Theorem mrcval 17578

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 17576	. . 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 3975	. . . . 5 ⊢ (𝑥 = 𝑈 → (𝑥 ⊆ 𝑠 ↔ 𝑈 ⊆ 𝑠))
5	4	rabbidv 3416	. . . 4 ⊢ (𝑥 = 𝑈 → {𝑠 ∈ 𝐶 ∣ 𝑥 ⊆ 𝑠} = {𝑠 ∈ 𝐶 ∣ 𝑈 ⊆ 𝑠})
6	5	inteqd 4918	. . 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 17562	. . . 4 ⊢ (𝐶 ∈ (Moore‘𝑋) → 𝑋 ∈ 𝐶)
9		elpw2g 5291	. . . 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 3976	. . . . 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 3664	. . . 4 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝑋) → 𝑋 ∈ {𝑠 ∈ 𝐶 ∣ 𝑈 ⊆ 𝑠})
16	15	ne0d 4308	. . 3 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝑋) → {𝑠 ∈ 𝐶 ∣ 𝑈 ⊆ 𝑠} ≠ ∅)
17		intex 5302	. . 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 6978	1 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝑋) → (𝐹‘𝑈) = ∩ {𝑠 ∈ 𝐶 ∣ 𝑈 ⊆ 𝑠})

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109   ≠ wne 2926  {crab 3408  Vcvv 3450   ⊆ wss 3917  ∅c0 4299  𝒫 cpw 4566  ∩ cint 4913   ↦ cmpt 5191  ‘cfv 6514  Moorecmre 17550  mrClscmrc 17551

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 2702  ax-sep 5254  ax-nul 5264  ax-pow 5323  ax-pr 5390  ax-un 7714

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 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-int 4914  df-br 5111  df-opab 5173  df-mpt 5192  df-id 5536  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-fv 6522  df-mre 17554  df-mrc 17555

This theorem is referenced by:  mrcid  17581  mrcss  17584  mrcssid  17585  cycsubg2  19149  aspval2  21814  mrelatlubALT  48987  mreclat  48989