Theorem pclvalN 40526

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 6885	. . 3 ⊢ 𝐴 ∈ V
3	2	elpw2 5295	. 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 40525	. . . . 5 ⊢ (𝐾 ∈ 𝑉 → 𝑈 = (𝑥 ∈ 𝒫 𝐴 ↦ ∩ {𝑦 ∈ 𝑆 ∣ 𝑥 ⊆ 𝑦}))
7	6	fveq1d 6873	. . . 4 ⊢ (𝐾 ∈ 𝑉 → (𝑈‘𝑋) = ((𝑥 ∈ 𝒫 𝐴 ↦ ∩ {𝑦 ∈ 𝑆 ∣ 𝑥 ⊆ 𝑦})‘𝑋))
8	7	adantr 485	. . 3 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑋 ∈ 𝒫 𝐴) → (𝑈‘𝑋) = ((𝑥 ∈ 𝒫 𝐴 ↦ ∩ {𝑦 ∈ 𝑆 ∣ 𝑥 ⊆ 𝑦})‘𝑋))
9		eqid 2765	. . . 4 ⊢ (𝑥 ∈ 𝒫 𝐴 ↦ ∩ {𝑦 ∈ 𝑆 ∣ 𝑥 ⊆ 𝑦}) = (𝑥 ∈ 𝒫 𝐴 ↦ ∩ {𝑦 ∈ 𝑆 ∣ 𝑥 ⊆ 𝑦})
10		sseq1 3964	. . . . . 6 ⊢ (𝑥 = 𝑋 → (𝑥 ⊆ 𝑦 ↔ 𝑋 ⊆ 𝑦))
11	10	rabbidv 3424	. . . . 5 ⊢ (𝑥 = 𝑋 → {𝑦 ∈ 𝑆 ∣ 𝑥 ⊆ 𝑦} = {𝑦 ∈ 𝑆 ∣ 𝑋 ⊆ 𝑦})
12	11	inteqd 4913	. . . 4 ⊢ (𝑥 = 𝑋 → ∩ {𝑦 ∈ 𝑆 ∣ 𝑥 ⊆ 𝑦} = ∩ {𝑦 ∈ 𝑆 ∣ 𝑋 ⊆ 𝑦})
13		simpr 489	. . . 4 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑋 ∈ 𝒫 𝐴) → 𝑋 ∈ 𝒫 𝐴)
14		elpwi 4565	. . . . . . . 8 ⊢ (𝑋 ∈ 𝒫 𝐴 → 𝑋 ⊆ 𝐴)
15	14	adantl 486	. . . . . . 7 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑋 ∈ 𝒫 𝐴) → 𝑋 ⊆ 𝐴)
16	1, 4	atpsubN 40389	. . . . . . . . 9 ⊢ (𝐾 ∈ 𝑉 → 𝐴 ∈ 𝑆)
17	16	adantr 485	. . . . . . . 8 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑋 ∈ 𝒫 𝐴) → 𝐴 ∈ 𝑆)
18		sseq2 3965	. . . . . . . . 9 ⊢ (𝑦 = 𝐴 → (𝑋 ⊆ 𝑦 ↔ 𝑋 ⊆ 𝐴))
19	18	elrab3 3654	. . . . . . . 8 ⊢ (𝐴 ∈ 𝑆 → (𝐴 ∈ {𝑦 ∈ 𝑆 ∣ 𝑋 ⊆ 𝑦} ↔ 𝑋 ⊆ 𝐴))
20	17, 19	syl 18	. . . . . . 7 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑋 ∈ 𝒫 𝐴) → (𝐴 ∈ {𝑦 ∈ 𝑆 ∣ 𝑋 ⊆ 𝑦} ↔ 𝑋 ⊆ 𝐴))
21	15, 20	mpbird 260	. . . . . 6 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑋 ∈ 𝒫 𝐴) → 𝐴 ∈ {𝑦 ∈ 𝑆 ∣ 𝑋 ⊆ 𝑦})
22	21	ne0d 4297	. . . . 5 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑋 ∈ 𝒫 𝐴) → {𝑦 ∈ 𝑆 ∣ 𝑋 ⊆ 𝑦} ≠ ∅)
23		intex 5305	. . . . 5 ⊢ ({𝑦 ∈ 𝑆 ∣ 𝑋 ⊆ 𝑦} ≠ ∅ ↔ ∩ {𝑦 ∈ 𝑆 ∣ 𝑋 ⊆ 𝑦} ∈ V)
24	22, 23	sylib 221	. . . 4 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑋 ∈ 𝒫 𝐴) → ∩ {𝑦 ∈ 𝑆 ∣ 𝑋 ⊆ 𝑦} ∈ V)
25	9, 12, 13, 24	fvmptd3 7003	. . 3 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑋 ∈ 𝒫 𝐴) → ((𝑥 ∈ 𝒫 𝐴 ↦ ∩ {𝑦 ∈ 𝑆 ∣ 𝑥 ⊆ 𝑦})‘𝑋) = ∩ {𝑦 ∈ 𝑆 ∣ 𝑋 ⊆ 𝑦})
26	8, 25	eqtrd 2800	. 2 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑋 ∈ 𝒫 𝐴) → (𝑈‘𝑋) = ∩ {𝑦 ∈ 𝑆 ∣ 𝑋 ⊆ 𝑦})
27	3, 26	sylan2br 606	1 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑋 ⊆ 𝐴) → (𝑈‘𝑋) = ∩ {𝑦 ∈ 𝑆 ∣ 𝑋 ⊆ 𝑦})

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1563   ∈ wcel 2145   ≠ wne 2960  {crab 3417  Vcvv 3457   ⊆ wss 3907  ∅c0 4288  𝒫 cpw 4558  ∩ cint 4908   ↦ cmpt 5186  ‘cfv 6525  Atomscatm 39899  PSubSpcpsubsp 40132  PClcpclN 40523

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737  ax-rep 5232  ax-sep 5251  ax-nul 5261  ax-pow 5327  ax-pr 5395

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3080  df-rex 3090  df-reu 3371  df-rab 3418  df-v 3459  df-sbc 3748  df-csb 3856  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-nul 4289  df-if 4484  df-pw 4560  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4869  df-int 4909  df-iun 4954  df-br 5106  df-opab 5168  df-mpt 5187  df-id 5547  df-xp 5658  df-rel 5659  df-cnv 5660  df-co 5661  df-dm 5662  df-rn 5663  df-res 5664  df-ima 5665  df-iota 6481  df-fun 6527  df-fn 6528  df-f 6529  df-f1 6530  df-fo 6531  df-f1o 6532  df-fv 6533  df-ov 7403  df-psubsp 40139  df-pclN 40524

This theorem is referenced by:  pclclN  40527  elpclN  40528  elpcliN  40529  pclssN  40530  pclssidN  40531  pclidN  40532