Theorem mrcfval 17689

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 6919	. . . . 5 ⊢ (Moore‘𝑋) ⊆ ∪ ran Moore
3	2	sseli 3936	. . . 4 ⊢ (𝐶 ∈ (Moore‘𝑋) → 𝐶 ∈ ∪ ran Moore)
4		unieq 4888	. . . . . . 7 ⊢ (𝑐 = 𝐶 → ∪ 𝑐 = ∪ 𝐶)
5	4	pweqd 4584	. . . . . 6 ⊢ (𝑐 = 𝐶 → 𝒫 ∪ 𝑐 = 𝒫 ∪ 𝐶)
6		rabeq 3433	. . . . . . 7 ⊢ (𝑐 = 𝐶 → {𝑠 ∈ 𝑐 ∣ 𝑥 ⊆ 𝑠} = {𝑠 ∈ 𝐶 ∣ 𝑥 ⊆ 𝑠})
7	6	inteqd 4922	. . . . . 6 ⊢ (𝑐 = 𝐶 → ∩ {𝑠 ∈ 𝑐 ∣ 𝑥 ⊆ 𝑠} = ∩ {𝑠 ∈ 𝐶 ∣ 𝑥 ⊆ 𝑠})
8	5, 7	mpteq12dv 5203	. . . . 5 ⊢ (𝑐 = 𝐶 → (𝑥 ∈ 𝒫 ∪ 𝑐 ↦ ∩ {𝑠 ∈ 𝑐 ∣ 𝑥 ⊆ 𝑠}) = (𝑥 ∈ 𝒫 ∪ 𝐶 ↦ ∩ {𝑠 ∈ 𝐶 ∣ 𝑥 ⊆ 𝑠}))
9		df-mrc 17664	. . . . 5 ⊢ mrCls = (𝑐 ∈ ∪ ran Moore ↦ (𝑥 ∈ 𝒫 ∪ 𝑐 ↦ ∩ {𝑠 ∈ 𝑐 ∣ 𝑥 ⊆ 𝑠}))
10		mreunirn 17678	. . . . . . . 8 ⊢ (𝑐 ∈ ∪ ran Moore ↔ 𝑐 ∈ (Moore‘∪ 𝑐))
11		mrcflem 17687	. . . . . . . 8 ⊢ (𝑐 ∈ (Moore‘∪ 𝑐) → (𝑥 ∈ 𝒫 ∪ 𝑐 ↦ ∩ {𝑠 ∈ 𝑐 ∣ 𝑥 ⊆ 𝑠}):𝒫 ∪ 𝑐⟶𝑐)
12	10, 11	sylbi 220	. . . . . . 7 ⊢ (𝑐 ∈ ∪ ran Moore → (𝑥 ∈ 𝒫 ∪ 𝑐 ↦ ∩ {𝑠 ∈ 𝑐 ∣ 𝑥 ⊆ 𝑠}):𝒫 ∪ 𝑐⟶𝑐)
13		fssxp 6740	. . . . . . 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 18	. . . . . 6 ⊢ (𝑐 ∈ ∪ ran Moore → (𝑥 ∈ 𝒫 ∪ 𝑐 ↦ ∩ {𝑠 ∈ 𝑐 ∣ 𝑥 ⊆ 𝑠}) ⊆ (𝒫 ∪ 𝑐 × 𝑐))
15		vuniex 7750	. . . . . . . 8 ⊢ ∪ 𝑐 ∈ V
16	15	pwex 5356	. . . . . . 7 ⊢ 𝒫 ∪ 𝑐 ∈ V
17		vex 3462	. . . . . . 7 ⊢ 𝑐 ∈ V
18	16, 17	xpex 7761	. . . . . 6 ⊢ (𝒫 ∪ 𝑐 × 𝑐) ∈ V
19		ssexg 5295	. . . . . 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 598	. . . . 5 ⊢ (𝑐 ∈ ∪ ran Moore → (𝑥 ∈ 𝒫 ∪ 𝑐 ↦ ∩ {𝑠 ∈ 𝑐 ∣ 𝑥 ⊆ 𝑠}) ∈ V)
21	8, 9, 20	fvmpt3 7001	. . . 4 ⊢ (𝐶 ∈ ∪ ran Moore → (mrCls‘𝐶) = (𝑥 ∈ 𝒫 ∪ 𝐶 ↦ ∩ {𝑠 ∈ 𝐶 ∣ 𝑥 ⊆ 𝑠}))
22	3, 21	syl 18	. . 3 ⊢ (𝐶 ∈ (Moore‘𝑋) → (mrCls‘𝐶) = (𝑥 ∈ 𝒫 ∪ 𝐶 ↦ ∩ {𝑠 ∈ 𝐶 ∣ 𝑥 ⊆ 𝑠}))
23		mreuni 17677	. . . . 5 ⊢ (𝐶 ∈ (Moore‘𝑋) → ∪ 𝐶 = 𝑋)
24	23	pweqd 4584	. . . 4 ⊢ (𝐶 ∈ (Moore‘𝑋) → 𝒫 ∪ 𝐶 = 𝒫 𝑋)
25	24	mpteq1d 5206	. . 3 ⊢ (𝐶 ∈ (Moore‘𝑋) → (𝑥 ∈ 𝒫 ∪ 𝐶 ↦ ∩ {𝑠 ∈ 𝐶 ∣ 𝑥 ⊆ 𝑠}) = (𝑥 ∈ 𝒫 𝑋 ↦ ∩ {𝑠 ∈ 𝐶 ∣ 𝑥 ⊆ 𝑠}))
26	22, 25	eqtrd 2801	. 2 ⊢ (𝐶 ∈ (Moore‘𝑋) → (mrCls‘𝐶) = (𝑥 ∈ 𝒫 𝑋 ↦ ∩ {𝑠 ∈ 𝐶 ∣ 𝑥 ⊆ 𝑠}))
27	1, 26	eqtrid 2813	1 ⊢ (𝐶 ∈ (Moore‘𝑋) → 𝐹 = (𝑥 ∈ 𝒫 𝑋 ↦ ∩ {𝑠 ∈ 𝐶 ∣ 𝑥 ⊆ 𝑠}))

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2146  {crab 3419  Vcvv 3458   ⊆ wss 3908  𝒫 cpw 4567  ∪ cuni 4877  ∩ cint 4917   ↦ cmpt 5197   × cxp 5664  ran crn 5667  ⟶wf 6539  ‘cfv 6543  Moorecmre 17659  mrClscmrc 17660

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2738  ax-sep 5262  ax-nul 5274  ax-pow 5341  ax-pr 5409  ax-un 7745

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2570  df-eu 2600  df-clab 2745  df-cleq 2758  df-clel 2841  df-nfc 2915  df-ne 2962  df-ral 3083  df-rex 3093  df-rab 3420  df-v 3460  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4290  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4878  df-int 4918  df-br 5115  df-opab 5179  df-mpt 5198  df-id 5561  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-iota 6499  df-fun 6545  df-fn 6546  df-f 6547  df-fv 6551  df-mre 17663  df-mrc 17664

This theorem is used by:  mrcf  17690  mrcval  17691  acsficl2d  18633  mrclsp  21147  mrccls  23273