Theorem elpclN 38763

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 38761	. . 3 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑋 ⊆ 𝐴) → (𝑈‘𝑋) = ∩ {𝑦 ∈ 𝑆 ∣ 𝑋 ⊆ 𝑦})
5	4	eleq2d 2820	. 2 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑋 ⊆ 𝐴) → (𝑄 ∈ (𝑈‘𝑋) ↔ 𝑄 ∈ ∩ {𝑦 ∈ 𝑆 ∣ 𝑋 ⊆ 𝑦}))
6		elpcl.q	. . 3 ⊢ 𝑄 ∈ V
7	6	elintrab 4965	. 2 ⊢ (𝑄 ∈ ∩ {𝑦 ∈ 𝑆 ∣ 𝑋 ⊆ 𝑦} ↔ ∀𝑦 ∈ 𝑆 (𝑋 ⊆ 𝑦 → 𝑄 ∈ 𝑦))
8	5, 7	bitrdi 287	1 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑋 ⊆ 𝐴) → (𝑄 ∈ (𝑈‘𝑋) ↔ ∀𝑦 ∈ 𝑆 (𝑋 ⊆ 𝑦 → 𝑄 ∈ 𝑦)))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 397   = wceq 1542   ∈ wcel 2107  ∀wral 3062  {crab 3433  Vcvv 3475   ⊆ wss 3949  ∩ cint 4951  ‘cfv 6544  Atomscatm 38133  PSubSpcpsubsp 38367  PClcpclN 38758

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-int 4952  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-ov 7412  df-psubsp 38374  df-pclN 38759

This theorem is referenced by:  pclfinclN  38821