Theorem mrcfval 17552

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 6925	. . . . 5 ⊢ (Moore‘𝑋) ⊆ ∪ ran Moore
3	2	sseli 3979	. . . 4 ⊢ (𝐶 ∈ (Moore‘𝑋) → 𝐶 ∈ ∪ ran Moore)
4		unieq 4920	. . . . . . 7 ⊢ (𝑐 = 𝐶 → ∪ 𝑐 = ∪ 𝐶)
5	4	pweqd 4620	. . . . . 6 ⊢ (𝑐 = 𝐶 → 𝒫 ∪ 𝑐 = 𝒫 ∪ 𝐶)
6		rabeq 3447	. . . . . . 7 ⊢ (𝑐 = 𝐶 → {𝑠 ∈ 𝑐 ∣ 𝑥 ⊆ 𝑠} = {𝑠 ∈ 𝐶 ∣ 𝑥 ⊆ 𝑠})
7	6	inteqd 4956	. . . . . 6 ⊢ (𝑐 = 𝐶 → ∩ {𝑠 ∈ 𝑐 ∣ 𝑥 ⊆ 𝑠} = ∩ {𝑠 ∈ 𝐶 ∣ 𝑥 ⊆ 𝑠})
8	5, 7	mpteq12dv 5240	. . . . 5 ⊢ (𝑐 = 𝐶 → (𝑥 ∈ 𝒫 ∪ 𝑐 ↦ ∩ {𝑠 ∈ 𝑐 ∣ 𝑥 ⊆ 𝑠}) = (𝑥 ∈ 𝒫 ∪ 𝐶 ↦ ∩ {𝑠 ∈ 𝐶 ∣ 𝑥 ⊆ 𝑠}))
9		df-mrc 17531	. . . . 5 ⊢ mrCls = (𝑐 ∈ ∪ ran Moore ↦ (𝑥 ∈ 𝒫 ∪ 𝑐 ↦ ∩ {𝑠 ∈ 𝑐 ∣ 𝑥 ⊆ 𝑠}))
10		mreunirn 17545	. . . . . . . 8 ⊢ (𝑐 ∈ ∪ ran Moore ↔ 𝑐 ∈ (Moore‘∪ 𝑐))
11		mrcflem 17550	. . . . . . . 8 ⊢ (𝑐 ∈ (Moore‘∪ 𝑐) → (𝑥 ∈ 𝒫 ∪ 𝑐 ↦ ∩ {𝑠 ∈ 𝑐 ∣ 𝑥 ⊆ 𝑠}):𝒫 ∪ 𝑐⟶𝑐)
12	10, 11	sylbi 216	. . . . . . 7 ⊢ (𝑐 ∈ ∪ ran Moore → (𝑥 ∈ 𝒫 ∪ 𝑐 ↦ ∩ {𝑠 ∈ 𝑐 ∣ 𝑥 ⊆ 𝑠}):𝒫 ∪ 𝑐⟶𝑐)
13		fssxp 6746	. . . . . . 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 7729	. . . . . . . 8 ⊢ ∪ 𝑐 ∈ V
16	15	pwex 5379	. . . . . . 7 ⊢ 𝒫 ∪ 𝑐 ∈ V
17		vex 3479	. . . . . . 7 ⊢ 𝑐 ∈ V
18	16, 17	xpex 7740	. . . . . 6 ⊢ (𝒫 ∪ 𝑐 × 𝑐) ∈ V
19		ssexg 5324	. . . . . 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 587	. . . . 5 ⊢ (𝑐 ∈ ∪ ran Moore → (𝑥 ∈ 𝒫 ∪ 𝑐 ↦ ∩ {𝑠 ∈ 𝑐 ∣ 𝑥 ⊆ 𝑠}) ∈ V)
21	8, 9, 20	fvmpt3 7003	. . . 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 17544	. . . . 5 ⊢ (𝐶 ∈ (Moore‘𝑋) → ∪ 𝐶 = 𝑋)
24	23	pweqd 4620	. . . 4 ⊢ (𝐶 ∈ (Moore‘𝑋) → 𝒫 ∪ 𝐶 = 𝒫 𝑋)
25	24	mpteq1d 5244	. . 3 ⊢ (𝐶 ∈ (Moore‘𝑋) → (𝑥 ∈ 𝒫 ∪ 𝐶 ↦ ∩ {𝑠 ∈ 𝐶 ∣ 𝑥 ⊆ 𝑠}) = (𝑥 ∈ 𝒫 𝑋 ↦ ∩ {𝑠 ∈ 𝐶 ∣ 𝑥 ⊆ 𝑠}))
26	22, 25	eqtrd 2773	. 2 ⊢ (𝐶 ∈ (Moore‘𝑋) → (mrCls‘𝐶) = (𝑥 ∈ 𝒫 𝑋 ↦ ∩ {𝑠 ∈ 𝐶 ∣ 𝑥 ⊆ 𝑠}))
27	1, 26	eqtrid 2785	1 ⊢ (𝐶 ∈ (Moore‘𝑋) → 𝐹 = (𝑥 ∈ 𝒫 𝑋 ↦ ∩ {𝑠 ∈ 𝐶 ∣ 𝑥 ⊆ 𝑠}))

Syntax hints:   → wi 4   = wceq 1542   ∈ wcel 2107  {crab 3433  Vcvv 3475   ⊆ wss 3949  𝒫 cpw 4603  ∪ cuni 4909  ∩ cint 4951   ↦ cmpt 5232   × cxp 5675  ran crn 5678  ⟶wf 6540  ‘cfv 6544  Moorecmre 17526  mrClscmrc 17527

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-int 4952  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-fv 6552  df-mre 17530  df-mrc 17531

This theorem is referenced by:  mrcf  17553  mrcval  17554  acsficl2d  18505  mrclsp  20600  mrccls  22583