Theorem mrcfval 17569

Description: Value of the function expression for the Moore closure. (Contributed by Stefan O'Rear, 31-Jan-2015.)

Hypothesis

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

Assertion

Ref	Expression
mrcfval	⊢ (𝐶 ∈ (Moore‘𝑋) → 𝐹 = (𝑥 ∈ 𝒫 𝑋 ↦ ∩ {𝑠 ∈ 𝐶 ∣ 𝑥 ⊆ 𝑠}))

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

Proof of Theorem mrcfval

Dummy variable 𝑐 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		mrcfval.f	. 2 ⊢ 𝐹 = (mrCls‘𝐶)
2		fvssunirn 6891	. . . . 5 ⊢ (Moore‘𝑋) ⊆ ∪ ran Moore
3	2	sseli 3942	. . . 4 ⊢ (𝐶 ∈ (Moore‘𝑋) → 𝐶 ∈ ∪ ran Moore)
4		unieq 4882	. . . . . . 7 ⊢ (𝑐 = 𝐶 → ∪ 𝑐 = ∪ 𝐶)
5	4	pweqd 4580	. . . . . 6 ⊢ (𝑐 = 𝐶 → 𝒫 ∪ 𝑐 = 𝒫 ∪ 𝐶)
6		rabeq 3420	. . . . . . 7 ⊢ (𝑐 = 𝐶 → {𝑠 ∈ 𝑐 ∣ 𝑥 ⊆ 𝑠} = {𝑠 ∈ 𝐶 ∣ 𝑥 ⊆ 𝑠})
7	6	inteqd 4915	. . . . . 6 ⊢ (𝑐 = 𝐶 → ∩ {𝑠 ∈ 𝑐 ∣ 𝑥 ⊆ 𝑠} = ∩ {𝑠 ∈ 𝐶 ∣ 𝑥 ⊆ 𝑠})
8	5, 7	mpteq12dv 5194	. . . . 5 ⊢ (𝑐 = 𝐶 → (𝑥 ∈ 𝒫 ∪ 𝑐 ↦ ∩ {𝑠 ∈ 𝑐 ∣ 𝑥 ⊆ 𝑠}) = (𝑥 ∈ 𝒫 ∪ 𝐶 ↦ ∩ {𝑠 ∈ 𝐶 ∣ 𝑥 ⊆ 𝑠}))
9		df-mrc 17548	. . . . 5 ⊢ mrCls = (𝑐 ∈ ∪ ran Moore ↦ (𝑥 ∈ 𝒫 ∪ 𝑐 ↦ ∩ {𝑠 ∈ 𝑐 ∣ 𝑥 ⊆ 𝑠}))
10		mreunirn 17562	. . . . . . . 8 ⊢ (𝑐 ∈ ∪ ran Moore ↔ 𝑐 ∈ (Moore‘∪ 𝑐))
11		mrcflem 17567	. . . . . . . 8 ⊢ (𝑐 ∈ (Moore‘∪ 𝑐) → (𝑥 ∈ 𝒫 ∪ 𝑐 ↦ ∩ {𝑠 ∈ 𝑐 ∣ 𝑥 ⊆ 𝑠}):𝒫 ∪ 𝑐⟶𝑐)
12	10, 11	sylbi 217	. . . . . . 7 ⊢ (𝑐 ∈ ∪ ran Moore → (𝑥 ∈ 𝒫 ∪ 𝑐 ↦ ∩ {𝑠 ∈ 𝑐 ∣ 𝑥 ⊆ 𝑠}):𝒫 ∪ 𝑐⟶𝑐)
13		fssxp 6715	. . . . . . 7 ⊢ ((𝑥 ∈ 𝒫 ∪ 𝑐 ↦ ∩ {𝑠 ∈ 𝑐 ∣ 𝑥 ⊆ 𝑠}):𝒫 ∪ 𝑐⟶𝑐 → (𝑥 ∈ 𝒫 ∪ 𝑐 ↦ ∩ {𝑠 ∈ 𝑐 ∣ 𝑥 ⊆ 𝑠}) ⊆ (𝒫 ∪ 𝑐 × 𝑐))
14	12, 13	syl 17	. . . . . 6 ⊢ (𝑐 ∈ ∪ ran Moore → (𝑥 ∈ 𝒫 ∪ 𝑐 ↦ ∩ {𝑠 ∈ 𝑐 ∣ 𝑥 ⊆ 𝑠}) ⊆ (𝒫 ∪ 𝑐 × 𝑐))
15		vuniex 7715	. . . . . . . 8 ⊢ ∪ 𝑐 ∈ V
16	15	pwex 5335	. . . . . . 7 ⊢ 𝒫 ∪ 𝑐 ∈ V
17		vex 3451	. . . . . . 7 ⊢ 𝑐 ∈ V
18	16, 17	xpex 7729	. . . . . 6 ⊢ (𝒫 ∪ 𝑐 × 𝑐) ∈ V
19		ssexg 5278	. . . . . 6 ⊢ (((𝑥 ∈ 𝒫 ∪ 𝑐 ↦ ∩ {𝑠 ∈ 𝑐 ∣ 𝑥 ⊆ 𝑠}) ⊆ (𝒫 ∪ 𝑐 × 𝑐) ∧ (𝒫 ∪ 𝑐 × 𝑐) ∈ V) → (𝑥 ∈ 𝒫 ∪ 𝑐 ↦ ∩ {𝑠 ∈ 𝑐 ∣ 𝑥 ⊆ 𝑠}) ∈ V)
20	14, 18, 19	sylancl 586	. . . . 5 ⊢ (𝑐 ∈ ∪ ran Moore → (𝑥 ∈ 𝒫 ∪ 𝑐 ↦ ∩ {𝑠 ∈ 𝑐 ∣ 𝑥 ⊆ 𝑠}) ∈ V)
21	8, 9, 20	fvmpt3 6972	. . . 4 ⊢ (𝐶 ∈ ∪ ran Moore → (mrCls‘𝐶) = (𝑥 ∈ 𝒫 ∪ 𝐶 ↦ ∩ {𝑠 ∈ 𝐶 ∣ 𝑥 ⊆ 𝑠}))
22	3, 21	syl 17	. . 3 ⊢ (𝐶 ∈ (Moore‘𝑋) → (mrCls‘𝐶) = (𝑥 ∈ 𝒫 ∪ 𝐶 ↦ ∩ {𝑠 ∈ 𝐶 ∣ 𝑥 ⊆ 𝑠}))
23		mreuni 17561	. . . . 5 ⊢ (𝐶 ∈ (Moore‘𝑋) → ∪ 𝐶 = 𝑋)
24	23	pweqd 4580	. . . 4 ⊢ (𝐶 ∈ (Moore‘𝑋) → 𝒫 ∪ 𝐶 = 𝒫 𝑋)
25	24	mpteq1d 5197	. . 3 ⊢ (𝐶 ∈ (Moore‘𝑋) → (𝑥 ∈ 𝒫 ∪ 𝐶 ↦ ∩ {𝑠 ∈ 𝐶 ∣ 𝑥 ⊆ 𝑠}) = (𝑥 ∈ 𝒫 𝑋 ↦ ∩ {𝑠 ∈ 𝐶 ∣ 𝑥 ⊆ 𝑠}))
26	22, 25	eqtrd 2764	. 2 ⊢ (𝐶 ∈ (Moore‘𝑋) → (mrCls‘𝐶) = (𝑥 ∈ 𝒫 𝑋 ↦ ∩ {𝑠 ∈ 𝐶 ∣ 𝑥 ⊆ 𝑠}))
27	1, 26	eqtrid 2776	1 ⊢ (𝐶 ∈ (Moore‘𝑋) → 𝐹 = (𝑥 ∈ 𝒫 𝑋 ↦ ∩ {𝑠 ∈ 𝐶 ∣ 𝑥 ⊆ 𝑠}))

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2109  {crab 3405  Vcvv 3447   ⊆ wss 3914  𝒫 cpw 4563  ∪ cuni 4871  ∩ cint 4910   ↦ cmpt 5188   × cxp 5636  ran crn 5639  ⟶wf 6507  ‘cfv 6511  Moorecmre 17543  mrClscmrc 17544

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 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387  ax-un 7711

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 3406  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-int 4911  df-br 5108  df-opab 5170  df-mpt 5189  df-id 5533  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-fv 6519  df-mre 17547  df-mrc 17548

This theorem is referenced by:  mrcf  17570  mrcval  17571  acsficl2d  18511  mrclsp  20895  mrccls  22966