Theorem mrisval 17596

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 6871	. . . . 5 ⊢ (Moore‘𝑋) ⊆ ∪ ran Moore
3	2	sseli 3917	. . . 4 ⊢ (𝐴 ∈ (Moore‘𝑋) → 𝐴 ∈ ∪ ran Moore)
4		unieq 4861	. . . . . . 7 ⊢ (𝑐 = 𝐴 → ∪ 𝑐 = ∪ 𝐴)
5	4	pweqd 4558	. . . . . 6 ⊢ (𝑐 = 𝐴 → 𝒫 ∪ 𝑐 = 𝒫 ∪ 𝐴)
6		fveq2 6840	. . . . . . . . . . 11 ⊢ (𝑐 = 𝐴 → (mrCls‘𝑐) = (mrCls‘𝐴))
7		mrisval.1	. . . . . . . . . . 11 ⊢ 𝑁 = (mrCls‘𝐴)
8	6, 7	eqtr4di 2789	. . . . . . . . . 10 ⊢ (𝑐 = 𝐴 → (mrCls‘𝑐) = 𝑁)
9	8	fveq1d 6842	. . . . . . . . 9 ⊢ (𝑐 = 𝐴 → ((mrCls‘𝑐)‘(𝑠 ∖ {𝑥})) = (𝑁‘(𝑠 ∖ {𝑥})))
10	9	eleq2d 2822	. . . . . . . 8 ⊢ (𝑐 = 𝐴 → (𝑥 ∈ ((mrCls‘𝑐)‘(𝑠 ∖ {𝑥})) ↔ 𝑥 ∈ (𝑁‘(𝑠 ∖ {𝑥}))))
11	10	notbid 318	. . . . . . 7 ⊢ (𝑐 = 𝐴 → (¬ 𝑥 ∈ ((mrCls‘𝑐)‘(𝑠 ∖ {𝑥})) ↔ ¬ 𝑥 ∈ (𝑁‘(𝑠 ∖ {𝑥}))))
12	11	ralbidv 3160	. . . . . 6 ⊢ (𝑐 = 𝐴 → (∀𝑥 ∈ 𝑠 ¬ 𝑥 ∈ ((mrCls‘𝑐)‘(𝑠 ∖ {𝑥})) ↔ ∀𝑥 ∈ 𝑠 ¬ 𝑥 ∈ (𝑁‘(𝑠 ∖ {𝑥}))))
13	5, 12	rabeqbidv 3407	. . . . 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 17550	. . . . 5 ⊢ mrInd = (𝑐 ∈ ∪ ran Moore ↦ {𝑠 ∈ 𝒫 ∪ 𝑐 ∣ ∀𝑥 ∈ 𝑠 ¬ 𝑥 ∈ ((mrCls‘𝑐)‘(𝑠 ∖ {𝑥}))})
15		vuniex 7693	. . . . . . 7 ⊢ ∪ 𝑐 ∈ V
16	15	pwex 5322	. . . . . 6 ⊢ 𝒫 ∪ 𝑐 ∈ V
17	16	rabex 5280	. . . . 5 ⊢ {𝑠 ∈ 𝒫 ∪ 𝑐 ∣ ∀𝑥 ∈ 𝑠 ¬ 𝑥 ∈ ((mrCls‘𝑐)‘(𝑠 ∖ {𝑥}))} ∈ V
18	13, 14, 17	fvmpt3i 6953	. . . 4 ⊢ (𝐴 ∈ ∪ ran Moore → (mrInd‘𝐴) = {𝑠 ∈ 𝒫 ∪ 𝐴 ∣ ∀𝑥 ∈ 𝑠 ¬ 𝑥 ∈ (𝑁‘(𝑠 ∖ {𝑥}))})
19	3, 18	syl 17	. . 3 ⊢ (𝐴 ∈ (Moore‘𝑋) → (mrInd‘𝐴) = {𝑠 ∈ 𝒫 ∪ 𝐴 ∣ ∀𝑥 ∈ 𝑠 ¬ 𝑥 ∈ (𝑁‘(𝑠 ∖ {𝑥}))})
20	1, 19	eqtrid 2783	. 2 ⊢ (𝐴 ∈ (Moore‘𝑋) → 𝐼 = {𝑠 ∈ 𝒫 ∪ 𝐴 ∣ ∀𝑥 ∈ 𝑠 ¬ 𝑥 ∈ (𝑁‘(𝑠 ∖ {𝑥}))})
21		mreuni 17562	. . . 4 ⊢ (𝐴 ∈ (Moore‘𝑋) → ∪ 𝐴 = 𝑋)
22	21	pweqd 4558	. . 3 ⊢ (𝐴 ∈ (Moore‘𝑋) → 𝒫 ∪ 𝐴 = 𝒫 𝑋)
23	22	rabeqdv 3404	. 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 2771	1 ⊢ (𝐴 ∈ (Moore‘𝑋) → 𝐼 = {𝑠 ∈ 𝒫 𝑋 ∣ ∀𝑥 ∈ 𝑠 ¬ 𝑥 ∈ (𝑁‘(𝑠 ∖ {𝑥}))})

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1542   ∈ wcel 2114  ∀wral 3051  {crab 3389   ∖ cdif 3886  𝒫 cpw 4541  {csn 4567  ∪ cuni 4850  ran crn 5632  ‘cfv 6498  Moorecmre 17544  mrClscmrc 17545  mrIndcmri 17546

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375  ax-un 7689

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3062  df-rab 3390  df-v 3431  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-br 5086  df-opab 5148  df-mpt 5167  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-iota 6454  df-fun 6500  df-fv 6506  df-mre 17548  df-mri 17550

This theorem is referenced by:  ismri  17597