Theorem mrisval 17681

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 ∣ ∀𝑥 ∈ 𝑠 ¬ 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.

Step	Hyp	Ref	Expression
1		mrisval.2	. . 3 ⊢ 𝐼 = (mrInd‘𝐴)
2		fvssunirn 6912	. . . . 5 ⊢ (Moore‘𝑋) ⊆ ∪ ran Moore
3	2	sseli 3933	. . . 4 ⊢ (𝐴 ∈ (Moore‘𝑋) → 𝐴 ∈ ∪ ran Moore)
4		unieq 4883	. . . . . . 7 ⊢ (𝑐 = 𝐴 → ∪ 𝑐 = ∪ 𝐴)
5	4	pweqd 4579	. . . . . 6 ⊢ (𝑐 = 𝐴 → 𝒫 ∪ 𝑐 = 𝒫 ∪ 𝐴)
6		fveq2 6881	. . . . . . . . . . 11 ⊢ (𝑐 = 𝐴 → (mrCls‘𝑐) = (mrCls‘𝐴))
7		mrisval.1	. . . . . . . . . . 11 ⊢ 𝑁 = (mrCls‘𝐴)
8	6, 7	eqtr4di 2816	. . . . . . . . . 10 ⊢ (𝑐 = 𝐴 → (mrCls‘𝑐) = 𝑁)
9	8	fveq1d 6883	. . . . . . . . 9 ⊢ (𝑐 = 𝐴 → ((mrCls‘𝑐)‘(𝑠 ∖ {𝑥})) = (𝑁‘(𝑠 ∖ {𝑥})))
10	9	eleq2d 2849	. . . . . . . 8 ⊢ (𝑐 = 𝐴 → (𝑥 ∈ ((mrCls‘𝑐)‘(𝑠 ∖ {𝑥})) ↔ 𝑥 ∈ (𝑁‘(𝑠 ∖ {𝑥}))))
11	10	notbid 321	. . . . . . 7 ⊢ (𝑐 = 𝐴 → (¬ 𝑥 ∈ ((mrCls‘𝑐)‘(𝑠 ∖ {𝑥})) ↔ ¬ 𝑥 ∈ (𝑁‘(𝑠 ∖ {𝑥}))))
12	11	ralbidv 3188	. . . . . 6 ⊢ (𝑐 = 𝐴 → (∀𝑥 ∈ 𝑠 ¬ 𝑥 ∈ ((mrCls‘𝑐)‘(𝑠 ∖ {𝑥})) ↔ ∀𝑥 ∈ 𝑠 ¬ 𝑥 ∈ (𝑁‘(𝑠 ∖ {𝑥}))))
13	5, 12	rabeqbidv 3434	. . . . 5 ⊢ (𝑐 = 𝐴 → {𝑠 ∈ 𝒫 ∪ 𝑐 ∣ ∀𝑥 ∈ 𝑠 ¬ 𝑥 ∈ ((mrCls‘𝑐)‘(𝑠 ∖ {𝑥}))} = {𝑠 ∈ 𝒫 ∪ 𝐴 ∣ ∀𝑥 ∈ 𝑠 ¬ 𝑥 ∈ (𝑁‘(𝑠 ∖ {𝑥}))})
14		df-mri 17635	. . . . 5 ⊢ mrInd = (𝑐 ∈ ∪ ran Moore ↦ {𝑠 ∈ 𝒫 ∪ 𝑐 ∣ ∀𝑥 ∈ 𝑠 ¬ 𝑥 ∈ ((mrCls‘𝑐)‘(𝑠 ∖ {𝑥}))})
15		vuniex 7737	. . . . . . 7 ⊢ ∪ 𝑐 ∈ V
16	15	pwex 5351	. . . . . 6 ⊢ 𝒫 ∪ 𝑐 ∈ V
17	16	rabex 5309	. . . . 5 ⊢ {𝑠 ∈ 𝒫 ∪ 𝑐 ∣ ∀𝑥 ∈ 𝑠 ¬ 𝑥 ∈ ((mrCls‘𝑐)‘(𝑠 ∖ {𝑥}))} ∈ V
18	13, 14, 17	fvmpt3i 6995	. . . 4 ⊢ (𝐴 ∈ ∪ ran Moore → (mrInd‘𝐴) = {𝑠 ∈ 𝒫 ∪ 𝐴 ∣ ∀𝑥 ∈ 𝑠 ¬ 𝑥 ∈ (𝑁‘(𝑠 ∖ {𝑥}))})
19	3, 18	syl 18	. . 3 ⊢ (𝐴 ∈ (Moore‘𝑋) → (mrInd‘𝐴) = {𝑠 ∈ 𝒫 ∪ 𝐴 ∣ ∀𝑥 ∈ 𝑠 ¬ 𝑥 ∈ (𝑁‘(𝑠 ∖ {𝑥}))})
20	1, 19	eqtrid 2810	. 2 ⊢ (𝐴 ∈ (Moore‘𝑋) → 𝐼 = {𝑠 ∈ 𝒫 ∪ 𝐴 ∣ ∀𝑥 ∈ 𝑠 ¬ 𝑥 ∈ (𝑁‘(𝑠 ∖ {𝑥}))})
21		mreuni 17647	. . . 4 ⊢ (𝐴 ∈ (Moore‘𝑋) → ∪ 𝐴 = 𝑋)
22	21	pweqd 4579	. . 3 ⊢ (𝐴 ∈ (Moore‘𝑋) → 𝒫 ∪ 𝐴 = 𝒫 𝑋)
23	22	rabeqdv 3431	. 2 ⊢ (𝐴 ∈ (Moore‘𝑋) → {𝑠 ∈ 𝒫 ∪ 𝐴 ∣ ∀𝑥 ∈ 𝑠 ¬ 𝑥 ∈ (𝑁‘(𝑠 ∖ {𝑥}))} = {𝑠 ∈ 𝒫 𝑋 ∣ ∀𝑥 ∈ 𝑠 ¬ 𝑥 ∈ (𝑁‘(𝑠 ∖ {𝑥}))})
24	20, 23	eqtrd 2798	1 ⊢ (𝐴 ∈ (Moore‘𝑋) → 𝐼 = {𝑠 ∈ 𝒫 𝑋 ∣ ∀𝑥 ∈ 𝑠 ¬ 𝑥 ∈ (𝑁‘(𝑠 ∖ {𝑥}))})

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1570   ∈ wcel 2143  ∀wral 3079  {crab 3416   ∖ cdif 3902  𝒫 cpw 4562  {csn 4589  ∪ cuni 4872  ran crn 5662  ‘cfv 6536  Moorecmre 17629  mrClscmrc 17630  mrIndcmri 17631

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5257  ax-nul 5269  ax-pow 5336  ax-pr 5404  ax-un 7732

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-pw 4564  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-br 5110  df-opab 5174  df-mpt 5193  df-id 5556  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-iota 6492  df-fun 6538  df-fv 6544  df-mre 17633  df-mri 17635

This theorem is referenced by:  ismri  17682