Theorem pclvalN 39273

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 6898	. . 3 ⊢ 𝐴 ∈ V
3	2	elpw2 5338	. 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 39272	. . . . 5 ⊢ (𝐾 ∈ 𝑉 → 𝑈 = (𝑥 ∈ 𝒫 𝐴 ↦ ∩ {𝑦 ∈ 𝑆 ∣ 𝑥 ⊆ 𝑦}))
7	6	fveq1d 6886	. . . 4 ⊢ (𝐾 ∈ 𝑉 → (𝑈‘𝑋) = ((𝑥 ∈ 𝒫 𝐴 ↦ ∩ {𝑦 ∈ 𝑆 ∣ 𝑥 ⊆ 𝑦})‘𝑋))
8	7	adantr 480	. . 3 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑋 ∈ 𝒫 𝐴) → (𝑈‘𝑋) = ((𝑥 ∈ 𝒫 𝐴 ↦ ∩ {𝑦 ∈ 𝑆 ∣ 𝑥 ⊆ 𝑦})‘𝑋))
9		eqid 2726	. . . 4 ⊢ (𝑥 ∈ 𝒫 𝐴 ↦ ∩ {𝑦 ∈ 𝑆 ∣ 𝑥 ⊆ 𝑦}) = (𝑥 ∈ 𝒫 𝐴 ↦ ∩ {𝑦 ∈ 𝑆 ∣ 𝑥 ⊆ 𝑦})
10		sseq1 4002	. . . . . 6 ⊢ (𝑥 = 𝑋 → (𝑥 ⊆ 𝑦 ↔ 𝑋 ⊆ 𝑦))
11	10	rabbidv 3434	. . . . 5 ⊢ (𝑥 = 𝑋 → {𝑦 ∈ 𝑆 ∣ 𝑥 ⊆ 𝑦} = {𝑦 ∈ 𝑆 ∣ 𝑋 ⊆ 𝑦})
12	11	inteqd 4948	. . . 4 ⊢ (𝑥 = 𝑋 → ∩ {𝑦 ∈ 𝑆 ∣ 𝑥 ⊆ 𝑦} = ∩ {𝑦 ∈ 𝑆 ∣ 𝑋 ⊆ 𝑦})
13		simpr 484	. . . 4 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑋 ∈ 𝒫 𝐴) → 𝑋 ∈ 𝒫 𝐴)
14		elpwi 4604	. . . . . . . 8 ⊢ (𝑋 ∈ 𝒫 𝐴 → 𝑋 ⊆ 𝐴)
15	14	adantl 481	. . . . . . 7 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑋 ∈ 𝒫 𝐴) → 𝑋 ⊆ 𝐴)
16	1, 4	atpsubN 39136	. . . . . . . . 9 ⊢ (𝐾 ∈ 𝑉 → 𝐴 ∈ 𝑆)
17	16	adantr 480	. . . . . . . 8 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑋 ∈ 𝒫 𝐴) → 𝐴 ∈ 𝑆)
18		sseq2 4003	. . . . . . . . 9 ⊢ (𝑦 = 𝐴 → (𝑋 ⊆ 𝑦 ↔ 𝑋 ⊆ 𝐴))
19	18	elrab3 3679	. . . . . . . 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 257	. . . . . 6 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑋 ∈ 𝒫 𝐴) → 𝐴 ∈ {𝑦 ∈ 𝑆 ∣ 𝑋 ⊆ 𝑦})
22	21	ne0d 4330	. . . . 5 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑋 ∈ 𝒫 𝐴) → {𝑦 ∈ 𝑆 ∣ 𝑋 ⊆ 𝑦} ≠ ∅)
23		intex 5330	. . . . 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 7014	. . 3 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑋 ∈ 𝒫 𝐴) → ((𝑥 ∈ 𝒫 𝐴 ↦ ∩ {𝑦 ∈ 𝑆 ∣ 𝑥 ⊆ 𝑦})‘𝑋) = ∩ {𝑦 ∈ 𝑆 ∣ 𝑋 ⊆ 𝑦})
26	8, 25	eqtrd 2766	. 2 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑋 ∈ 𝒫 𝐴) → (𝑈‘𝑋) = ∩ {𝑦 ∈ 𝑆 ∣ 𝑋 ⊆ 𝑦})
27	3, 26	sylan2br 594	1 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑋 ⊆ 𝐴) → (𝑈‘𝑋) = ∩ {𝑦 ∈ 𝑆 ∣ 𝑋 ⊆ 𝑦})

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   = wceq 1533   ∈ wcel 2098   ≠ wne 2934  {crab 3426  Vcvv 3468   ⊆ wss 3943  ∅c0 4317  𝒫 cpw 4597  ∩ cint 4943   ↦ cmpt 5224  ‘cfv 6536  Atomscatm 38645  PSubSpcpsubsp 38879  PClcpclN 39270

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-int 4944  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6488  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-ov 7407  df-psubsp 38886  df-pclN 39271

This theorem is referenced by:  pclclN  39274  elpclN  39275  elpcliN  39276  pclssN  39277  pclssidN  39278  pclidN  39279