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Theorem isline 39742
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 39741 . . 3 (𝐾𝐷𝑁 = {𝑥 ∣ ∃𝑞𝐴𝑟𝐴 (𝑞𝑟𝑥 = {𝑝𝐴𝑝 (𝑞 𝑟)})})
65eleq2d 2826 . 2 (𝐾𝐷 → (𝑋𝑁𝑋 ∈ {𝑥 ∣ ∃𝑞𝐴𝑟𝐴 (𝑞𝑟𝑥 = {𝑝𝐴𝑝 (𝑞 𝑟)})}))
73fvexi 6919 . . . . . . . 8 𝐴 ∈ V
87rabex 5338 . . . . . . 7 {𝑝𝐴𝑝 (𝑞 𝑟)} ∈ V
9 eleq1 2828 . . . . . . 7 (𝑋 = {𝑝𝐴𝑝 (𝑞 𝑟)} → (𝑋 ∈ V ↔ {𝑝𝐴𝑝 (𝑞 𝑟)} ∈ V))
108, 9mpbiri 258 . . . . . 6 (𝑋 = {𝑝𝐴𝑝 (𝑞 𝑟)} → 𝑋 ∈ V)
1110adantl 481 . . . . 5 ((𝑞𝑟𝑋 = {𝑝𝐴𝑝 (𝑞 𝑟)}) → 𝑋 ∈ V)
1211a1i 11 . . . 4 ((𝑞𝐴𝑟𝐴) → ((𝑞𝑟𝑋 = {𝑝𝐴𝑝 (𝑞 𝑟)}) → 𝑋 ∈ V))
1312rexlimivv 3200 . . 3 (∃𝑞𝐴𝑟𝐴 (𝑞𝑟𝑋 = {𝑝𝐴𝑝 (𝑞 𝑟)}) → 𝑋 ∈ V)
14 eqeq1 2740 . . . . 5 (𝑥 = 𝑋 → (𝑥 = {𝑝𝐴𝑝 (𝑞 𝑟)} ↔ 𝑋 = {𝑝𝐴𝑝 (𝑞 𝑟)}))
1514anbi2d 630 . . . 4 (𝑥 = 𝑋 → ((𝑞𝑟𝑥 = {𝑝𝐴𝑝 (𝑞 𝑟)}) ↔ (𝑞𝑟𝑋 = {𝑝𝐴𝑝 (𝑞 𝑟)})))
16152rexbidv 3221 . . 3 (𝑥 = 𝑋 → (∃𝑞𝐴𝑟𝐴 (𝑞𝑟𝑥 = {𝑝𝐴𝑝 (𝑞 𝑟)}) ↔ ∃𝑞𝐴𝑟𝐴 (𝑞𝑟𝑋 = {𝑝𝐴𝑝 (𝑞 𝑟)})))
1713, 16elab3 3685 . 2 (𝑋 ∈ {𝑥 ∣ ∃𝑞𝐴𝑟𝐴 (𝑞𝑟𝑥 = {𝑝𝐴𝑝 (𝑞 𝑟)})} ↔ ∃𝑞𝐴𝑟𝐴 (𝑞𝑟𝑋 = {𝑝𝐴𝑝 (𝑞 𝑟)}))
186, 17bitrdi 287 1 (𝐾𝐷 → (𝑋𝑁 ↔ ∃𝑞𝐴𝑟𝐴 (𝑞𝑟𝑋 = {𝑝𝐴𝑝 (𝑞 𝑟)})))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1539  wcel 2107  {cab 2713  wne 2939  wrex 3069  {crab 3435  Vcvv 3479   class class class wbr 5142  cfv 6560  (class class class)co 7432  lecple 17305  joincjn 18358  Atomscatm 39265  Linesclines 39497
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-rep 5278  ax-sep 5295  ax-nul 5305  ax-pr 5431  ax-un 7756
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-iun 4992  df-br 5143  df-opab 5205  df-mpt 5225  df-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-iota 6513  df-fun 6562  df-fv 6568  df-ov 7435  df-lines 39504
This theorem is referenced by:  islinei  39743  linepsubN  39755  isline2  39777
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