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Theorem mrisval 17510
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 6875 . . . . 5 (Moore‘𝑋) ⊆ ran Moore
32sseli 3940 . . . 4 (𝐴 ∈ (Moore‘𝑋) → 𝐴 ran Moore)
4 unieq 4876 . . . . . . 7 (𝑐 = 𝐴 𝑐 = 𝐴)
54pweqd 4577 . . . . . 6 (𝑐 = 𝐴 → 𝒫 𝑐 = 𝒫 𝐴)
6 fveq2 6842 . . . . . . . . . . 11 (𝑐 = 𝐴 → (mrCls‘𝑐) = (mrCls‘𝐴))
7 mrisval.1 . . . . . . . . . . 11 𝑁 = (mrCls‘𝐴)
86, 7eqtr4di 2794 . . . . . . . . . 10 (𝑐 = 𝐴 → (mrCls‘𝑐) = 𝑁)
98fveq1d 6844 . . . . . . . . 9 (𝑐 = 𝐴 → ((mrCls‘𝑐)‘(𝑠 ∖ {𝑥})) = (𝑁‘(𝑠 ∖ {𝑥})))
109eleq2d 2823 . . . . . . . 8 (𝑐 = 𝐴 → (𝑥 ∈ ((mrCls‘𝑐)‘(𝑠 ∖ {𝑥})) ↔ 𝑥 ∈ (𝑁‘(𝑠 ∖ {𝑥}))))
1110notbid 317 . . . . . . 7 (𝑐 = 𝐴 → (¬ 𝑥 ∈ ((mrCls‘𝑐)‘(𝑠 ∖ {𝑥})) ↔ ¬ 𝑥 ∈ (𝑁‘(𝑠 ∖ {𝑥}))))
1211ralbidv 3174 . . . . . 6 (𝑐 = 𝐴 → (∀𝑥𝑠 ¬ 𝑥 ∈ ((mrCls‘𝑐)‘(𝑠 ∖ {𝑥})) ↔ ∀𝑥𝑠 ¬ 𝑥 ∈ (𝑁‘(𝑠 ∖ {𝑥}))))
135, 12rabeqbidv 3424 . . . . 5 (𝑐 = 𝐴 → {𝑠 ∈ 𝒫 𝑐 ∣ ∀𝑥𝑠 ¬ 𝑥 ∈ ((mrCls‘𝑐)‘(𝑠 ∖ {𝑥}))} = {𝑠 ∈ 𝒫 𝐴 ∣ ∀𝑥𝑠 ¬ 𝑥 ∈ (𝑁‘(𝑠 ∖ {𝑥}))})
14 df-mri 17468 . . . . 5 mrInd = (𝑐 ran Moore ↦ {𝑠 ∈ 𝒫 𝑐 ∣ ∀𝑥𝑠 ¬ 𝑥 ∈ ((mrCls‘𝑐)‘(𝑠 ∖ {𝑥}))})
15 vuniex 7676 . . . . . . 7 𝑐 ∈ V
1615pwex 5335 . . . . . 6 𝒫 𝑐 ∈ V
1716rabex 5289 . . . . 5 {𝑠 ∈ 𝒫 𝑐 ∣ ∀𝑥𝑠 ¬ 𝑥 ∈ ((mrCls‘𝑐)‘(𝑠 ∖ {𝑥}))} ∈ V
1813, 14, 17fvmpt3i 6953 . . . 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, 19eqtrid 2788 . 2 (𝐴 ∈ (Moore‘𝑋) → 𝐼 = {𝑠 ∈ 𝒫 𝐴 ∣ ∀𝑥𝑠 ¬ 𝑥 ∈ (𝑁‘(𝑠 ∖ {𝑥}))})
21 mreuni 17480 . . . 4 (𝐴 ∈ (Moore‘𝑋) → 𝐴 = 𝑋)
2221pweqd 4577 . . 3 (𝐴 ∈ (Moore‘𝑋) → 𝒫 𝐴 = 𝒫 𝑋)
2322rabeqdv 3422 . 2 (𝐴 ∈ (Moore‘𝑋) → {𝑠 ∈ 𝒫 𝐴 ∣ ∀𝑥𝑠 ¬ 𝑥 ∈ (𝑁‘(𝑠 ∖ {𝑥}))} = {𝑠 ∈ 𝒫 𝑋 ∣ ∀𝑥𝑠 ¬ 𝑥 ∈ (𝑁‘(𝑠 ∖ {𝑥}))})
2420, 23eqtrd 2776 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 1541  wcel 2106  wral 3064  {crab 3407  cdif 3907  𝒫 cpw 4560  {csn 4586   cuni 4865  ran crn 5634  cfv 6496  Moorecmre 17462  mrClscmrc 17463  mrIndcmri 17464
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-rab 3408  df-v 3447  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-br 5106  df-opab 5168  df-mpt 5189  df-id 5531  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-iota 6448  df-fun 6498  df-fv 6504  df-mre 17466  df-mri 17468
This theorem is referenced by:  ismri  17511
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