Theorem pclfvalN 40770

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 3474	. 2 ⊢ (𝐾 ∈ 𝑉 → 𝐾 ∈ V)
2		pclfval.c	. . 3 ⊢ 𝑈 = (PCl‘𝐾)
3		fveq2 6882	. . . . . . 7 ⊢ (𝑘 = 𝐾 → (Atoms‘𝑘) = (Atoms‘𝐾))
4		pclfval.a	. . . . . . 7 ⊢ 𝐴 = (Atoms‘𝐾)
5	3, 4	eqtr4di 2815	. . . . . 6 ⊢ (𝑘 = 𝐾 → (Atoms‘𝑘) = 𝐴)
6	5	pweqd 4577	. . . . 5 ⊢ (𝑘 = 𝐾 → 𝒫 (Atoms‘𝑘) = 𝒫 𝐴)
7		fveq2 6882	. . . . . . . 8 ⊢ (𝑘 = 𝐾 → (PSubSp‘𝑘) = (PSubSp‘𝐾))
8		pclfval.s	. . . . . . . 8 ⊢ 𝑆 = (PSubSp‘𝐾)
9	7, 8	eqtr4di 2815	. . . . . . 7 ⊢ (𝑘 = 𝐾 → (PSubSp‘𝑘) = 𝑆)
10	9	rabeqdv 3429	. . . . . 6 ⊢ (𝑘 = 𝐾 → {𝑦 ∈ (PSubSp‘𝑘) ∣ 𝑥 ⊆ 𝑦} = {𝑦 ∈ 𝑆 ∣ 𝑥 ⊆ 𝑦})
11	10	inteqd 4915	. . . . 5 ⊢ (𝑘 = 𝐾 → ∩ {𝑦 ∈ (PSubSp‘𝑘) ∣ 𝑥 ⊆ 𝑦} = ∩ {𝑦 ∈ 𝑆 ∣ 𝑥 ⊆ 𝑦})
12	6, 11	mpteq12dv 5196	. . . 4 ⊢ (𝑘 = 𝐾 → (𝑥 ∈ 𝒫 (Atoms‘𝑘) ↦ ∩ {𝑦 ∈ (PSubSp‘𝑘) ∣ 𝑥 ⊆ 𝑦}) = (𝑥 ∈ 𝒫 𝐴 ↦ ∩ {𝑦 ∈ 𝑆 ∣ 𝑥 ⊆ 𝑦}))
13		df-pclN 40769	. . . 4 ⊢ PCl = (𝑘 ∈ V ↦ (𝑥 ∈ 𝒫 (Atoms‘𝑘) ↦ ∩ {𝑦 ∈ (PSubSp‘𝑘) ∣ 𝑥 ⊆ 𝑦}))
14	4	fvexi 6896	. . . . . 6 ⊢ 𝐴 ∈ V
15	14	pwex 5349	. . . . 5 ⊢ 𝒫 𝐴 ∈ V
16	15	mptex 7226	. . . 4 ⊢ (𝑥 ∈ 𝒫 𝐴 ↦ ∩ {𝑦 ∈ 𝑆 ∣ 𝑥 ⊆ 𝑦}) ∈ V
17	12, 13, 16	fvmpt 6990	. . 3 ⊢ (𝐾 ∈ V → (PCl‘𝐾) = (𝑥 ∈ 𝒫 𝐴 ↦ ∩ {𝑦 ∈ 𝑆 ∣ 𝑥 ⊆ 𝑦}))
18	2, 17	eqtrid 2809	. 2 ⊢ (𝐾 ∈ V → 𝑈 = (𝑥 ∈ 𝒫 𝐴 ↦ ∩ {𝑦 ∈ 𝑆 ∣ 𝑥 ⊆ 𝑦}))
19	1, 18	syl 18	1 ⊢ (𝐾 ∈ 𝑉 → 𝑈 = (𝑥 ∈ 𝒫 𝐴 ↦ ∩ {𝑦 ∈ 𝑆 ∣ 𝑥 ⊆ 𝑦}))

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2145  {crab 3414  Vcvv 3453   ⊆ wss 3902  𝒫 cpw 4560  ∩ cint 4910   ↦ cmpt 5190  ‘cfv 6537  Atomscatm 40144  PSubSpcpsubsp 40377  PClcpclN 40768

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-rep 5236  ax-sep 5255  ax-nul 5267  ax-pow 5334  ax-pr 5402

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-reu 3368  df-rab 3415  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-int 4911  df-iun 4956  df-br 5108  df-opab 5172  df-mpt 5191  df-id 5554  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-pclN 40769

This theorem is used by:  pclvalN  40771