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Theorem tglnunirn 25488
Description: Lines are sets of points. (Contributed by Thierry Arnoux, 25-May-2019.)
Hypotheses
Ref Expression
tglng.p 𝑃 = (Base‘𝐺)
tglng.l 𝐿 = (LineG‘𝐺)
tglng.i 𝐼 = (Itv‘𝐺)
Assertion
Ref Expression
tglnunirn (𝐺 ∈ TarskiG → ran 𝐿𝑃)

Proof of Theorem tglnunirn
Dummy variables 𝑝 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 tglng.p . . . . . . . 8 𝑃 = (Base‘𝐺)
2 tglng.l . . . . . . . 8 𝐿 = (LineG‘𝐺)
3 tglng.i . . . . . . . 8 𝐼 = (Itv‘𝐺)
41, 2, 3tglng 25486 . . . . . . 7 (𝐺 ∈ TarskiG → 𝐿 = (𝑥𝑃, 𝑦 ∈ (𝑃 ∖ {𝑥}) ↦ {𝑧𝑃 ∣ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))}))
54rneqd 5385 . . . . . 6 (𝐺 ∈ TarskiG → ran 𝐿 = ran (𝑥𝑃, 𝑦 ∈ (𝑃 ∖ {𝑥}) ↦ {𝑧𝑃 ∣ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))}))
65eleq2d 2716 . . . . 5 (𝐺 ∈ TarskiG → (𝑝 ∈ ran 𝐿𝑝 ∈ ran (𝑥𝑃, 𝑦 ∈ (𝑃 ∖ {𝑥}) ↦ {𝑧𝑃 ∣ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))})))
76biimpa 500 . . . 4 ((𝐺 ∈ TarskiG ∧ 𝑝 ∈ ran 𝐿) → 𝑝 ∈ ran (𝑥𝑃, 𝑦 ∈ (𝑃 ∖ {𝑥}) ↦ {𝑧𝑃 ∣ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))}))
8 eqid 2651 . . . . . 6 (𝑥𝑃, 𝑦 ∈ (𝑃 ∖ {𝑥}) ↦ {𝑧𝑃 ∣ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))}) = (𝑥𝑃, 𝑦 ∈ (𝑃 ∖ {𝑥}) ↦ {𝑧𝑃 ∣ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))})
9 fvex 6239 . . . . . . . 8 (Base‘𝐺) ∈ V
101, 9eqeltri 2726 . . . . . . 7 𝑃 ∈ V
1110rabex 4845 . . . . . 6 {𝑧𝑃 ∣ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))} ∈ V
128, 11elrnmpt2 6815 . . . . 5 (𝑝 ∈ ran (𝑥𝑃, 𝑦 ∈ (𝑃 ∖ {𝑥}) ↦ {𝑧𝑃 ∣ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))}) ↔ ∃𝑥𝑃𝑦 ∈ (𝑃 ∖ {𝑥})𝑝 = {𝑧𝑃 ∣ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))})
13 ssrab2 3720 . . . . . . . 8 {𝑧𝑃 ∣ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))} ⊆ 𝑃
14 sseq1 3659 . . . . . . . 8 (𝑝 = {𝑧𝑃 ∣ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))} → (𝑝𝑃 ↔ {𝑧𝑃 ∣ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))} ⊆ 𝑃))
1513, 14mpbiri 248 . . . . . . 7 (𝑝 = {𝑧𝑃 ∣ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))} → 𝑝𝑃)
1615rexlimivw 3058 . . . . . 6 (∃𝑦 ∈ (𝑃 ∖ {𝑥})𝑝 = {𝑧𝑃 ∣ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))} → 𝑝𝑃)
1716rexlimivw 3058 . . . . 5 (∃𝑥𝑃𝑦 ∈ (𝑃 ∖ {𝑥})𝑝 = {𝑧𝑃 ∣ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))} → 𝑝𝑃)
1812, 17sylbi 207 . . . 4 (𝑝 ∈ ran (𝑥𝑃, 𝑦 ∈ (𝑃 ∖ {𝑥}) ↦ {𝑧𝑃 ∣ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))}) → 𝑝𝑃)
197, 18syl 17 . . 3 ((𝐺 ∈ TarskiG ∧ 𝑝 ∈ ran 𝐿) → 𝑝𝑃)
2019ralrimiva 2995 . 2 (𝐺 ∈ TarskiG → ∀𝑝 ∈ ran 𝐿 𝑝𝑃)
21 unissb 4501 . 2 ( ran 𝐿𝑃 ↔ ∀𝑝 ∈ ran 𝐿 𝑝𝑃)
2220, 21sylibr 224 1 (𝐺 ∈ TarskiG → ran 𝐿𝑃)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  w3o 1053   = wceq 1523  wcel 2030  wral 2941  wrex 2942  {crab 2945  Vcvv 3231  cdif 3604  wss 3607  {csn 4210   cuni 4468  ran crn 5144  cfv 5926  (class class class)co 6690  cmpt2 6692  Basecbs 15904  TarskiGcstrkg 25374  Itvcitv 25380  LineGclng 25381
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-cnv 5151  df-dm 5153  df-rn 5154  df-iota 5889  df-fv 5934  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-trkg 25397
This theorem is referenced by:  tglnpt  25489  tglineintmo  25582
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