Theorem elpclN 35966

Description: Membership in the projective subspace closure function. (Contributed by NM, 13-Sep-2013.) (New usage is discouraged.)

Hypotheses

Ref	Expression
pclfval.a	⊢ 𝐴 = (Atoms‘𝐾)
pclfval.s	⊢ 𝑆 = (PSubSp‘𝐾)
pclfval.c	⊢ 𝑈 = (PCl‘𝐾)
elpcl.q	⊢ 𝑄 ∈ V

Assertion

Ref	Expression
elpclN	⊢ ((𝐾 ∈ 𝑉 ∧ 𝑋 ⊆ 𝐴) → (𝑄 ∈ (𝑈‘𝑋) ↔ ∀𝑦 ∈ 𝑆 (𝑋 ⊆ 𝑦 → 𝑄 ∈ 𝑦)))

Distinct variable groups:   𝑦,𝐴   𝑦,𝐾   𝑦,𝑆   𝑦,𝑋   𝑦,𝑉   𝑦,𝑄

Allowed substitution hint:   𝑈(𝑦)

Proof of Theorem elpclN

Step	Hyp	Ref	Expression
1		pclfval.a	. . . 4 ⊢ 𝐴 = (Atoms‘𝐾)
2		pclfval.s	. . . 4 ⊢ 𝑆 = (PSubSp‘𝐾)
3		pclfval.c	. . . 4 ⊢ 𝑈 = (PCl‘𝐾)
4	1, 2, 3	pclvalN 35964	. . 3 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑋 ⊆ 𝐴) → (𝑈‘𝑋) = ∩ {𝑦 ∈ 𝑆 ∣ 𝑋 ⊆ 𝑦})
5	4	eleq2d 2891	. 2 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑋 ⊆ 𝐴) → (𝑄 ∈ (𝑈‘𝑋) ↔ 𝑄 ∈ ∩ {𝑦 ∈ 𝑆 ∣ 𝑋 ⊆ 𝑦}))
6		elpcl.q	. . 3 ⊢ 𝑄 ∈ V
7	6	elintrab 4708	. 2 ⊢ (𝑄 ∈ ∩ {𝑦 ∈ 𝑆 ∣ 𝑋 ⊆ 𝑦} ↔ ∀𝑦 ∈ 𝑆 (𝑋 ⊆ 𝑦 → 𝑄 ∈ 𝑦))
8	5, 7	syl6bb 279	1 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑋 ⊆ 𝐴) → (𝑄 ∈ (𝑈‘𝑋) ↔ ∀𝑦 ∈ 𝑆 (𝑋 ⊆ 𝑦 → 𝑄 ∈ 𝑦)))

Syntax hints:   → wi 4   ↔ wb 198   ∧ wa 386   = wceq 1658   ∈ wcel 2166  ∀wral 3116  {crab 3120  Vcvv 3413   ⊆ wss 3797  ∩ cint 4696  ‘cfv 6122  Atomscatm 35337  PSubSpcpsubsp 35570  PClcpclN 35961

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-8 2168  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2390  ax-ext 2802  ax-rep 4993  ax-sep 5004  ax-nul 5012  ax-pow 5064  ax-pr 5126

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2604  df-eu 2639  df-clab 2811  df-cleq 2817  df-clel 2820  df-nfc 2957  df-ne 2999  df-ral 3121  df-rex 3122  df-reu 3123  df-rab 3125  df-v 3415  df-sbc 3662  df-csb 3757  df-dif 3800  df-un 3802  df-in 3804  df-ss 3811  df-nul 4144  df-if 4306  df-pw 4379  df-sn 4397  df-pr 4399  df-op 4403  df-uni 4658  df-int 4697  df-iun 4741  df-br 4873  df-opab 4935  df-mpt 4952  df-id 5249  df-xp 5347  df-rel 5348  df-cnv 5349  df-co 5350  df-dm 5351  df-rn 5352  df-res 5353  df-ima 5354  df-iota 6085  df-fun 6124  df-fn 6125  df-f 6126  df-f1 6127  df-fo 6128  df-f1o 6129  df-fv 6130  df-ov 6907  df-psubsp 35577  df-pclN 35962

This theorem is referenced by:  pclfinclN  36024