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Theorem isline 35814
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 35813 . . 3 (𝐾𝐷𝑁 = {𝑥 ∣ ∃𝑞𝐴𝑟𝐴 (𝑞𝑟𝑥 = {𝑝𝐴𝑝 (𝑞 𝑟)})})
65eleq2d 2892 . 2 (𝐾𝐷 → (𝑋𝑁𝑋 ∈ {𝑥 ∣ ∃𝑞𝐴𝑟𝐴 (𝑞𝑟𝑥 = {𝑝𝐴𝑝 (𝑞 𝑟)})}))
73fvexi 6447 . . . . . . . 8 𝐴 ∈ V
87rabex 5037 . . . . . . 7 {𝑝𝐴𝑝 (𝑞 𝑟)} ∈ V
9 eleq1 2894 . . . . . . 7 (𝑋 = {𝑝𝐴𝑝 (𝑞 𝑟)} → (𝑋 ∈ V ↔ {𝑝𝐴𝑝 (𝑞 𝑟)} ∈ V))
108, 9mpbiri 250 . . . . . 6 (𝑋 = {𝑝𝐴𝑝 (𝑞 𝑟)} → 𝑋 ∈ V)
1110adantl 475 . . . . 5 ((𝑞𝑟𝑋 = {𝑝𝐴𝑝 (𝑞 𝑟)}) → 𝑋 ∈ V)
1211a1i 11 . . . 4 ((𝑞𝐴𝑟𝐴) → ((𝑞𝑟𝑋 = {𝑝𝐴𝑝 (𝑞 𝑟)}) → 𝑋 ∈ V))
1312rexlimivv 3246 . . 3 (∃𝑞𝐴𝑟𝐴 (𝑞𝑟𝑋 = {𝑝𝐴𝑝 (𝑞 𝑟)}) → 𝑋 ∈ V)
14 eqeq1 2829 . . . . 5 (𝑥 = 𝑋 → (𝑥 = {𝑝𝐴𝑝 (𝑞 𝑟)} ↔ 𝑋 = {𝑝𝐴𝑝 (𝑞 𝑟)}))
1514anbi2d 624 . . . 4 (𝑥 = 𝑋 → ((𝑞𝑟𝑥 = {𝑝𝐴𝑝 (𝑞 𝑟)}) ↔ (𝑞𝑟𝑋 = {𝑝𝐴𝑝 (𝑞 𝑟)})))
16152rexbidv 3267 . . 3 (𝑥 = 𝑋 → (∃𝑞𝐴𝑟𝐴 (𝑞𝑟𝑥 = {𝑝𝐴𝑝 (𝑞 𝑟)}) ↔ ∃𝑞𝐴𝑟𝐴 (𝑞𝑟𝑋 = {𝑝𝐴𝑝 (𝑞 𝑟)})))
1713, 16elab3 3579 . 2 (𝑋 ∈ {𝑥 ∣ ∃𝑞𝐴𝑟𝐴 (𝑞𝑟𝑥 = {𝑝𝐴𝑝 (𝑞 𝑟)})} ↔ ∃𝑞𝐴𝑟𝐴 (𝑞𝑟𝑋 = {𝑝𝐴𝑝 (𝑞 𝑟)}))
186, 17syl6bb 279 1 (𝐾𝐷 → (𝑋𝑁 ↔ ∃𝑞𝐴𝑟𝐴 (𝑞𝑟𝑋 = {𝑝𝐴𝑝 (𝑞 𝑟)})))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 386   = wceq 1658  wcel 2166  {cab 2811  wne 2999  wrex 3118  {crab 3121  Vcvv 3414   class class class wbr 4873  cfv 6123  (class class class)co 6905  lecple 16312  joincjn 17297  Atomscatm 35338  Linesclines 35569
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-8 2168  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2803  ax-rep 4994  ax-sep 5005  ax-nul 5013  ax-pr 5127  ax-un 7209
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-ral 3122  df-rex 3123  df-reu 3124  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4145  df-if 4307  df-sn 4398  df-pr 4400  df-op 4404  df-uni 4659  df-iun 4742  df-br 4874  df-opab 4936  df-mpt 4953  df-id 5250  df-xp 5348  df-rel 5349  df-cnv 5350  df-co 5351  df-dm 5352  df-rn 5353  df-res 5354  df-ima 5355  df-iota 6086  df-fun 6125  df-fn 6126  df-f 6127  df-f1 6128  df-fo 6129  df-f1o 6130  df-fv 6131  df-ov 6908  df-lines 35576
This theorem is referenced by:  islinei  35815  linepsubN  35827  isline2  35849
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