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Theorem psubclsetN 35540
Description: The set of closed projective subspaces in a Hilbert lattice. (Contributed by NM, 23-Jan-2012.) (New usage is discouraged.)
Hypotheses
Ref Expression
psubclset.a 𝐴 = (Atoms‘𝐾)
psubclset.p = (⊥𝑃𝐾)
psubclset.c 𝐶 = (PSubCl‘𝐾)
Assertion
Ref Expression
psubclsetN (𝐾𝐵𝐶 = {𝑠 ∣ (𝑠𝐴 ∧ ( ‘( 𝑠)) = 𝑠)})
Distinct variable groups:   𝐴,𝑠   𝐾,𝑠
Allowed substitution hints:   𝐵(𝑠)   𝐶(𝑠)   (𝑠)

Proof of Theorem psubclsetN
Dummy variable 𝑘 is distinct from all other variables.
StepHypRef Expression
1 elex 3243 . 2 (𝐾𝐵𝐾 ∈ V)
2 psubclset.c . . 3 𝐶 = (PSubCl‘𝐾)
3 fveq2 6229 . . . . . . . 8 (𝑘 = 𝐾 → (Atoms‘𝑘) = (Atoms‘𝐾))
4 psubclset.a . . . . . . . 8 𝐴 = (Atoms‘𝐾)
53, 4syl6eqr 2703 . . . . . . 7 (𝑘 = 𝐾 → (Atoms‘𝑘) = 𝐴)
65sseq2d 3666 . . . . . 6 (𝑘 = 𝐾 → (𝑠 ⊆ (Atoms‘𝑘) ↔ 𝑠𝐴))
7 fveq2 6229 . . . . . . . . 9 (𝑘 = 𝐾 → (⊥𝑃𝑘) = (⊥𝑃𝐾))
8 psubclset.p . . . . . . . . 9 = (⊥𝑃𝐾)
97, 8syl6eqr 2703 . . . . . . . 8 (𝑘 = 𝐾 → (⊥𝑃𝑘) = )
109fveq1d 6231 . . . . . . . 8 (𝑘 = 𝐾 → ((⊥𝑃𝑘)‘𝑠) = ( 𝑠))
119, 10fveq12d 6235 . . . . . . 7 (𝑘 = 𝐾 → ((⊥𝑃𝑘)‘((⊥𝑃𝑘)‘𝑠)) = ( ‘( 𝑠)))
1211eqeq1d 2653 . . . . . 6 (𝑘 = 𝐾 → (((⊥𝑃𝑘)‘((⊥𝑃𝑘)‘𝑠)) = 𝑠 ↔ ( ‘( 𝑠)) = 𝑠))
136, 12anbi12d 747 . . . . 5 (𝑘 = 𝐾 → ((𝑠 ⊆ (Atoms‘𝑘) ∧ ((⊥𝑃𝑘)‘((⊥𝑃𝑘)‘𝑠)) = 𝑠) ↔ (𝑠𝐴 ∧ ( ‘( 𝑠)) = 𝑠)))
1413abbidv 2770 . . . 4 (𝑘 = 𝐾 → {𝑠 ∣ (𝑠 ⊆ (Atoms‘𝑘) ∧ ((⊥𝑃𝑘)‘((⊥𝑃𝑘)‘𝑠)) = 𝑠)} = {𝑠 ∣ (𝑠𝐴 ∧ ( ‘( 𝑠)) = 𝑠)})
15 df-psubclN 35539 . . . 4 PSubCl = (𝑘 ∈ V ↦ {𝑠 ∣ (𝑠 ⊆ (Atoms‘𝑘) ∧ ((⊥𝑃𝑘)‘((⊥𝑃𝑘)‘𝑠)) = 𝑠)})
16 fvex 6239 . . . . . . 7 (Atoms‘𝐾) ∈ V
174, 16eqeltri 2726 . . . . . 6 𝐴 ∈ V
1817pwex 4878 . . . . 5 𝒫 𝐴 ∈ V
19 selpw 4198 . . . . . . . 8 (𝑠 ∈ 𝒫 𝐴𝑠𝐴)
2019anbi1i 731 . . . . . . 7 ((𝑠 ∈ 𝒫 𝐴 ∧ ( ‘( 𝑠)) = 𝑠) ↔ (𝑠𝐴 ∧ ( ‘( 𝑠)) = 𝑠))
2120abbii 2768 . . . . . 6 {𝑠 ∣ (𝑠 ∈ 𝒫 𝐴 ∧ ( ‘( 𝑠)) = 𝑠)} = {𝑠 ∣ (𝑠𝐴 ∧ ( ‘( 𝑠)) = 𝑠)}
22 ssab2 3719 . . . . . 6 {𝑠 ∣ (𝑠 ∈ 𝒫 𝐴 ∧ ( ‘( 𝑠)) = 𝑠)} ⊆ 𝒫 𝐴
2321, 22eqsstr3i 3669 . . . . 5 {𝑠 ∣ (𝑠𝐴 ∧ ( ‘( 𝑠)) = 𝑠)} ⊆ 𝒫 𝐴
2418, 23ssexi 4836 . . . 4 {𝑠 ∣ (𝑠𝐴 ∧ ( ‘( 𝑠)) = 𝑠)} ∈ V
2514, 15, 24fvmpt 6321 . . 3 (𝐾 ∈ V → (PSubCl‘𝐾) = {𝑠 ∣ (𝑠𝐴 ∧ ( ‘( 𝑠)) = 𝑠)})
262, 25syl5eq 2697 . 2 (𝐾 ∈ V → 𝐶 = {𝑠 ∣ (𝑠𝐴 ∧ ( ‘( 𝑠)) = 𝑠)})
271, 26syl 17 1 (𝐾𝐵𝐶 = {𝑠 ∣ (𝑠𝐴 ∧ ( ‘( 𝑠)) = 𝑠)})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1523  wcel 2030  {cab 2637  Vcvv 3231  wss 3607  𝒫 cpw 4191  cfv 5926  Atomscatm 34868  𝑃cpolN 35506  PSubClcpscN 35538
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-iota 5889  df-fun 5928  df-fv 5934  df-psubclN 35539
This theorem is referenced by:  ispsubclN  35541
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