Theorem pclssidN 35703

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 4629	. 2 ⊢ 𝑋 ⊆ ∩ {𝑦 ∈ (PSubSp‘𝐾) ∣ 𝑋 ⊆ 𝑦}
2		pclss.a	. . 3 ⊢ 𝐴 = (Atoms‘𝐾)
3		eqid 2771	. . 3 ⊢ (PSubSp‘𝐾) = (PSubSp‘𝐾)
4		pclss.c	. . 3 ⊢ 𝑈 = (PCl‘𝐾)
5	2, 3, 4	pclvalN 35698	. 2 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑋 ⊆ 𝐴) → (𝑈‘𝑋) = ∩ {𝑦 ∈ (PSubSp‘𝐾) ∣ 𝑋 ⊆ 𝑦})
6	1, 5	syl5sseqr 3803	1 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑋 ⊆ 𝐴) → 𝑋 ⊆ (𝑈‘𝑋))

Syntax hints:   → wi 4   ∧ wa 382   = wceq 1631   ∈ wcel 2145  {crab 3065   ⊆ wss 3723  ∩ cint 4611  ‘cfv 6031  Atomscatm 35072  PSubSpcpsubsp 35304  PClcpclN 35695

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4904  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034

This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-int 4612  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-ov 6796  df-psubsp 35311  df-pclN 35696

This theorem is referenced by:  pclunN  35706  pcl0bN  35731  pclfinclN  35758