Theorem pclvalN 38353

Description: Value of the projective subspace closure function. (Contributed by NM, 7-Sep-2013.) (New usage is discouraged.)

Hypotheses

Ref	Expression
pclfval.a	⊢ 𝐴 = (Atoms‘𝐾)
pclfval.s	⊢ 𝑆 = (PSubSp‘𝐾)
pclfval.c	⊢ 𝑈 = (PCl‘𝐾)

Assertion

Ref	Expression
pclvalN	⊢ ((𝐾 ∈ 𝑉 ∧ 𝑋 ⊆ 𝐴) → (𝑈‘𝑋) = ∩ {𝑦 ∈ 𝑆 ∣ 𝑋 ⊆ 𝑦})

Distinct variable groups:   𝑦,𝐴   𝑦,𝐾   𝑦,𝑆   𝑦,𝑋

Allowed substitution hints:   𝑈(𝑦)   𝑉(𝑦)

Proof of Theorem pclvalN

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		pclfval.a	. . . 4 ⊢ 𝐴 = (Atoms‘𝐾)
2	1	fvexi 6856	. . 3 ⊢ 𝐴 ∈ V
3	2	elpw2 5302	. 2 ⊢ (𝑋 ∈ 𝒫 𝐴 ↔ 𝑋 ⊆ 𝐴)
4		pclfval.s	. . . . . 6 ⊢ 𝑆 = (PSubSp‘𝐾)
5		pclfval.c	. . . . . 6 ⊢ 𝑈 = (PCl‘𝐾)
6	1, 4, 5	pclfvalN 38352	. . . . 5 ⊢ (𝐾 ∈ 𝑉 → 𝑈 = (𝑥 ∈ 𝒫 𝐴 ↦ ∩ {𝑦 ∈ 𝑆 ∣ 𝑥 ⊆ 𝑦}))
7	6	fveq1d 6844	. . . 4 ⊢ (𝐾 ∈ 𝑉 → (𝑈‘𝑋) = ((𝑥 ∈ 𝒫 𝐴 ↦ ∩ {𝑦 ∈ 𝑆 ∣ 𝑥 ⊆ 𝑦})‘𝑋))
8	7	adantr 481	. . 3 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑋 ∈ 𝒫 𝐴) → (𝑈‘𝑋) = ((𝑥 ∈ 𝒫 𝐴 ↦ ∩ {𝑦 ∈ 𝑆 ∣ 𝑥 ⊆ 𝑦})‘𝑋))
9		eqid 2736	. . . 4 ⊢ (𝑥 ∈ 𝒫 𝐴 ↦ ∩ {𝑦 ∈ 𝑆 ∣ 𝑥 ⊆ 𝑦}) = (𝑥 ∈ 𝒫 𝐴 ↦ ∩ {𝑦 ∈ 𝑆 ∣ 𝑥 ⊆ 𝑦})
10		sseq1 3969	. . . . . 6 ⊢ (𝑥 = 𝑋 → (𝑥 ⊆ 𝑦 ↔ 𝑋 ⊆ 𝑦))
11	10	rabbidv 3415	. . . . 5 ⊢ (𝑥 = 𝑋 → {𝑦 ∈ 𝑆 ∣ 𝑥 ⊆ 𝑦} = {𝑦 ∈ 𝑆 ∣ 𝑋 ⊆ 𝑦})
12	11	inteqd 4912	. . . 4 ⊢ (𝑥 = 𝑋 → ∩ {𝑦 ∈ 𝑆 ∣ 𝑥 ⊆ 𝑦} = ∩ {𝑦 ∈ 𝑆 ∣ 𝑋 ⊆ 𝑦})
13		simpr 485	. . . 4 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑋 ∈ 𝒫 𝐴) → 𝑋 ∈ 𝒫 𝐴)
14		elpwi 4567	. . . . . . . 8 ⊢ (𝑋 ∈ 𝒫 𝐴 → 𝑋 ⊆ 𝐴)
15	14	adantl 482	. . . . . . 7 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑋 ∈ 𝒫 𝐴) → 𝑋 ⊆ 𝐴)
16	1, 4	atpsubN 38216	. . . . . . . . 9 ⊢ (𝐾 ∈ 𝑉 → 𝐴 ∈ 𝑆)
17	16	adantr 481	. . . . . . . 8 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑋 ∈ 𝒫 𝐴) → 𝐴 ∈ 𝑆)
18		sseq2 3970	. . . . . . . . 9 ⊢ (𝑦 = 𝐴 → (𝑋 ⊆ 𝑦 ↔ 𝑋 ⊆ 𝐴))
19	18	elrab3 3646	. . . . . . . 8 ⊢ (𝐴 ∈ 𝑆 → (𝐴 ∈ {𝑦 ∈ 𝑆 ∣ 𝑋 ⊆ 𝑦} ↔ 𝑋 ⊆ 𝐴))
20	17, 19	syl 17	. . . . . . 7 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑋 ∈ 𝒫 𝐴) → (𝐴 ∈ {𝑦 ∈ 𝑆 ∣ 𝑋 ⊆ 𝑦} ↔ 𝑋 ⊆ 𝐴))
21	15, 20	mpbird 256	. . . . . 6 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑋 ∈ 𝒫 𝐴) → 𝐴 ∈ {𝑦 ∈ 𝑆 ∣ 𝑋 ⊆ 𝑦})
22	21	ne0d 4295	. . . . 5 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑋 ∈ 𝒫 𝐴) → {𝑦 ∈ 𝑆 ∣ 𝑋 ⊆ 𝑦} ≠ ∅)
23		intex 5294	. . . . 5 ⊢ ({𝑦 ∈ 𝑆 ∣ 𝑋 ⊆ 𝑦} ≠ ∅ ↔ ∩ {𝑦 ∈ 𝑆 ∣ 𝑋 ⊆ 𝑦} ∈ V)
24	22, 23	sylib 217	. . . 4 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑋 ∈ 𝒫 𝐴) → ∩ {𝑦 ∈ 𝑆 ∣ 𝑋 ⊆ 𝑦} ∈ V)
25	9, 12, 13, 24	fvmptd3 6971	. . 3 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑋 ∈ 𝒫 𝐴) → ((𝑥 ∈ 𝒫 𝐴 ↦ ∩ {𝑦 ∈ 𝑆 ∣ 𝑥 ⊆ 𝑦})‘𝑋) = ∩ {𝑦 ∈ 𝑆 ∣ 𝑋 ⊆ 𝑦})
26	8, 25	eqtrd 2776	. 2 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑋 ∈ 𝒫 𝐴) → (𝑈‘𝑋) = ∩ {𝑦 ∈ 𝑆 ∣ 𝑋 ⊆ 𝑦})
27	3, 26	sylan2br 595	1 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑋 ⊆ 𝐴) → (𝑈‘𝑋) = ∩ {𝑦 ∈ 𝑆 ∣ 𝑋 ⊆ 𝑦})

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1541   ∈ wcel 2106   ≠ wne 2943  {crab 3407  Vcvv 3445   ⊆ wss 3910  ∅c0 4282  𝒫 cpw 4560  ∩ cint 4907   ↦ cmpt 5188  ‘cfv 6496  Atomscatm 37725  PSubSpcpsubsp 37959  PClcpclN 38350

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-int 4908  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-id 5531  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-ov 7360  df-psubsp 37966  df-pclN 38351

This theorem is referenced by:  pclclN  38354  elpclN  38355  elpcliN  38356  pclssN  38357  pclssidN  38358  pclidN  38359