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Theorem ispsubclN 39939
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 = (⊥𝑃𝐾)
psubclset.c 𝐶 = (PSubCl‘𝐾)
Assertion
Ref Expression
ispsubclN (𝐾𝐷 → (𝑋𝐶 ↔ (𝑋𝐴 ∧ ( ‘( 𝑋)) = 𝑋)))

Proof of Theorem ispsubclN
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 psubclset.a . . . 4 𝐴 = (Atoms‘𝐾)
2 psubclset.p . . . 4 = (⊥𝑃𝐾)
3 psubclset.c . . . 4 𝐶 = (PSubCl‘𝐾)
41, 2, 3psubclsetN 39938 . . 3 (𝐾𝐷𝐶 = {𝑥 ∣ (𝑥𝐴 ∧ ( ‘( 𝑥)) = 𝑥)})
54eleq2d 2827 . 2 (𝐾𝐷 → (𝑋𝐶𝑋 ∈ {𝑥 ∣ (𝑥𝐴 ∧ ( ‘( 𝑥)) = 𝑥)}))
61fvexi 6920 . . . . 5 𝐴 ∈ V
76ssex 5321 . . . 4 (𝑋𝐴𝑋 ∈ V)
87adantr 480 . . 3 ((𝑋𝐴 ∧ ( ‘( 𝑋)) = 𝑋) → 𝑋 ∈ V)
9 sseq1 4009 . . . 4 (𝑥 = 𝑋 → (𝑥𝐴𝑋𝐴))
10 2fveq3 6911 . . . . 5 (𝑥 = 𝑋 → ( ‘( 𝑥)) = ( ‘( 𝑋)))
11 id 22 . . . . 5 (𝑥 = 𝑋𝑥 = 𝑋)
1210, 11eqeq12d 2753 . . . 4 (𝑥 = 𝑋 → (( ‘( 𝑥)) = 𝑥 ↔ ( ‘( 𝑋)) = 𝑋))
139, 12anbi12d 632 . . 3 (𝑥 = 𝑋 → ((𝑥𝐴 ∧ ( ‘( 𝑥)) = 𝑥) ↔ (𝑋𝐴 ∧ ( ‘( 𝑋)) = 𝑋)))
148, 13elab3 3686 . 2 (𝑋 ∈ {𝑥 ∣ (𝑥𝐴 ∧ ( ‘( 𝑥)) = 𝑥)} ↔ (𝑋𝐴 ∧ ( ‘( 𝑋)) = 𝑋))
155, 14bitrdi 287 1 (𝐾𝐷 → (𝑋𝐶 ↔ (𝑋𝐴 ∧ ( ‘( 𝑋)) = 𝑋)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1540  wcel 2108  {cab 2714  Vcvv 3480  wss 3951  cfv 6561  Atomscatm 39264  𝑃cpolN 39904  PSubClcpscN 39936
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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-iota 6514  df-fun 6563  df-fv 6569  df-psubclN 39937
This theorem is referenced by:  psubcliN  39940  psubcli2N  39941  0psubclN  39945  1psubclN  39946  atpsubclN  39947  pmapsubclN  39948  ispsubcl2N  39949  osumclN  39969  pexmidN  39971  pexmidlem6N  39977
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