Theorem mrcfval 17234

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 6785	. . . . 5 ⊢ (Moore‘𝑋) ⊆ ∪ ran Moore
3	2	sseli 3913	. . . 4 ⊢ (𝐶 ∈ (Moore‘𝑋) → 𝐶 ∈ ∪ ran Moore)
4		unieq 4847	. . . . . . 7 ⊢ (𝑐 = 𝐶 → ∪ 𝑐 = ∪ 𝐶)
5	4	pweqd 4549	. . . . . 6 ⊢ (𝑐 = 𝐶 → 𝒫 ∪ 𝑐 = 𝒫 ∪ 𝐶)
6		rabeq 3408	. . . . . . 7 ⊢ (𝑐 = 𝐶 → {𝑠 ∈ 𝑐 ∣ 𝑥 ⊆ 𝑠} = {𝑠 ∈ 𝐶 ∣ 𝑥 ⊆ 𝑠})
7	6	inteqd 4881	. . . . . 6 ⊢ (𝑐 = 𝐶 → ∩ {𝑠 ∈ 𝑐 ∣ 𝑥 ⊆ 𝑠} = ∩ {𝑠 ∈ 𝐶 ∣ 𝑥 ⊆ 𝑠})
8	5, 7	mpteq12dv 5161	. . . . 5 ⊢ (𝑐 = 𝐶 → (𝑥 ∈ 𝒫 ∪ 𝑐 ↦ ∩ {𝑠 ∈ 𝑐 ∣ 𝑥 ⊆ 𝑠}) = (𝑥 ∈ 𝒫 ∪ 𝐶 ↦ ∩ {𝑠 ∈ 𝐶 ∣ 𝑥 ⊆ 𝑠}))
9		df-mrc 17213	. . . . 5 ⊢ mrCls = (𝑐 ∈ ∪ ran Moore ↦ (𝑥 ∈ 𝒫 ∪ 𝑐 ↦ ∩ {𝑠 ∈ 𝑐 ∣ 𝑥 ⊆ 𝑠}))
10		mreunirn 17227	. . . . . . . 8 ⊢ (𝑐 ∈ ∪ ran Moore ↔ 𝑐 ∈ (Moore‘∪ 𝑐))
11		mrcflem 17232	. . . . . . . 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 6612	. . . . . . 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 7570	. . . . . . . 8 ⊢ ∪ 𝑐 ∈ V
16	15	pwex 5298	. . . . . . 7 ⊢ 𝒫 ∪ 𝑐 ∈ V
17		vex 3426	. . . . . . 7 ⊢ 𝑐 ∈ V
18	16, 17	xpex 7581	. . . . . 6 ⊢ (𝒫 ∪ 𝑐 × 𝑐) ∈ V
19		ssexg 5242	. . . . . 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 585	. . . . 5 ⊢ (𝑐 ∈ ∪ ran Moore → (𝑥 ∈ 𝒫 ∪ 𝑐 ↦ ∩ {𝑠 ∈ 𝑐 ∣ 𝑥 ⊆ 𝑠}) ∈ V)
21	8, 9, 20	fvmpt3 6861	. . . 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 17226	. . . . 5 ⊢ (𝐶 ∈ (Moore‘𝑋) → ∪ 𝐶 = 𝑋)
24	23	pweqd 4549	. . . 4 ⊢ (𝐶 ∈ (Moore‘𝑋) → 𝒫 ∪ 𝐶 = 𝒫 𝑋)
25	24	mpteq1d 5165	. . 3 ⊢ (𝐶 ∈ (Moore‘𝑋) → (𝑥 ∈ 𝒫 ∪ 𝐶 ↦ ∩ {𝑠 ∈ 𝐶 ∣ 𝑥 ⊆ 𝑠}) = (𝑥 ∈ 𝒫 𝑋 ↦ ∩ {𝑠 ∈ 𝐶 ∣ 𝑥 ⊆ 𝑠}))
26	22, 25	eqtrd 2778	. 2 ⊢ (𝐶 ∈ (Moore‘𝑋) → (mrCls‘𝐶) = (𝑥 ∈ 𝒫 𝑋 ↦ ∩ {𝑠 ∈ 𝐶 ∣ 𝑥 ⊆ 𝑠}))
27	1, 26	eqtrid 2790	1 ⊢ (𝐶 ∈ (Moore‘𝑋) → 𝐹 = (𝑥 ∈ 𝒫 𝑋 ↦ ∩ {𝑠 ∈ 𝐶 ∣ 𝑥 ⊆ 𝑠}))

Syntax hints:   → wi 4   = wceq 1539   ∈ wcel 2108  {crab 3067  Vcvv 3422   ⊆ wss 3883  𝒫 cpw 4530  ∪ cuni 4836  ∩ cint 4876   ↦ cmpt 5153   × cxp 5578  ran crn 5581  ⟶wf 6414  ‘cfv 6418  Moorecmre 17208  mrClscmrc 17209

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-int 4877  df-br 5071  df-opab 5133  df-mpt 5154  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-fv 6426  df-mre 17212  df-mrc 17213

This theorem is referenced by:  mrcf  17235  mrcval  17236  acsficl2d  18185  mrclsp  20166  mrccls  22138