Theorem mrcfval 17516

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 6859	. . . . 5 ⊢ (Moore‘𝑋) ⊆ ∪ ran Moore
3	2	sseli 3926	. . . 4 ⊢ (𝐶 ∈ (Moore‘𝑋) → 𝐶 ∈ ∪ ran Moore)
4		unieq 4869	. . . . . . 7 ⊢ (𝑐 = 𝐶 → ∪ 𝑐 = ∪ 𝐶)
5	4	pweqd 4566	. . . . . 6 ⊢ (𝑐 = 𝐶 → 𝒫 ∪ 𝑐 = 𝒫 ∪ 𝐶)
6		rabeq 3410	. . . . . . 7 ⊢ (𝑐 = 𝐶 → {𝑠 ∈ 𝑐 ∣ 𝑥 ⊆ 𝑠} = {𝑠 ∈ 𝐶 ∣ 𝑥 ⊆ 𝑠})
7	6	inteqd 4902	. . . . . 6 ⊢ (𝑐 = 𝐶 → ∩ {𝑠 ∈ 𝑐 ∣ 𝑥 ⊆ 𝑠} = ∩ {𝑠 ∈ 𝐶 ∣ 𝑥 ⊆ 𝑠})
8	5, 7	mpteq12dv 5180	. . . . 5 ⊢ (𝑐 = 𝐶 → (𝑥 ∈ 𝒫 ∪ 𝑐 ↦ ∩ {𝑠 ∈ 𝑐 ∣ 𝑥 ⊆ 𝑠}) = (𝑥 ∈ 𝒫 ∪ 𝐶 ↦ ∩ {𝑠 ∈ 𝐶 ∣ 𝑥 ⊆ 𝑠}))
9		df-mrc 17491	. . . . 5 ⊢ mrCls = (𝑐 ∈ ∪ ran Moore ↦ (𝑥 ∈ 𝒫 ∪ 𝑐 ↦ ∩ {𝑠 ∈ 𝑐 ∣ 𝑥 ⊆ 𝑠}))
10		mreunirn 17505	. . . . . . . 8 ⊢ (𝑐 ∈ ∪ ran Moore ↔ 𝑐 ∈ (Moore‘∪ 𝑐))
11		mrcflem 17514	. . . . . . . 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 6683	. . . . . . 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 7678	. . . . . . . 8 ⊢ ∪ 𝑐 ∈ V
16	15	pwex 5320	. . . . . . 7 ⊢ 𝒫 ∪ 𝑐 ∈ V
17		vex 3441	. . . . . . 7 ⊢ 𝑐 ∈ V
18	16, 17	xpex 7692	. . . . . 6 ⊢ (𝒫 ∪ 𝑐 × 𝑐) ∈ V
19		ssexg 5263	. . . . . 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 6939	. . . 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 17504	. . . . 5 ⊢ (𝐶 ∈ (Moore‘𝑋) → ∪ 𝐶 = 𝑋)
24	23	pweqd 4566	. . . 4 ⊢ (𝐶 ∈ (Moore‘𝑋) → 𝒫 ∪ 𝐶 = 𝒫 𝑋)
25	24	mpteq1d 5183	. . 3 ⊢ (𝐶 ∈ (Moore‘𝑋) → (𝑥 ∈ 𝒫 ∪ 𝐶 ↦ ∩ {𝑠 ∈ 𝐶 ∣ 𝑥 ⊆ 𝑠}) = (𝑥 ∈ 𝒫 𝑋 ↦ ∩ {𝑠 ∈ 𝐶 ∣ 𝑥 ⊆ 𝑠}))
26	22, 25	eqtrd 2768	. 2 ⊢ (𝐶 ∈ (Moore‘𝑋) → (mrCls‘𝐶) = (𝑥 ∈ 𝒫 𝑋 ↦ ∩ {𝑠 ∈ 𝐶 ∣ 𝑥 ⊆ 𝑠}))
27	1, 26	eqtrid 2780	1 ⊢ (𝐶 ∈ (Moore‘𝑋) → 𝐹 = (𝑥 ∈ 𝒫 𝑋 ↦ ∩ {𝑠 ∈ 𝐶 ∣ 𝑥 ⊆ 𝑠}))

Syntax hints:   → wi 4   = wceq 1541   ∈ wcel 2113  {crab 3396  Vcvv 3437   ⊆ wss 3898  𝒫 cpw 4549  ∪ cuni 4858  ∩ cint 4897   ↦ cmpt 5174   × cxp 5617  ran crn 5620  ⟶wf 6482  ‘cfv 6486  Moorecmre 17486  mrClscmrc 17487

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-sep 5236  ax-nul 5246  ax-pow 5305  ax-pr 5372  ax-un 7674

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ne 2930  df-ral 3049  df-rex 3058  df-rab 3397  df-v 3439  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4283  df-if 4475  df-pw 4551  df-sn 4576  df-pr 4578  df-op 4582  df-uni 4859  df-int 4898  df-br 5094  df-opab 5156  df-mpt 5175  df-id 5514  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-fv 6494  df-mre 17490  df-mrc 17491

This theorem is referenced by:  mrcf  17517  mrcval  17518  acsficl2d  18460  mrclsp  20924  mrccls  22995