Theorem mrcval 17653

Description: Evaluation of the Moore closure of a set. (Contributed by Stefan O'Rear, 31-Jan-2015.) (Proof shortened by Fan Zheng, 6-Jun-2016.)

Hypothesis

Ref	Expression
mrcfval.f	⊢ 𝐹 = (mrCls‘𝐶)

Assertion

Ref	Expression
mrcval	⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝑋) → (𝐹‘𝑈) = ∩ {𝑠 ∈ 𝐶 ∣ 𝑈 ⊆ 𝑠})

Distinct variable groups:   𝐹,𝑠   𝐶,𝑠   𝑋,𝑠   𝑈,𝑠

Proof of Theorem mrcval

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		mrcfval.f	. . . 4 ⊢ 𝐹 = (mrCls‘𝐶)
2	1	mrcfval 17651	. . 3 ⊢ (𝐶 ∈ (Moore‘𝑋) → 𝐹 = (𝑥 ∈ 𝒫 𝑋 ↦ ∩ {𝑠 ∈ 𝐶 ∣ 𝑥 ⊆ 𝑠}))
3	2	adantr 480	. 2 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝑋) → 𝐹 = (𝑥 ∈ 𝒫 𝑋 ↦ ∩ {𝑠 ∈ 𝐶 ∣ 𝑥 ⊆ 𝑠}))
4		sseq1 4009	. . . . 5 ⊢ (𝑥 = 𝑈 → (𝑥 ⊆ 𝑠 ↔ 𝑈 ⊆ 𝑠))
5	4	rabbidv 3444	. . . 4 ⊢ (𝑥 = 𝑈 → {𝑠 ∈ 𝐶 ∣ 𝑥 ⊆ 𝑠} = {𝑠 ∈ 𝐶 ∣ 𝑈 ⊆ 𝑠})
6	5	inteqd 4951	. . 3 ⊢ (𝑥 = 𝑈 → ∩ {𝑠 ∈ 𝐶 ∣ 𝑥 ⊆ 𝑠} = ∩ {𝑠 ∈ 𝐶 ∣ 𝑈 ⊆ 𝑠})
7	6	adantl 481	. 2 ⊢ (((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝑋) ∧ 𝑥 = 𝑈) → ∩ {𝑠 ∈ 𝐶 ∣ 𝑥 ⊆ 𝑠} = ∩ {𝑠 ∈ 𝐶 ∣ 𝑈 ⊆ 𝑠})
8		mre1cl 17637	. . . 4 ⊢ (𝐶 ∈ (Moore‘𝑋) → 𝑋 ∈ 𝐶)
9		elpw2g 5333	. . . 4 ⊢ (𝑋 ∈ 𝐶 → (𝑈 ∈ 𝒫 𝑋 ↔ 𝑈 ⊆ 𝑋))
10	8, 9	syl 17	. . 3 ⊢ (𝐶 ∈ (Moore‘𝑋) → (𝑈 ∈ 𝒫 𝑋 ↔ 𝑈 ⊆ 𝑋))
11	10	biimpar 477	. 2 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝑋) → 𝑈 ∈ 𝒫 𝑋)
12		sseq2 4010	. . . . 5 ⊢ (𝑠 = 𝑋 → (𝑈 ⊆ 𝑠 ↔ 𝑈 ⊆ 𝑋))
13	8	adantr 480	. . . . 5 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝑋) → 𝑋 ∈ 𝐶)
14		simpr 484	. . . . 5 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝑋) → 𝑈 ⊆ 𝑋)
15	12, 13, 14	elrabd 3694	. . . 4 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝑋) → 𝑋 ∈ {𝑠 ∈ 𝐶 ∣ 𝑈 ⊆ 𝑠})
16	15	ne0d 4342	. . 3 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝑋) → {𝑠 ∈ 𝐶 ∣ 𝑈 ⊆ 𝑠} ≠ ∅)
17		intex 5344	. . 3 ⊢ ({𝑠 ∈ 𝐶 ∣ 𝑈 ⊆ 𝑠} ≠ ∅ ↔ ∩ {𝑠 ∈ 𝐶 ∣ 𝑈 ⊆ 𝑠} ∈ V)
18	16, 17	sylib 218	. 2 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝑋) → ∩ {𝑠 ∈ 𝐶 ∣ 𝑈 ⊆ 𝑠} ∈ V)
19	3, 7, 11, 18	fvmptd 7023	1 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝑋) → (𝐹‘𝑈) = ∩ {𝑠 ∈ 𝐶 ∣ 𝑈 ⊆ 𝑠})

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2108   ≠ wne 2940  {crab 3436  Vcvv 3480   ⊆ wss 3951  ∅c0 4333  𝒫 cpw 4600  ∩ cint 4946   ↦ cmpt 5225  ‘cfv 6561  Moorecmre 17625  mrClscmrc 17626

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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-fv 6569  df-mre 17629  df-mrc 17630

This theorem is referenced by:  mrcid  17656  mrcss  17659  mrcssid  17660  cycsubg2  19228  aspval2  21918  mrelatlubALT  48884  mreclat  48886