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Theorem ispsubclN 34041
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 34040 . . 3 (𝐾𝐷𝐶 = {𝑥 ∣ (𝑥𝐴 ∧ ( ‘( 𝑥)) = 𝑥)})
54eleq2d 2669 . 2 (𝐾𝐷 → (𝑋𝐶𝑋 ∈ {𝑥 ∣ (𝑥𝐴 ∧ ( ‘( 𝑥)) = 𝑥)}))
6 fvex 6095 . . . . . 6 (Atoms‘𝐾) ∈ V
71, 6eqeltri 2680 . . . . 5 𝐴 ∈ V
87ssex 4722 . . . 4 (𝑋𝐴𝑋 ∈ V)
98adantr 479 . . 3 ((𝑋𝐴 ∧ ( ‘( 𝑋)) = 𝑋) → 𝑋 ∈ V)
10 sseq1 3585 . . . 4 (𝑥 = 𝑋 → (𝑥𝐴𝑋𝐴))
11 fveq2 6085 . . . . . 6 (𝑥 = 𝑋 → ( 𝑥) = ( 𝑋))
1211fveq2d 6089 . . . . 5 (𝑥 = 𝑋 → ( ‘( 𝑥)) = ( ‘( 𝑋)))
13 id 22 . . . . 5 (𝑥 = 𝑋𝑥 = 𝑋)
1412, 13eqeq12d 2621 . . . 4 (𝑥 = 𝑋 → (( ‘( 𝑥)) = 𝑥 ↔ ( ‘( 𝑋)) = 𝑋))
1510, 14anbi12d 742 . . 3 (𝑥 = 𝑋 → ((𝑥𝐴 ∧ ( ‘( 𝑥)) = 𝑥) ↔ (𝑋𝐴 ∧ ( ‘( 𝑋)) = 𝑋)))
169, 15elab3 3323 . 2 (𝑋 ∈ {𝑥 ∣ (𝑥𝐴 ∧ ( ‘( 𝑥)) = 𝑥)} ↔ (𝑋𝐴 ∧ ( ‘( 𝑋)) = 𝑋))
175, 16syl6bb 274 1 (𝐾𝐷 → (𝑋𝐶 ↔ (𝑋𝐴 ∧ ( ‘( 𝑋)) = 𝑋)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 194  wa 382   = wceq 1474  wcel 1976  {cab 2592  Vcvv 3169  wss 3536  cfv 5787  Atomscatm 33368  𝑃cpolN 34006  PSubClcpscN 34038
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2032  ax-13 2229  ax-ext 2586  ax-sep 4700  ax-nul 4709  ax-pow 4761  ax-pr 4825
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2458  df-mo 2459  df-clab 2593  df-cleq 2599  df-clel 2602  df-nfc 2736  df-ral 2897  df-rex 2898  df-rab 2901  df-v 3171  df-sbc 3399  df-dif 3539  df-un 3541  df-in 3543  df-ss 3550  df-nul 3871  df-if 4033  df-pw 4106  df-sn 4122  df-pr 4124  df-op 4128  df-uni 4364  df-br 4575  df-opab 4635  df-mpt 4636  df-id 4940  df-xp 5031  df-rel 5032  df-cnv 5033  df-co 5034  df-dm 5035  df-iota 5751  df-fun 5789  df-fv 5795  df-psubclN 34039
This theorem is referenced by:  psubcliN  34042  psubcli2N  34043  0psubclN  34047  1psubclN  34048  atpsubclN  34049  pmapsubclN  34050  ispsubcl2N  34051  osumclN  34071  pexmidN  34073  pexmidlem6N  34079
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