Mathbox for Norm Megill < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  ispsubclN Structured version   Visualization version   GIF version

Theorem ispsubclN 37513
 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 37512 . . 3 (𝐾𝐷𝐶 = {𝑥 ∣ (𝑥𝐴 ∧ ( ‘( 𝑥)) = 𝑥)})
54eleq2d 2837 . 2 (𝐾𝐷 → (𝑋𝐶𝑋 ∈ {𝑥 ∣ (𝑥𝐴 ∧ ( ‘( 𝑥)) = 𝑥)}))
61fvexi 6672 . . . . 5 𝐴 ∈ V
76ssex 5191 . . . 4 (𝑋𝐴𝑋 ∈ V)
87adantr 484 . . 3 ((𝑋𝐴 ∧ ( ‘( 𝑋)) = 𝑋) → 𝑋 ∈ V)
9 sseq1 3917 . . . 4 (𝑥 = 𝑋 → (𝑥𝐴𝑋𝐴))
10 2fveq3 6663 . . . . 5 (𝑥 = 𝑋 → ( ‘( 𝑥)) = ( ‘( 𝑋)))
11 id 22 . . . . 5 (𝑥 = 𝑋𝑥 = 𝑋)
1210, 11eqeq12d 2774 . . . 4 (𝑥 = 𝑋 → (( ‘( 𝑥)) = 𝑥 ↔ ( ‘( 𝑋)) = 𝑋))
139, 12anbi12d 633 . . 3 (𝑥 = 𝑋 → ((𝑥𝐴 ∧ ( ‘( 𝑥)) = 𝑥) ↔ (𝑋𝐴 ∧ ( ‘( 𝑋)) = 𝑋)))
148, 13elab3 3595 . 2 (𝑋 ∈ {𝑥 ∣ (𝑥𝐴 ∧ ( ‘( 𝑥)) = 𝑥)} ↔ (𝑋𝐴 ∧ ( ‘( 𝑋)) = 𝑋))
155, 14bitrdi 290 1 (𝐾𝐷 → (𝑋𝐶 ↔ (𝑋𝐴 ∧ ( ‘( 𝑋)) = 𝑋)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1538   ∈ wcel 2111  {cab 2735  Vcvv 3409   ⊆ wss 3858  ‘cfv 6335  Atomscatm 36839  ⊥𝑃cpolN 37478  PSubClcpscN 37510 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-sep 5169  ax-nul 5176  ax-pow 5234  ax-pr 5298 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ral 3075  df-rex 3076  df-rab 3079  df-v 3411  df-sbc 3697  df-dif 3861  df-un 3863  df-in 3865  df-ss 3875  df-nul 4226  df-if 4421  df-pw 4496  df-sn 4523  df-pr 4525  df-op 4529  df-uni 4799  df-br 5033  df-opab 5095  df-mpt 5113  df-id 5430  df-xp 5530  df-rel 5531  df-cnv 5532  df-co 5533  df-dm 5534  df-iota 6294  df-fun 6337  df-fv 6343  df-psubclN 37511 This theorem is referenced by:  psubcliN  37514  psubcli2N  37515  0psubclN  37519  1psubclN  37520  atpsubclN  37521  pmapsubclN  37522  ispsubcl2N  37523  osumclN  37543  pexmidN  37545  pexmidlem6N  37551
 Copyright terms: Public domain W3C validator