Theorem ispsubclN 39920

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 39919	. . 3 ⊢ (𝐾 ∈ 𝐷 → 𝐶 = {𝑥 ∣ (𝑥 ⊆ 𝐴 ∧ ( ⊥ ‘( ⊥ ‘𝑥)) = 𝑥)})
5	4	eleq2d 2814	. 2 ⊢ (𝐾 ∈ 𝐷 → (𝑋 ∈ 𝐶 ↔ 𝑋 ∈ {𝑥 ∣ (𝑥 ⊆ 𝐴 ∧ ( ⊥ ‘( ⊥ ‘𝑥)) = 𝑥)}))
6	1	fvexi 6836	. . . . 5 ⊢ 𝐴 ∈ V
7	6	ssex 5260	. . . 4 ⊢ (𝑋 ⊆ 𝐴 → 𝑋 ∈ V)
8	7	adantr 480	. . 3 ⊢ ((𝑋 ⊆ 𝐴 ∧ ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋) → 𝑋 ∈ V)
9		sseq1 3961	. . . 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 2745	. . . 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 3642	. 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 1540   ∈ wcel 2109  {cab 2707  Vcvv 3436   ⊆ wss 3903  ‘cfv 6482  Atomscatm 39246  ⊥_𝑃cpolN 39885  PSubClcpscN 39917

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5235  ax-nul 5245  ax-pow 5304  ax-pr 5371

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3395  df-v 3438  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4859  df-br 5093  df-opab 5155  df-mpt 5174  df-id 5514  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-iota 6438  df-fun 6484  df-fv 6490  df-psubclN 39918

This theorem is referenced by:  psubcliN  39921  psubcli2N  39922  0psubclN  39926  1psubclN  39927  atpsubclN  39928  pmapsubclN  39929  ispsubcl2N  39930  osumclN  39950  pexmidN  39952  pexmidlem6N  39958