Theorem pclfvalN 40349

Description: 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
pclfvalN	⊢ (𝐾 ∈ 𝑉 → 𝑈 = (𝑥 ∈ 𝒫 𝐴 ↦ ∩ {𝑦 ∈ 𝑆 ∣ 𝑥 ⊆ 𝑦}))

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

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

Proof of Theorem pclfvalN

Dummy variable 𝑘 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elex 3451	. 2 ⊢ (𝐾 ∈ 𝑉 → 𝐾 ∈ V)
2		pclfval.c	. . 3 ⊢ 𝑈 = (PCl‘𝐾)
3		fveq2 6834	. . . . . . 7 ⊢ (𝑘 = 𝐾 → (Atoms‘𝑘) = (Atoms‘𝐾))
4		pclfval.a	. . . . . . 7 ⊢ 𝐴 = (Atoms‘𝐾)
5	3, 4	eqtr4di 2790	. . . . . 6 ⊢ (𝑘 = 𝐾 → (Atoms‘𝑘) = 𝐴)
6	5	pweqd 4559	. . . . 5 ⊢ (𝑘 = 𝐾 → 𝒫 (Atoms‘𝑘) = 𝒫 𝐴)
7		fveq2 6834	. . . . . . . 8 ⊢ (𝑘 = 𝐾 → (PSubSp‘𝑘) = (PSubSp‘𝐾))
8		pclfval.s	. . . . . . . 8 ⊢ 𝑆 = (PSubSp‘𝐾)
9	7, 8	eqtr4di 2790	. . . . . . 7 ⊢ (𝑘 = 𝐾 → (PSubSp‘𝑘) = 𝑆)
10	9	rabeqdv 3405	. . . . . 6 ⊢ (𝑘 = 𝐾 → {𝑦 ∈ (PSubSp‘𝑘) ∣ 𝑥 ⊆ 𝑦} = {𝑦 ∈ 𝑆 ∣ 𝑥 ⊆ 𝑦})
11	10	inteqd 4895	. . . . 5 ⊢ (𝑘 = 𝐾 → ∩ {𝑦 ∈ (PSubSp‘𝑘) ∣ 𝑥 ⊆ 𝑦} = ∩ {𝑦 ∈ 𝑆 ∣ 𝑥 ⊆ 𝑦})
12	6, 11	mpteq12dv 5173	. . . 4 ⊢ (𝑘 = 𝐾 → (𝑥 ∈ 𝒫 (Atoms‘𝑘) ↦ ∩ {𝑦 ∈ (PSubSp‘𝑘) ∣ 𝑥 ⊆ 𝑦}) = (𝑥 ∈ 𝒫 𝐴 ↦ ∩ {𝑦 ∈ 𝑆 ∣ 𝑥 ⊆ 𝑦}))
13		df-pclN 40348	. . . 4 ⊢ PCl = (𝑘 ∈ V ↦ (𝑥 ∈ 𝒫 (Atoms‘𝑘) ↦ ∩ {𝑦 ∈ (PSubSp‘𝑘) ∣ 𝑥 ⊆ 𝑦}))
14	4	fvexi 6848	. . . . . 6 ⊢ 𝐴 ∈ V
15	14	pwex 5317	. . . . 5 ⊢ 𝒫 𝐴 ∈ V
16	15	mptex 7171	. . . 4 ⊢ (𝑥 ∈ 𝒫 𝐴 ↦ ∩ {𝑦 ∈ 𝑆 ∣ 𝑥 ⊆ 𝑦}) ∈ V
17	12, 13, 16	fvmpt 6941	. . 3 ⊢ (𝐾 ∈ V → (PCl‘𝐾) = (𝑥 ∈ 𝒫 𝐴 ↦ ∩ {𝑦 ∈ 𝑆 ∣ 𝑥 ⊆ 𝑦}))
18	2, 17	eqtrid 2784	. 2 ⊢ (𝐾 ∈ V → 𝑈 = (𝑥 ∈ 𝒫 𝐴 ↦ ∩ {𝑦 ∈ 𝑆 ∣ 𝑥 ⊆ 𝑦}))
19	1, 18	syl 17	1 ⊢ (𝐾 ∈ 𝑉 → 𝑈 = (𝑥 ∈ 𝒫 𝐴 ↦ ∩ {𝑦 ∈ 𝑆 ∣ 𝑥 ⊆ 𝑦}))

Syntax hints:   → wi 4   = wceq 1542   ∈ wcel 2114  {crab 3390  Vcvv 3430   ⊆ wss 3890  𝒫 cpw 4542  ∩ cint 4890   ↦ cmpt 5167  ‘cfv 6492  Atomscatm 39723  PSubSpcpsubsp 39956  PClcpclN 40347

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5302  ax-pr 5370

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-int 4891  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-pclN 40348

This theorem is referenced by:  pclvalN  40350