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Theorem ispsubclN 38613
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 38612 . . 3 (𝐾𝐷𝐶 = {𝑥 ∣ (𝑥𝐴 ∧ ( ‘( 𝑥)) = 𝑥)})
54eleq2d 2818 . 2 (𝐾𝐷 → (𝑋𝐶𝑋 ∈ {𝑥 ∣ (𝑥𝐴 ∧ ( ‘( 𝑥)) = 𝑥)}))
61fvexi 6892 . . . . 5 𝐴 ∈ V
76ssex 5314 . . . 4 (𝑋𝐴𝑋 ∈ V)
87adantr 481 . . 3 ((𝑋𝐴 ∧ ( ‘( 𝑋)) = 𝑋) → 𝑋 ∈ V)
9 sseq1 4003 . . . 4 (𝑥 = 𝑋 → (𝑥𝐴𝑋𝐴))
10 2fveq3 6883 . . . . 5 (𝑥 = 𝑋 → ( ‘( 𝑥)) = ( ‘( 𝑋)))
11 id 22 . . . . 5 (𝑥 = 𝑋𝑥 = 𝑋)
1210, 11eqeq12d 2747 . . . 4 (𝑥 = 𝑋 → (( ‘( 𝑥)) = 𝑥 ↔ ( ‘( 𝑋)) = 𝑋))
139, 12anbi12d 631 . . 3 (𝑥 = 𝑋 → ((𝑥𝐴 ∧ ( ‘( 𝑥)) = 𝑥) ↔ (𝑋𝐴 ∧ ( ‘( 𝑋)) = 𝑋)))
148, 13elab3 3672 . 2 (𝑋 ∈ {𝑥 ∣ (𝑥𝐴 ∧ ( ‘( 𝑥)) = 𝑥)} ↔ (𝑋𝐴 ∧ ( ‘( 𝑋)) = 𝑋))
155, 14bitrdi 286 1 (𝐾𝐷 → (𝑋𝐶 ↔ (𝑋𝐴 ∧ ( ‘( 𝑋)) = 𝑋)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396   = wceq 1541  wcel 2106  {cab 2708  Vcvv 3473  wss 3944  cfv 6532  Atomscatm 37938  𝑃cpolN 38578  PSubClcpscN 38610
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2702  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4523  df-pw 4598  df-sn 4623  df-pr 4625  df-op 4629  df-uni 4902  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-iota 6484  df-fun 6534  df-fv 6540  df-psubclN 38611
This theorem is referenced by:  psubcliN  38614  psubcli2N  38615  0psubclN  38619  1psubclN  38620  atpsubclN  38621  pmapsubclN  38622  ispsubcl2N  38623  osumclN  38643  pexmidN  38645  pexmidlem6N  38651
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