Theorem mrisval 17554

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 6857	. . . . 5 ⊢ (Moore‘𝑋) ⊆ ∪ ran Moore
3	2	sseli 3933	. . . 4 ⊢ (𝐴 ∈ (Moore‘𝑋) → 𝐴 ∈ ∪ ran Moore)
4		unieq 4872	. . . . . . 7 ⊢ (𝑐 = 𝐴 → ∪ 𝑐 = ∪ 𝐴)
5	4	pweqd 4570	. . . . . 6 ⊢ (𝑐 = 𝐴 → 𝒫 ∪ 𝑐 = 𝒫 ∪ 𝐴)
6		fveq2 6826	. . . . . . . . . . 11 ⊢ (𝑐 = 𝐴 → (mrCls‘𝑐) = (mrCls‘𝐴))
7		mrisval.1	. . . . . . . . . . 11 ⊢ 𝑁 = (mrCls‘𝐴)
8	6, 7	eqtr4di 2782	. . . . . . . . . 10 ⊢ (𝑐 = 𝐴 → (mrCls‘𝑐) = 𝑁)
9	8	fveq1d 6828	. . . . . . . . 9 ⊢ (𝑐 = 𝐴 → ((mrCls‘𝑐)‘(𝑠 ∖ {𝑥})) = (𝑁‘(𝑠 ∖ {𝑥})))
10	9	eleq2d 2814	. . . . . . . 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 3152	. . . . . 6 ⊢ (𝑐 = 𝐴 → (∀𝑥 ∈ 𝑠 ¬ 𝑥 ∈ ((mrCls‘𝑐)‘(𝑠 ∖ {𝑥})) ↔ ∀𝑥 ∈ 𝑠 ¬ 𝑥 ∈ (𝑁‘(𝑠 ∖ {𝑥}))))
13	5, 12	rabeqbidv 3415	. . . . 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 17508	. . . . 5 ⊢ mrInd = (𝑐 ∈ ∪ ran Moore ↦ {𝑠 ∈ 𝒫 ∪ 𝑐 ∣ ∀𝑥 ∈ 𝑠 ¬ 𝑥 ∈ ((mrCls‘𝑐)‘(𝑠 ∖ {𝑥}))})
15		vuniex 7679	. . . . . . 7 ⊢ ∪ 𝑐 ∈ V
16	15	pwex 5322	. . . . . 6 ⊢ 𝒫 ∪ 𝑐 ∈ V
17	16	rabex 5281	. . . . 5 ⊢ {𝑠 ∈ 𝒫 ∪ 𝑐 ∣ ∀𝑥 ∈ 𝑠 ¬ 𝑥 ∈ ((mrCls‘𝑐)‘(𝑠 ∖ {𝑥}))} ∈ V
18	13, 14, 17	fvmpt3i 6939	. . . 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 2776	. 2 ⊢ (𝐴 ∈ (Moore‘𝑋) → 𝐼 = {𝑠 ∈ 𝒫 ∪ 𝐴 ∣ ∀𝑥 ∈ 𝑠 ¬ 𝑥 ∈ (𝑁‘(𝑠 ∖ {𝑥}))})
21		mreuni 17520	. . . 4 ⊢ (𝐴 ∈ (Moore‘𝑋) → ∪ 𝐴 = 𝑋)
22	21	pweqd 4570	. . 3 ⊢ (𝐴 ∈ (Moore‘𝑋) → 𝒫 ∪ 𝐴 = 𝒫 𝑋)
23	22	rabeqdv 3412	. 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 2764	1 ⊢ (𝐴 ∈ (Moore‘𝑋) → 𝐼 = {𝑠 ∈ 𝒫 𝑋 ∣ ∀𝑥 ∈ 𝑠 ¬ 𝑥 ∈ (𝑁‘(𝑠 ∖ {𝑥}))})

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1540   ∈ wcel 2109  ∀wral 3044  {crab 3396   ∖ cdif 3902  𝒫 cpw 4553  {csn 4579  ∪ cuni 4861  ran crn 5624  ‘cfv 6486  Moorecmre 17502  mrClscmrc 17503  mrIndcmri 17504

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5238  ax-nul 5248  ax-pow 5307  ax-pr 5374  ax-un 7675

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3397  df-v 3440  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4479  df-pw 4555  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4862  df-br 5096  df-opab 5158  df-mpt 5177  df-id 5518  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-iota 6442  df-fun 6488  df-fv 6494  df-mre 17506  df-mri 17508

This theorem is referenced by:  ismri  17555