Theorem pclvalN 38756

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 6905	. . 3 ⊢ 𝐴 ∈ V
3	2	elpw2 5345	. 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 38755	. . . . 5 ⊢ (𝐾 ∈ 𝑉 → 𝑈 = (𝑥 ∈ 𝒫 𝐴 ↦ ∩ {𝑦 ∈ 𝑆 ∣ 𝑥 ⊆ 𝑦}))
7	6	fveq1d 6893	. . . 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 2732	. . . 4 ⊢ (𝑥 ∈ 𝒫 𝐴 ↦ ∩ {𝑦 ∈ 𝑆 ∣ 𝑥 ⊆ 𝑦}) = (𝑥 ∈ 𝒫 𝐴 ↦ ∩ {𝑦 ∈ 𝑆 ∣ 𝑥 ⊆ 𝑦})
10		sseq1 4007	. . . . . 6 ⊢ (𝑥 = 𝑋 → (𝑥 ⊆ 𝑦 ↔ 𝑋 ⊆ 𝑦))
11	10	rabbidv 3440	. . . . 5 ⊢ (𝑥 = 𝑋 → {𝑦 ∈ 𝑆 ∣ 𝑥 ⊆ 𝑦} = {𝑦 ∈ 𝑆 ∣ 𝑋 ⊆ 𝑦})
12	11	inteqd 4955	. . . 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 4609	. . . . . . . 8 ⊢ (𝑋 ∈ 𝒫 𝐴 → 𝑋 ⊆ 𝐴)
15	14	adantl 482	. . . . . . 7 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑋 ∈ 𝒫 𝐴) → 𝑋 ⊆ 𝐴)
16	1, 4	atpsubN 38619	. . . . . . . . 9 ⊢ (𝐾 ∈ 𝑉 → 𝐴 ∈ 𝑆)
17	16	adantr 481	. . . . . . . 8 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑋 ∈ 𝒫 𝐴) → 𝐴 ∈ 𝑆)
18		sseq2 4008	. . . . . . . . 9 ⊢ (𝑦 = 𝐴 → (𝑋 ⊆ 𝑦 ↔ 𝑋 ⊆ 𝐴))
19	18	elrab3 3684	. . . . . . . 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 4335	. . . . 5 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑋 ∈ 𝒫 𝐴) → {𝑦 ∈ 𝑆 ∣ 𝑋 ⊆ 𝑦} ≠ ∅)
23		intex 5337	. . . . 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 7021	. . 3 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑋 ∈ 𝒫 𝐴) → ((𝑥 ∈ 𝒫 𝐴 ↦ ∩ {𝑦 ∈ 𝑆 ∣ 𝑥 ⊆ 𝑦})‘𝑋) = ∩ {𝑦 ∈ 𝑆 ∣ 𝑋 ⊆ 𝑦})
26	8, 25	eqtrd 2772	. 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 2940  {crab 3432  Vcvv 3474   ⊆ wss 3948  ∅c0 4322  𝒫 cpw 4602  ∩ cint 4950   ↦ cmpt 5231  ‘cfv 6543  Atomscatm 38128  PSubSpcpsubsp 38362  PClcpclN 38753

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 2703  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427

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 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-int 4951  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7411  df-psubsp 38369  df-pclN 38754

This theorem is referenced by:  pclclN  38757  elpclN  38758  elpcliN  38759  pclssN  38760  pclssidN  38761  pclidN  38762