Theorem ispsubclN 40811

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 40810	. . 3 ⊢ (𝐾 ∈ 𝐷 → 𝐶 = {𝑥 ∣ (𝑥 ⊆ 𝐴 ∧ ( ⊥ ‘( ⊥ ‘𝑥)) = 𝑥)})
5	4	eleq2d 2846	. 2 ⊢ (𝐾 ∈ 𝐷 → (𝑋 ∈ 𝐶 ↔ 𝑋 ∈ {𝑥 ∣ (𝑥 ⊆ 𝐴 ∧ ( ⊥ ‘( ⊥ ‘𝑥)) = 𝑥)}))
6	1	fvexi 6893	. . . . 5 ⊢ 𝐴 ∈ V
7	6	ssex 5285	. . . 4 ⊢ (𝑋 ⊆ 𝐴 → 𝑋 ∈ V)
8	7	adantr 486	. . 3 ⊢ ((𝑋 ⊆ 𝐴 ∧ ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋) → 𝑋 ∈ V)
9		sseq1 3956	. . . 4 ⊢ (𝑥 = 𝑋 → (𝑥 ⊆ 𝐴 ↔ 𝑋 ⊆ 𝐴))
10		2fveq3 6884	. . . . 5 ⊢ (𝑥 = 𝑋 → ( ⊥ ‘( ⊥ ‘𝑥)) = ( ⊥ ‘( ⊥ ‘𝑋)))
11		id 23	. . . . 5 ⊢ (𝑥 = 𝑋 → 𝑥 = 𝑋)
12	10, 11	eqeq12d 2776	. . . 4 ⊢ (𝑥 = 𝑋 → (( ⊥ ‘( ⊥ ‘𝑥)) = 𝑥 ↔ ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋))
13	9, 12	anbi12d 644	. . 3 ⊢ (𝑥 = 𝑋 → ((𝑥 ⊆ 𝐴 ∧ ( ⊥ ‘( ⊥ ‘𝑥)) = 𝑥) ↔ (𝑋 ⊆ 𝐴 ∧ ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋)))
14	8, 13	elab3 3640	. 2 ⊢ (𝑋 ∈ {𝑥 ∣ (𝑥 ⊆ 𝐴 ∧ ( ⊥ ‘( ⊥ ‘𝑥)) = 𝑥)} ↔ (𝑋 ⊆ 𝐴 ∧ ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋))
15	5, 14	bitrdi 290	1 ⊢ (𝐾 ∈ 𝐷 → (𝑋 ∈ 𝐶 ↔ (𝑋 ⊆ 𝐴 ∧ ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2145  {cab 2738  Vcvv 3450   ⊆ wss 3899  ‘cfv 6533  Atomscatm 40137  ⊥_𝑃cpolN 40776  PSubClcpscN 40808

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-sep 5251  ax-nul 5263  ax-pow 5330  ax-pr 5398

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-iota 6489  df-fun 6535  df-fv 6541  df-psubclN 40809

This theorem is used by:  psubcliN  40812  psubcli2N  40813  0psubclN  40817  1psubclN  40818  atpsubclN  40819  pmapsubclN  40820  ispsubcl2N  40821  osumclN  40841  pexmidN  40843  pexmidlem6N  40849