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Theorem isline 36983
Description: The predicate "is a line". (Contributed by NM, 19-Sep-2011.)
Hypotheses
Ref Expression
isline.l = (le‘𝐾)
isline.j = (join‘𝐾)
isline.a 𝐴 = (Atoms‘𝐾)
isline.n 𝑁 = (Lines‘𝐾)
Assertion
Ref Expression
isline (𝐾𝐷 → (𝑋𝑁 ↔ ∃𝑞𝐴𝑟𝐴 (𝑞𝑟𝑋 = {𝑝𝐴𝑝 (𝑞 𝑟)})))
Distinct variable groups:   𝑞,𝑝,𝑟,𝐴   𝐾,𝑝,𝑞,𝑟   𝑋,𝑞,𝑟
Allowed substitution hints:   𝐷(𝑟,𝑞,𝑝)   (𝑟,𝑞,𝑝)   (𝑟,𝑞,𝑝)   𝑁(𝑟,𝑞,𝑝)   𝑋(𝑝)

Proof of Theorem isline
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 isline.l . . . 4 = (le‘𝐾)
2 isline.j . . . 4 = (join‘𝐾)
3 isline.a . . . 4 𝐴 = (Atoms‘𝐾)
4 isline.n . . . 4 𝑁 = (Lines‘𝐾)
51, 2, 3, 4lineset 36982 . . 3 (𝐾𝐷𝑁 = {𝑥 ∣ ∃𝑞𝐴𝑟𝐴 (𝑞𝑟𝑥 = {𝑝𝐴𝑝 (𝑞 𝑟)})})
65eleq2d 2901 . 2 (𝐾𝐷 → (𝑋𝑁𝑋 ∈ {𝑥 ∣ ∃𝑞𝐴𝑟𝐴 (𝑞𝑟𝑥 = {𝑝𝐴𝑝 (𝑞 𝑟)})}))
73fvexi 6675 . . . . . . . 8 𝐴 ∈ V
87rabex 5221 . . . . . . 7 {𝑝𝐴𝑝 (𝑞 𝑟)} ∈ V
9 eleq1 2903 . . . . . . 7 (𝑋 = {𝑝𝐴𝑝 (𝑞 𝑟)} → (𝑋 ∈ V ↔ {𝑝𝐴𝑝 (𝑞 𝑟)} ∈ V))
108, 9mpbiri 261 . . . . . 6 (𝑋 = {𝑝𝐴𝑝 (𝑞 𝑟)} → 𝑋 ∈ V)
1110adantl 485 . . . . 5 ((𝑞𝑟𝑋 = {𝑝𝐴𝑝 (𝑞 𝑟)}) → 𝑋 ∈ V)
1211a1i 11 . . . 4 ((𝑞𝐴𝑟𝐴) → ((𝑞𝑟𝑋 = {𝑝𝐴𝑝 (𝑞 𝑟)}) → 𝑋 ∈ V))
1312rexlimivv 3284 . . 3 (∃𝑞𝐴𝑟𝐴 (𝑞𝑟𝑋 = {𝑝𝐴𝑝 (𝑞 𝑟)}) → 𝑋 ∈ V)
14 eqeq1 2828 . . . . 5 (𝑥 = 𝑋 → (𝑥 = {𝑝𝐴𝑝 (𝑞 𝑟)} ↔ 𝑋 = {𝑝𝐴𝑝 (𝑞 𝑟)}))
1514anbi2d 631 . . . 4 (𝑥 = 𝑋 → ((𝑞𝑟𝑥 = {𝑝𝐴𝑝 (𝑞 𝑟)}) ↔ (𝑞𝑟𝑋 = {𝑝𝐴𝑝 (𝑞 𝑟)})))
16152rexbidv 3292 . . 3 (𝑥 = 𝑋 → (∃𝑞𝐴𝑟𝐴 (𝑞𝑟𝑥 = {𝑝𝐴𝑝 (𝑞 𝑟)}) ↔ ∃𝑞𝐴𝑟𝐴 (𝑞𝑟𝑋 = {𝑝𝐴𝑝 (𝑞 𝑟)})))
1713, 16elab3 3660 . 2 (𝑋 ∈ {𝑥 ∣ ∃𝑞𝐴𝑟𝐴 (𝑞𝑟𝑥 = {𝑝𝐴𝑝 (𝑞 𝑟)})} ↔ ∃𝑞𝐴𝑟𝐴 (𝑞𝑟𝑋 = {𝑝𝐴𝑝 (𝑞 𝑟)}))
186, 17syl6bb 290 1 (𝐾𝐷 → (𝑋𝑁 ↔ ∃𝑞𝐴𝑟𝐴 (𝑞𝑟𝑋 = {𝑝𝐴𝑝 (𝑞 𝑟)})))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1538  wcel 2115  {cab 2802  wne 3014  wrex 3134  {crab 3137  Vcvv 3480   class class class wbr 5052  cfv 6343  (class class class)co 7149  lecple 16572  joincjn 17554  Atomscatm 36507  Linesclines 36738
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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-rep 5176  ax-sep 5189  ax-nul 5196  ax-pr 5317  ax-un 7455
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-reu 3140  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4825  df-iun 4907  df-br 5053  df-opab 5115  df-mpt 5133  df-id 5447  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-rn 5553  df-res 5554  df-ima 5555  df-iota 6302  df-fun 6345  df-fn 6346  df-f 6347  df-f1 6348  df-fo 6349  df-f1o 6350  df-fv 6351  df-ov 7152  df-lines 36745
This theorem is referenced by:  islinei  36984  linepsubN  36996  isline2  37018
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