Theorem ispsubclN 40046

Description: The predicate "is a closed projective subspace". (Contributed by NM, 23-Jan-2012.) (New usage is discouraged.)

Hypotheses

Ref	Expression
psubclset.a	⊢ 𝐴 = (Atoms‘𝐾)
psubclset.p	⊢ ⊥ = (⊥𝑃‘𝐾)
psubclset.c	⊢ 𝐶 = (PSubCl‘𝐾)

Assertion

Ref	Expression
ispsubclN	⊢ (𝐾 ∈ 𝐷 → (𝑋 ∈ 𝐶 ↔ (𝑋 ⊆ 𝐴 ∧ ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋)))

Proof of Theorem ispsubclN

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		psubclset.a	. . . 4 ⊢ 𝐴 = (Atoms‘𝐾)
2		psubclset.p	. . . 4 ⊢ ⊥ = (⊥𝑃‘𝐾)
3		psubclset.c	. . . 4 ⊢ 𝐶 = (PSubCl‘𝐾)
4	1, 2, 3	psubclsetN 40045	. . 3 ⊢ (𝐾 ∈ 𝐷 → 𝐶 = {𝑥 ∣ (𝑥 ⊆ 𝐴 ∧ ( ⊥ ‘( ⊥ ‘𝑥)) = 𝑥)})
5	4	eleq2d 2817	. 2 ⊢ (𝐾 ∈ 𝐷 → (𝑋 ∈ 𝐶 ↔ 𝑋 ∈ {𝑥 ∣ (𝑥 ⊆ 𝐴 ∧ ( ⊥ ‘( ⊥ ‘𝑥)) = 𝑥)}))
6	1	fvexi 6836	. . . . 5 ⊢ 𝐴 ∈ V
7	6	ssex 5257	. . . 4 ⊢ (𝑋 ⊆ 𝐴 → 𝑋 ∈ V)
8	7	adantr 480	. . 3 ⊢ ((𝑋 ⊆ 𝐴 ∧ ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋) → 𝑋 ∈ V)
9		sseq1 3955	. . . 4 ⊢ (𝑥 = 𝑋 → (𝑥 ⊆ 𝐴 ↔ 𝑋 ⊆ 𝐴))
10		2fveq3 6827	. . . . 5 ⊢ (𝑥 = 𝑋 → ( ⊥ ‘( ⊥ ‘𝑥)) = ( ⊥ ‘( ⊥ ‘𝑋)))
11		id 22	. . . . 5 ⊢ (𝑥 = 𝑋 → 𝑥 = 𝑋)
12	10, 11	eqeq12d 2747	. . . 4 ⊢ (𝑥 = 𝑋 → (( ⊥ ‘( ⊥ ‘𝑥)) = 𝑥 ↔ ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋))
13	9, 12	anbi12d 632	. . 3 ⊢ (𝑥 = 𝑋 → ((𝑥 ⊆ 𝐴 ∧ ( ⊥ ‘( ⊥ ‘𝑥)) = 𝑥) ↔ (𝑋 ⊆ 𝐴 ∧ ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋)))
14	8, 13	elab3 3637	. 2 ⊢ (𝑋 ∈ {𝑥 ∣ (𝑥 ⊆ 𝐴 ∧ ( ⊥ ‘( ⊥ ‘𝑥)) = 𝑥)} ↔ (𝑋 ⊆ 𝐴 ∧ ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋))
15	5, 14	bitrdi 287	1 ⊢ (𝐾 ∈ 𝐷 → (𝑋 ∈ 𝐶 ↔ (𝑋 ⊆ 𝐴 ∧ ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1541   ∈ wcel 2111  {cab 2709  Vcvv 3436   ⊆ wss 3897  ‘cfv 6481  Atomscatm 39372  ⊥_𝑃cpolN 40011  PSubClcpscN 40043

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 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5232  ax-nul 5242  ax-pow 5301  ax-pr 5368

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 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4281  df-if 4473  df-pw 4549  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4857  df-br 5090  df-opab 5152  df-mpt 5171  df-id 5509  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-iota 6437  df-fun 6483  df-fv 6489  df-psubclN 40044

This theorem is referenced by:  psubcliN  40047  psubcli2N  40048  0psubclN  40052  1psubclN  40053  atpsubclN  40054  pmapsubclN  40055  ispsubcl2N  40056  osumclN  40076  pexmidN  40078  pexmidlem6N  40084