Theorem mrcval 17574

Description: Evaluation of the Moore closure of a set. (Contributed by Stefan O'Rear, 31-Jan-2015.) (Proof shortened by Fan Zheng, 6-Jun-2016.)

Hypothesis

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

Assertion

Ref	Expression
mrcval	⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝑋) → (𝐹‘𝑈) = ∩ {𝑠 ∈ 𝐶 ∣ 𝑈 ⊆ 𝑠})

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

Proof of Theorem mrcval

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		mrcfval.f	. . . 4 ⊢ 𝐹 = (mrCls‘𝐶)
2	1	mrcfval 17572	. . 3 ⊢ (𝐶 ∈ (Moore‘𝑋) → 𝐹 = (𝑥 ∈ 𝒫 𝑋 ↦ ∩ {𝑠 ∈ 𝐶 ∣ 𝑥 ⊆ 𝑠}))
3	2	adantr 481	. 2 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝑋) → 𝐹 = (𝑥 ∈ 𝒫 𝑋 ↦ ∩ {𝑠 ∈ 𝐶 ∣ 𝑥 ⊆ 𝑠}))
4		sseq1 3947	. . . . 5 ⊢ (𝑥 = 𝑈 → (𝑥 ⊆ 𝑠 ↔ 𝑈 ⊆ 𝑠))
5	4	rabbidv 3399	. . . 4 ⊢ (𝑥 = 𝑈 → {𝑠 ∈ 𝐶 ∣ 𝑥 ⊆ 𝑠} = {𝑠 ∈ 𝐶 ∣ 𝑈 ⊆ 𝑠})
6	5	inteqd 4889	. . 3 ⊢ (𝑥 = 𝑈 → ∩ {𝑠 ∈ 𝐶 ∣ 𝑥 ⊆ 𝑠} = ∩ {𝑠 ∈ 𝐶 ∣ 𝑈 ⊆ 𝑠})
7	6	adantl 482	. 2 ⊢ (((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝑋) ∧ 𝑥 = 𝑈) → ∩ {𝑠 ∈ 𝐶 ∣ 𝑥 ⊆ 𝑠} = ∩ {𝑠 ∈ 𝐶 ∣ 𝑈 ⊆ 𝑠})
8		mre1cl 17554	. . . 4 ⊢ (𝐶 ∈ (Moore‘𝑋) → 𝑋 ∈ 𝐶)
9		elpw2g 5268	. . . 4 ⊢ (𝑋 ∈ 𝐶 → (𝑈 ∈ 𝒫 𝑋 ↔ 𝑈 ⊆ 𝑋))
10	8, 9	syl 17	. . 3 ⊢ (𝐶 ∈ (Moore‘𝑋) → (𝑈 ∈ 𝒫 𝑋 ↔ 𝑈 ⊆ 𝑋))
11	10	biimpar 478	. 2 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝑋) → 𝑈 ∈ 𝒫 𝑋)
12		sseq2 3948	. . . . 5 ⊢ (𝑠 = 𝑋 → (𝑈 ⊆ 𝑠 ↔ 𝑈 ⊆ 𝑋))
13	8	adantr 481	. . . . 5 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝑋) → 𝑋 ∈ 𝐶)
14		simpr 485	. . . . 5 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝑋) → 𝑈 ⊆ 𝑋)
15	12, 13, 14	elrabd 3638	. . . 4 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝑋) → 𝑋 ∈ {𝑠 ∈ 𝐶 ∣ 𝑈 ⊆ 𝑠})
16	15	ne0d 4277	. . 3 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝑋) → {𝑠 ∈ 𝐶 ∣ 𝑈 ⊆ 𝑠} ≠ ∅)
17		intex 5279	. . 3 ⊢ ({𝑠 ∈ 𝐶 ∣ 𝑈 ⊆ 𝑠} ≠ ∅ ↔ ∩ {𝑠 ∈ 𝐶 ∣ 𝑈 ⊆ 𝑠} ∈ V)
18	16, 17	sylib 219	. 2 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝑋) → ∩ {𝑠 ∈ 𝐶 ∣ 𝑈 ⊆ 𝑠} ∈ V)
19	3, 7, 11, 18	fvmptd 6950	1 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝑋) → (𝐹‘𝑈) = ∩ {𝑠 ∈ 𝐶 ∣ 𝑈 ⊆ 𝑠})

Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396   = wceq 1547   ∈ wcel 2119   ≠ wne 2935  {crab 3392  Vcvv 3432   ⊆ wss 3890  ∅c0 4268  𝒫 cpw 4536  ∩ cint 4884   ↦ cmpt 5160  ‘cfv 6492  Moorecmre 17542  mrClscmrc 17543

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2712  ax-sep 5225  ax-nul 5235  ax-pow 5301  ax-pr 5369  ax-un 7685

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2719  df-cleq 2732  df-clel 2815  df-nfc 2889  df-ne 2936  df-ral 3055  df-rex 3065  df-rab 3393  df-v 3434  df-sbc 3731  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4269  df-if 4462  df-pw 4538  df-sn 4563  df-pr 4565  df-op 4569  df-uni 4846  df-int 4885  df-br 5080  df-opab 5142  df-mpt 5161  df-id 5520  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-fv 6500  df-mre 17546  df-mrc 17547

This theorem is referenced by:  mrcid  17577  mrcss  17580  mrcssid  17581  cycsubg2  19183  aspval2  21880  mrelatlubALT  49486  mreclat  49488