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Theorem tglnunirn 26039
 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 26037 . . . . . . 7 (𝐺 ∈ TarskiG → 𝐿 = (𝑥𝑃, 𝑦 ∈ (𝑃 ∖ {𝑥}) ↦ {𝑧𝑃 ∣ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))}))
54rneqd 5652 . . . . . 6 (𝐺 ∈ TarskiG → ran 𝐿 = ran (𝑥𝑃, 𝑦 ∈ (𝑃 ∖ {𝑥}) ↦ {𝑧𝑃 ∣ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))}))
65eleq2d 2851 . . . . 5 (𝐺 ∈ TarskiG → (𝑝 ∈ ran 𝐿𝑝 ∈ ran (𝑥𝑃, 𝑦 ∈ (𝑃 ∖ {𝑥}) ↦ {𝑧𝑃 ∣ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))})))
76biimpa 469 . . . 4 ((𝐺 ∈ TarskiG ∧ 𝑝 ∈ ran 𝐿) → 𝑝 ∈ ran (𝑥𝑃, 𝑦 ∈ (𝑃 ∖ {𝑥}) ↦ {𝑧𝑃 ∣ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))}))
8 eqid 2778 . . . . . 6 (𝑥𝑃, 𝑦 ∈ (𝑃 ∖ {𝑥}) ↦ {𝑧𝑃 ∣ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))}) = (𝑥𝑃, 𝑦 ∈ (𝑃 ∖ {𝑥}) ↦ {𝑧𝑃 ∣ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))})
91fvexi 6515 . . . . . . 7 𝑃 ∈ V
109rabex 5092 . . . . . 6 {𝑧𝑃 ∣ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))} ∈ V
118, 10elrnmpo 7105 . . . . 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))})
12 ssrab2 3948 . . . . . . . 8 {𝑧𝑃 ∣ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))} ⊆ 𝑃
13 sseq1 3884 . . . . . . . 8 (𝑝 = {𝑧𝑃 ∣ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))} → (𝑝𝑃 ↔ {𝑧𝑃 ∣ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))} ⊆ 𝑃))
1412, 13mpbiri 250 . . . . . . 7 (𝑝 = {𝑧𝑃 ∣ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))} → 𝑝𝑃)
1514rexlimivw 3227 . . . . . 6 (∃𝑦 ∈ (𝑃 ∖ {𝑥})𝑝 = {𝑧𝑃 ∣ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))} → 𝑝𝑃)
1615rexlimivw 3227 . . . . 5 (∃𝑥𝑃𝑦 ∈ (𝑃 ∖ {𝑥})𝑝 = {𝑧𝑃 ∣ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))} → 𝑝𝑃)
1711, 16sylbi 209 . . . 4 (𝑝 ∈ ran (𝑥𝑃, 𝑦 ∈ (𝑃 ∖ {𝑥}) ↦ {𝑧𝑃 ∣ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))}) → 𝑝𝑃)
187, 17syl 17 . . 3 ((𝐺 ∈ TarskiG ∧ 𝑝 ∈ ran 𝐿) → 𝑝𝑃)
1918ralrimiva 3132 . 2 (𝐺 ∈ TarskiG → ∀𝑝 ∈ ran 𝐿 𝑝𝑃)
20 unissb 4744 . 2 ( ran 𝐿𝑃 ↔ ∀𝑝 ∈ ran 𝐿 𝑝𝑃)
2119, 20sylibr 226 1 (𝐺 ∈ TarskiG → ran 𝐿𝑃)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 387   ∨ w3o 1067   = wceq 1507   ∈ wcel 2050  ∀wral 3088  ∃wrex 3089  {crab 3092   ∖ cdif 3828   ⊆ wss 3831  {csn 4442  ∪ cuni 4713  ran crn 5409  ‘cfv 6190  (class class class)co 6978   ∈ cmpo 6980  Basecbs 16342  TarskiGcstrkg 25921  Itvcitv 25927  LineGclng 25928 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2750  ax-sep 5061  ax-nul 5068  ax-pr 5187 This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3or 1069  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-eu 2583  df-clab 2759  df-cleq 2771  df-clel 2846  df-nfc 2918  df-ral 3093  df-rex 3094  df-rab 3097  df-v 3417  df-sbc 3684  df-dif 3834  df-un 3836  df-in 3838  df-ss 3845  df-nul 4181  df-if 4352  df-sn 4443  df-pr 4445  df-op 4449  df-uni 4714  df-br 4931  df-opab 4993  df-cnv 5416  df-dm 5418  df-rn 5419  df-iota 6154  df-fv 6198  df-ov 6981  df-oprab 6982  df-mpo 6983  df-trkg 25944 This theorem is referenced by:  tglnpt  26040  tglineintmo  26133
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