𝑐 = 𝐴 → 𝒫 𝑐 = 𝒫 𝐴)
6 fveq2 6649 . . . . . . . . . . 11 (𝑐 = 𝐴 → (mrCls‘𝑐) = (mrCls‘𝐴))
7 mrisval.1 . . . . . . . . . . 11 𝑁 = (mrCls‘𝐴)
86, 7eqtr4di 2854 . . . . . . . . . 10 (𝑐 = 𝐴 → (mrCls‘𝑐) = 𝑁)
98fveq1d 6651 . . . . . . . . 9 (𝑐 = 𝐴 → ((mrCls‘𝑐)‘(𝑠 ∖ {𝑥})) = (𝑁‘(𝑠 ∖ {𝑥})))
109eleq2d 2878 . . . . . . . 8 (𝑐 = 𝐴 → (𝑥 ∈ ((mrCls‘𝑐)‘(𝑠 ∖ {𝑥})) ↔ 𝑥 ∈ (𝑁‘(𝑠 ∖ {𝑥}))))
1110notbid 321 . . . . . . 7 (𝑐 = 𝐴 → (¬ 𝑥 ∈ ((mrCls‘𝑐)‘(𝑠 ∖ {𝑥})) ↔ ¬ 𝑥 ∈ (𝑁‘(𝑠 ∖ {𝑥}))))
1211ralbidv 3165 . . . . . 6 (𝑐 = 𝐴 → (∀𝑥𝑠 ¬ 𝑥 ∈ ((mrCls‘𝑐)‘(𝑠 ∖ {𝑥})) ↔ ∀𝑥𝑠 ¬ 𝑥 ∈ (𝑁‘(𝑠 ∖ {𝑥}))))
135, 12rabeqbidv 3436 . . . . 5 (𝑐 = 𝐴 → {𝑠 ∈ 𝒫 𝑐 ∣ ∀𝑥𝑠 ¬ 𝑥 ∈ ((mrCls‘𝑐)‘(𝑠 ∖ {𝑥}))} = {𝑠 ∈ 𝒫 𝐴 ∣ ∀𝑥𝑠 ¬ 𝑥 ∈ (𝑁‘(𝑠 ∖ {𝑥}))})
14 df-mri 16855 . . . . 5 mrInd = (𝑐 ran Moore ↦ {𝑠 ∈ 𝒫 𝑐 ∣ ∀𝑥𝑠 ¬ 𝑥 ∈ ((mrCls‘𝑐)‘(𝑠 ∖ {𝑥}))})
15 vuniex 7449 . . . . . . 7 𝑐 ∈ V
1615pwex 5249 . . . . . 6 𝒫 𝑐 ∈ V
1716rabex 5202 . . . . 5 {𝑠 ∈ 𝒫 𝑐 ∣ ∀𝑥𝑠 ¬ 𝑥 ∈ ((mrCls‘𝑐)‘(𝑠 ∖ {𝑥}))} ∈ V
1813, 14, 17fvmpt3i 6754 . . . 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 2848 . 2 (𝐴 ∈ (Moore‘𝑋) → 𝐼 = {𝑠 ∈ 𝒫 𝐴 ∣ ∀𝑥𝑠 ¬ 𝑥 ∈ (𝑁‘(𝑠 ∖ {𝑥}))})
21 mreuni 16867 . . . 4 (𝐴 ∈ (Moore‘𝑋) → 𝐴 = 𝑋)
2221pweqd 4519 . . 3 (𝐴 ∈ (Moore‘𝑋) → 𝒫 𝐴 = 𝒫 𝑋)
2322rabeqdv 3435 . 2 (𝐴 ∈ (Moore‘𝑋) → {𝑠 ∈ 𝒫 𝐴 ∣ ∀𝑥𝑠 ¬ 𝑥 ∈ (𝑁‘(𝑠 ∖ {𝑥}))} = {𝑠 ∈ 𝒫 𝑋 ∣ ∀𝑥𝑠 ¬ 𝑥 ∈ (𝑁‘(𝑠 ∖ {𝑥}))})
2420, 23eqtrd 2836 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 1538   ∈ wcel 2112  ∀wral 3109  {crab 3113   ∖ cdif 3881  𝒫 cpw 4500  {csn 4528  ∪ cuni 4803  ran crn 5524  ‘cfv 6328  Moorecmre 16849  mrClscmrc 16850  mrIndcmri 16851 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 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445 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 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-rab 3118  df-v 3446  df-sbc 3724  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5428  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-iota 6287  df-fun 6330  df-fv 6336  df-mre 16853  df-mri 16855 This theorem is referenced by:  ismri  16898
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