Theorem pclssidN 40151

Description: A set of atoms is included in its projective subspace closure. (Contributed by NM, 12-Sep-2013.) (New usage is discouraged.)

Hypotheses

Ref	Expression
pclss.a	⊢ 𝐴 = (Atoms‘𝐾)
pclss.c	⊢ 𝑈 = (PCl‘𝐾)

Assertion

Ref	Expression
pclssidN	⊢ ((𝐾 ∈ 𝑉 ∧ 𝑋 ⊆ 𝐴) → 𝑋 ⊆ (𝑈‘𝑋))

Proof of Theorem pclssidN

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ssintub 4921	. 2 ⊢ 𝑋 ⊆ ∩ {𝑦 ∈ (PSubSp‘𝐾) ∣ 𝑋 ⊆ 𝑦}
2		pclss.a	. . 3 ⊢ 𝐴 = (Atoms‘𝐾)
3		eqid 2736	. . 3 ⊢ (PSubSp‘𝐾) = (PSubSp‘𝐾)
4		pclss.c	. . 3 ⊢ 𝑈 = (PCl‘𝐾)
5	2, 3, 4	pclvalN 40146	. 2 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑋 ⊆ 𝐴) → (𝑈‘𝑋) = ∩ {𝑦 ∈ (PSubSp‘𝐾) ∣ 𝑋 ⊆ 𝑦})
6	1, 5	sseqtrrid 3977	1 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑋 ⊆ 𝐴) → 𝑋 ⊆ (𝑈‘𝑋))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2113  {crab 3399   ⊆ wss 3901  ∩ cint 4902  ‘cfv 6492  Atomscatm 39519  PSubSpcpsubsp 39752  PClcpclN 40143

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-rep 5224  ax-sep 5241  ax-nul 5251  ax-pow 5310  ax-pr 5377

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-reu 3351  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-int 4903  df-iun 4948  df-br 5099  df-opab 5161  df-mpt 5180  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-ov 7361  df-psubsp 39759  df-pclN 40144

This theorem is referenced by:  pclunN  40154  pcl0bN  40179  pclfinclN  40206