Theorem mrisval 17682

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 6910	. . . . 5 ⊢ (Moore‘𝑋) ⊆ ∪ ran Moore
3	2	sseli 3941	. . . 4 ⊢ (𝐴 ∈ (Moore‘𝑋) → 𝐴 ∈ ∪ ran Moore)
4		unieq 4884	. . . . . . 7 ⊢ (𝑐 = 𝐴 → ∪ 𝑐 = ∪ 𝐴)
5	4	pweqd 4581	. . . . . 6 ⊢ (𝑐 = 𝐴 → 𝒫 ∪ 𝑐 = 𝒫 ∪ 𝐴)
6		fveq2 6879	. . . . . . . . . . 11 ⊢ (𝑐 = 𝐴 → (mrCls‘𝑐) = (mrCls‘𝐴))
7		mrisval.1	. . . . . . . . . . 11 ⊢ 𝑁 = (mrCls‘𝐴)
8	6, 7	eqtr4di 2822	. . . . . . . . . 10 ⊢ (𝑐 = 𝐴 → (mrCls‘𝑐) = 𝑁)
9	8	fveq1d 6881	. . . . . . . . 9 ⊢ (𝑐 = 𝐴 → ((mrCls‘𝑐)‘(𝑠 ∖ {𝑥})) = (𝑁‘(𝑠 ∖ {𝑥})))
10	9	eleq2d 2855	. . . . . . . 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 3194	. . . . . 6 ⊢ (𝑐 = 𝐴 → (∀𝑥 ∈ 𝑠 ¬ 𝑥 ∈ ((mrCls‘𝑐)‘(𝑠 ∖ {𝑥})) ↔ ∀𝑥 ∈ 𝑠 ¬ 𝑥 ∈ (𝑁‘(𝑠 ∖ {𝑥}))))
13	5, 12	rabeqbidv 3441	. . . . 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 17636	. . . . 5 ⊢ mrInd = (𝑐 ∈ ∪ ran Moore ↦ {𝑠 ∈ 𝒫 ∪ 𝑐 ∣ ∀𝑥 ∈ 𝑠 ¬ 𝑥 ∈ ((mrCls‘𝑐)‘(𝑠 ∖ {𝑥}))})
15		vuniex 7734	. . . . . . 7 ⊢ ∪ 𝑐 ∈ V
16	15	pwex 5349	. . . . . 6 ⊢ 𝒫 ∪ 𝑐 ∈ V
17	16	rabex 5307	. . . . 5 ⊢ {𝑠 ∈ 𝒫 ∪ 𝑐 ∣ ∀𝑥 ∈ 𝑠 ¬ 𝑥 ∈ ((mrCls‘𝑐)‘(𝑠 ∖ {𝑥}))} ∈ V
18	13, 14, 17	fvmpt3i 6993	. . . 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 2816	. 2 ⊢ (𝐴 ∈ (Moore‘𝑋) → 𝐼 = {𝑠 ∈ 𝒫 ∪ 𝐴 ∣ ∀𝑥 ∈ 𝑠 ¬ 𝑥 ∈ (𝑁‘(𝑠 ∖ {𝑥}))})
21		mreuni 17648	. . . 4 ⊢ (𝐴 ∈ (Moore‘𝑋) → ∪ 𝐴 = 𝑋)
22	21	pweqd 4581	. . 3 ⊢ (𝐴 ∈ (Moore‘𝑋) → 𝒫 ∪ 𝐴 = 𝒫 𝑋)
23	22	rabeqdv 3438	. 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 2804	1 ⊢ (𝐴 ∈ (Moore‘𝑋) → 𝐼 = {𝑠 ∈ 𝒫 𝑋 ∣ ∀𝑥 ∈ 𝑠 ¬ 𝑥 ∈ (𝑁‘(𝑠 ∖ {𝑥}))})

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1567   ∈ wcel 2149  ∀wral 3085  {crab 3423   ∖ cdif 3910  𝒫 cpw 4564  {csn 4591  ∪ cuni 4873  ran crn 5660  ‘cfv 6534  Moorecmre 17630  mrClscmrc 17631  mrIndcmri 17632

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-sep 5258  ax-nul 5268  ax-pow 5334  ax-pr 5402  ax-un 7730

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-rab 3424  df-v 3465  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4874  df-br 5111  df-opab 5175  df-mpt 5194  df-id 5554  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-iota 6490  df-fun 6536  df-fv 6542  df-mre 17634  df-mri 17636

This theorem is referenced by:  ismri  17683