Theorem mrisval 17718

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 6909	. . . . 5 ⊢ (Moore‘𝑋) ⊆ ∪ ran Moore
3	2	sseli 3927	. . . 4 ⊢ (𝐴 ∈ (Moore‘𝑋) → 𝐴 ∈ ∪ ran Moore)
4		unieq 4878	. . . . . . 7 ⊢ (𝑐 = 𝐴 → ∪ 𝑐 = ∪ 𝐴)
5	4	pweqd 4574	. . . . . 6 ⊢ (𝑐 = 𝐴 → 𝒫 ∪ 𝑐 = 𝒫 ∪ 𝐴)
6		fveq2 6878	. . . . . . . . . . 11 ⊢ (𝑐 = 𝐴 → (mrCls‘𝑐) = (mrCls‘𝐴))
7		mrisval.1	. . . . . . . . . . 11 ⊢ 𝑁 = (mrCls‘𝐴)
8	6, 7	eqtr4di 2813	. . . . . . . . . 10 ⊢ (𝑐 = 𝐴 → (mrCls‘𝑐) = 𝑁)
9	8	fveq1d 6880	. . . . . . . . 9 ⊢ (𝑐 = 𝐴 → ((mrCls‘𝑐)‘(𝑠 ∖ {𝑥})) = (𝑁‘(𝑠 ∖ {𝑥})))
10	9	eleq2d 2846	. . . . . . . 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 3185	. . . . . 6 ⊢ (𝑐 = 𝐴 → (∀𝑥 ∈ 𝑠 ¬ 𝑥 ∈ ((mrCls‘𝑐)‘(𝑠 ∖ {𝑥})) ↔ ∀𝑥 ∈ 𝑠 ¬ 𝑥 ∈ (𝑁‘(𝑠 ∖ {𝑥}))))
13	5, 12	rabeqbidv 3429	. . . . 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 17672	. . . . 5 ⊢ mrInd = (𝑐 ∈ ∪ ran Moore ↦ {𝑠 ∈ 𝒫 ∪ 𝑐 ∣ ∀𝑥 ∈ 𝑠 ¬ 𝑥 ∈ ((mrCls‘𝑐)‘(𝑠 ∖ {𝑥}))})
15		vuniex 7741	. . . . . . 7 ⊢ ∪ 𝑐 ∈ V
16	15	pwex 5345	. . . . . 6 ⊢ 𝒫 ∪ 𝑐 ∈ V
17	16	rabex 5303	. . . . 5 ⊢ {𝑠 ∈ 𝒫 ∪ 𝑐 ∣ ∀𝑥 ∈ 𝑠 ¬ 𝑥 ∈ ((mrCls‘𝑐)‘(𝑠 ∖ {𝑥}))} ∈ V
18	13, 14, 17	fvmpt3i 6992	. . . 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 2807	. 2 ⊢ (𝐴 ∈ (Moore‘𝑋) → 𝐼 = {𝑠 ∈ 𝒫 ∪ 𝐴 ∣ ∀𝑥 ∈ 𝑠 ¬ 𝑥 ∈ (𝑁‘(𝑠 ∖ {𝑥}))})
21		mreuni 17684	. . . 4 ⊢ (𝐴 ∈ (Moore‘𝑋) → ∪ 𝐴 = 𝑋)
22	21	pweqd 4574	. . 3 ⊢ (𝐴 ∈ (Moore‘𝑋) → 𝒫 ∪ 𝐴 = 𝒫 𝑋)
23	22	rabeqdv 3427	. 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 2795	1 ⊢ (𝐴 ∈ (Moore‘𝑋) → 𝐼 = {𝑠 ∈ 𝒫 𝑋 ∣ ∀𝑥 ∈ 𝑠 ¬ 𝑥 ∈ (𝑁‘(𝑠 ∖ {𝑥}))})

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   = wceq 1570   ∈ wcel 2145  ∀wral 3076  {crab 3412   ∖ cdif 3896  𝒫 cpw 4557  {csn 4584  ∪ cuni 4867  ran crn 5656  ‘cfv 6533  Moorecmre 17666  mrClscmrc 17667  mrIndcmri 17668

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-sep 5251  ax-nul 5263  ax-pow 5330  ax-pr 5398  ax-un 7736

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-iota 6489  df-fun 6535  df-fv 6541  df-mre 17670  df-mri 17672

This theorem is used by:  ismri  17719