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Theorem mrisval 16498
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 6360 . . . . 5 (Moore‘𝑋) ⊆ ran Moore
32sseli 3748 . . . 4 (𝐴 ∈ (Moore‘𝑋) → 𝐴 ran Moore)
4 unieq 4583 . . . . . . 7 (𝑐 = 𝐴 𝑐 = 𝐴)
54pweqd 4303 . . . . . 6 (𝑐 = 𝐴 → 𝒫 𝑐 = 𝒫 𝐴)
6 fveq2 6333 . . . . . . . . . . 11 (𝑐 = 𝐴 → (mrCls‘𝑐) = (mrCls‘𝐴))
7 mrisval.1 . . . . . . . . . . 11 𝑁 = (mrCls‘𝐴)
86, 7syl6eqr 2823 . . . . . . . . . 10 (𝑐 = 𝐴 → (mrCls‘𝑐) = 𝑁)
98fveq1d 6335 . . . . . . . . 9 (𝑐 = 𝐴 → ((mrCls‘𝑐)‘(𝑠 ∖ {𝑥})) = (𝑁‘(𝑠 ∖ {𝑥})))
109eleq2d 2836 . . . . . . . 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 3135 . . . . . 6 (𝑐 = 𝐴 → (∀𝑥𝑠 ¬ 𝑥 ∈ ((mrCls‘𝑐)‘(𝑠 ∖ {𝑥})) ↔ ∀𝑥𝑠 ¬ 𝑥 ∈ (𝑁‘(𝑠 ∖ {𝑥}))))
135, 12rabeqbidv 3345 . . . . 5 (𝑐 = 𝐴 → {𝑠 ∈ 𝒫 𝑐 ∣ ∀𝑥𝑠 ¬ 𝑥 ∈ ((mrCls‘𝑐)‘(𝑠 ∖ {𝑥}))} = {𝑠 ∈ 𝒫 𝐴 ∣ ∀𝑥𝑠 ¬ 𝑥 ∈ (𝑁‘(𝑠 ∖ {𝑥}))})
14 df-mri 16456 . . . . 5 mrInd = (𝑐 ran Moore ↦ {𝑠 ∈ 𝒫 𝑐 ∣ ∀𝑥𝑠 ¬ 𝑥 ∈ ((mrCls‘𝑐)‘(𝑠 ∖ {𝑥}))})
15 vuniex 7105 . . . . . . 7 𝑐 ∈ V
1615pwex 4982 . . . . . 6 𝒫 𝑐 ∈ V
1716rabex 4947 . . . . 5 {𝑠 ∈ 𝒫 𝑐 ∣ ∀𝑥𝑠 ¬ 𝑥 ∈ ((mrCls‘𝑐)‘(𝑠 ∖ {𝑥}))} ∈ V
1813, 14, 17fvmpt3i 6431 . . . 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 2817 . 2 (𝐴 ∈ (Moore‘𝑋) → 𝐼 = {𝑠 ∈ 𝒫 𝐴 ∣ ∀𝑥𝑠 ¬ 𝑥 ∈ (𝑁‘(𝑠 ∖ {𝑥}))})
21 mreuni 16468 . . . 4 (𝐴 ∈ (Moore‘𝑋) → 𝐴 = 𝑋)
2221pweqd 4303 . . 3 (𝐴 ∈ (Moore‘𝑋) → 𝒫 𝐴 = 𝒫 𝑋)
2322rabeqdv 3344 . 2 (𝐴 ∈ (Moore‘𝑋) → {𝑠 ∈ 𝒫 𝐴 ∣ ∀𝑥𝑠 ¬ 𝑥 ∈ (𝑁‘(𝑠 ∖ {𝑥}))} = {𝑠 ∈ 𝒫 𝑋 ∣ ∀𝑥𝑠 ¬ 𝑥 ∈ (𝑁‘(𝑠 ∖ {𝑥}))})
2420, 23eqtrd 2805 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 1631  wcel 2145  wral 3061  {crab 3065  cdif 3720  𝒫 cpw 4298  {csn 4317   cuni 4575  ran crn 5251  cfv 6030  Moorecmre 16450  mrClscmrc 16451  mrIndcmri 16452
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4916  ax-nul 4924  ax-pow 4975  ax-pr 5035  ax-un 7100
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4227  df-pw 4300  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4576  df-br 4788  df-opab 4848  df-mpt 4865  df-id 5158  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-rn 5261  df-iota 5993  df-fun 6032  df-fv 6038  df-mre 16454  df-mri 16456
This theorem is referenced by:  ismri  16499
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