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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 = (⊥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 39919 . . 3 (𝐾𝐷𝐶 = {𝑥 ∣ (𝑥𝐴 ∧ ( ‘( 𝑥)) = 𝑥)})
54eleq2d 2825 . 2 (𝐾𝐷 → (𝑋𝐶𝑋 ∈ {𝑥 ∣ (𝑥𝐴 ∧ ( ‘( 𝑥)) = 𝑥)}))
61fvexi 6921 . . . . 5 𝐴 ∈ V
76ssex 5327 . . . 4 (𝑋𝐴𝑋 ∈ V)
87adantr 480 . . 3 ((𝑋𝐴 ∧ ( ‘( 𝑋)) = 𝑋) → 𝑋 ∈ V)
9 sseq1 4021 . . . 4 (𝑥 = 𝑋 → (𝑥𝐴𝑋𝐴))
10 2fveq3 6912 . . . . 5 (𝑥 = 𝑋 → ( ‘( 𝑥)) = ( ‘( 𝑋)))
11 id 22 . . . . 5 (𝑥 = 𝑋𝑥 = 𝑋)
1210, 11eqeq12d 2751 . . . 4 (𝑥 = 𝑋 → (( ‘( 𝑥)) = 𝑥 ↔ ( ‘( 𝑋)) = 𝑋))
139, 12anbi12d 632 . . 3 (𝑥 = 𝑋 → ((𝑥𝐴 ∧ ( ‘( 𝑥)) = 𝑥) ↔ (𝑋𝐴 ∧ ( ‘( 𝑋)) = 𝑋)))
148, 13elab3 3689 . 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 1537  wcel 2106  {cab 2712  Vcvv 3478  wss 3963  cfv 6563  Atomscatm 39245  𝑃cpolN 39885  PSubClcpscN 39917
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-iota 6516  df-fun 6565  df-fv 6571  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
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