MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  tglnunirn Structured version   Visualization version   GIF version

Theorem tglnunirn 27796
Description: Lines are sets of points. (Contributed by Thierry Arnoux, 25-May-2019.)
Hypotheses
Ref Expression
tglng.p 𝑃 = (Baseβ€˜πΊ)
tglng.l 𝐿 = (LineGβ€˜πΊ)
tglng.i 𝐼 = (Itvβ€˜πΊ)
Assertion
Ref Expression
tglnunirn (𝐺 ∈ TarskiG β†’ βˆͺ ran 𝐿 βŠ† 𝑃)

Proof of Theorem tglnunirn
Dummy variables 𝑝 π‘₯ 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 tglng.p . . . . . . . 8 𝑃 = (Baseβ€˜πΊ)
2 tglng.l . . . . . . . 8 𝐿 = (LineGβ€˜πΊ)
3 tglng.i . . . . . . . 8 𝐼 = (Itvβ€˜πΊ)
41, 2, 3tglng 27794 . . . . . . 7 (𝐺 ∈ TarskiG β†’ 𝐿 = (π‘₯ ∈ 𝑃, 𝑦 ∈ (𝑃 βˆ– {π‘₯}) ↦ {𝑧 ∈ 𝑃 ∣ (𝑧 ∈ (π‘₯𝐼𝑦) ∨ π‘₯ ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (π‘₯𝐼𝑧))}))
54rneqd 5937 . . . . . 6 (𝐺 ∈ TarskiG β†’ ran 𝐿 = ran (π‘₯ ∈ 𝑃, 𝑦 ∈ (𝑃 βˆ– {π‘₯}) ↦ {𝑧 ∈ 𝑃 ∣ (𝑧 ∈ (π‘₯𝐼𝑦) ∨ π‘₯ ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (π‘₯𝐼𝑧))}))
65eleq2d 2819 . . . . 5 (𝐺 ∈ TarskiG β†’ (𝑝 ∈ ran 𝐿 ↔ 𝑝 ∈ ran (π‘₯ ∈ 𝑃, 𝑦 ∈ (𝑃 βˆ– {π‘₯}) ↦ {𝑧 ∈ 𝑃 ∣ (𝑧 ∈ (π‘₯𝐼𝑦) ∨ π‘₯ ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (π‘₯𝐼𝑧))})))
76biimpa 477 . . . 4 ((𝐺 ∈ TarskiG ∧ 𝑝 ∈ ran 𝐿) β†’ 𝑝 ∈ ran (π‘₯ ∈ 𝑃, 𝑦 ∈ (𝑃 βˆ– {π‘₯}) ↦ {𝑧 ∈ 𝑃 ∣ (𝑧 ∈ (π‘₯𝐼𝑦) ∨ π‘₯ ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (π‘₯𝐼𝑧))}))
8 eqid 2732 . . . . . 6 (π‘₯ ∈ 𝑃, 𝑦 ∈ (𝑃 βˆ– {π‘₯}) ↦ {𝑧 ∈ 𝑃 ∣ (𝑧 ∈ (π‘₯𝐼𝑦) ∨ π‘₯ ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (π‘₯𝐼𝑧))}) = (π‘₯ ∈ 𝑃, 𝑦 ∈ (𝑃 βˆ– {π‘₯}) ↦ {𝑧 ∈ 𝑃 ∣ (𝑧 ∈ (π‘₯𝐼𝑦) ∨ π‘₯ ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (π‘₯𝐼𝑧))})
91fvexi 6905 . . . . . . 7 𝑃 ∈ V
109rabex 5332 . . . . . 6 {𝑧 ∈ 𝑃 ∣ (𝑧 ∈ (π‘₯𝐼𝑦) ∨ π‘₯ ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (π‘₯𝐼𝑧))} ∈ V
118, 10elrnmpo 7544 . . . . 5 (𝑝 ∈ ran (π‘₯ ∈ 𝑃, 𝑦 ∈ (𝑃 βˆ– {π‘₯}) ↦ {𝑧 ∈ 𝑃 ∣ (𝑧 ∈ (π‘₯𝐼𝑦) ∨ π‘₯ ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (π‘₯𝐼𝑧))}) ↔ βˆƒπ‘₯ ∈ 𝑃 βˆƒπ‘¦ ∈ (𝑃 βˆ– {π‘₯})𝑝 = {𝑧 ∈ 𝑃 ∣ (𝑧 ∈ (π‘₯𝐼𝑦) ∨ π‘₯ ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (π‘₯𝐼𝑧))})
12 ssrab2 4077 . . . . . . . 8 {𝑧 ∈ 𝑃 ∣ (𝑧 ∈ (π‘₯𝐼𝑦) ∨ π‘₯ ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (π‘₯𝐼𝑧))} βŠ† 𝑃
13 sseq1 4007 . . . . . . . 8 (𝑝 = {𝑧 ∈ 𝑃 ∣ (𝑧 ∈ (π‘₯𝐼𝑦) ∨ π‘₯ ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (π‘₯𝐼𝑧))} β†’ (𝑝 βŠ† 𝑃 ↔ {𝑧 ∈ 𝑃 ∣ (𝑧 ∈ (π‘₯𝐼𝑦) ∨ π‘₯ ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (π‘₯𝐼𝑧))} βŠ† 𝑃))
1412, 13mpbiri 257 . . . . . . 7 (𝑝 = {𝑧 ∈ 𝑃 ∣ (𝑧 ∈ (π‘₯𝐼𝑦) ∨ π‘₯ ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (π‘₯𝐼𝑧))} β†’ 𝑝 βŠ† 𝑃)
1514rexlimivw 3151 . . . . . 6 (βˆƒπ‘¦ ∈ (𝑃 βˆ– {π‘₯})𝑝 = {𝑧 ∈ 𝑃 ∣ (𝑧 ∈ (π‘₯𝐼𝑦) ∨ π‘₯ ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (π‘₯𝐼𝑧))} β†’ 𝑝 βŠ† 𝑃)
1615rexlimivw 3151 . . . . 5 (βˆƒπ‘₯ ∈ 𝑃 βˆƒπ‘¦ ∈ (𝑃 βˆ– {π‘₯})𝑝 = {𝑧 ∈ 𝑃 ∣ (𝑧 ∈ (π‘₯𝐼𝑦) ∨ π‘₯ ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (π‘₯𝐼𝑧))} β†’ 𝑝 βŠ† 𝑃)
1711, 16sylbi 216 . . . 4 (𝑝 ∈ ran (π‘₯ ∈ 𝑃, 𝑦 ∈ (𝑃 βˆ– {π‘₯}) ↦ {𝑧 ∈ 𝑃 ∣ (𝑧 ∈ (π‘₯𝐼𝑦) ∨ π‘₯ ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (π‘₯𝐼𝑧))}) β†’ 𝑝 βŠ† 𝑃)
187, 17syl 17 . . 3 ((𝐺 ∈ TarskiG ∧ 𝑝 ∈ ran 𝐿) β†’ 𝑝 βŠ† 𝑃)
1918ralrimiva 3146 . 2 (𝐺 ∈ TarskiG β†’ βˆ€π‘ ∈ ran 𝐿 𝑝 βŠ† 𝑃)
20 unissb 4943 . 2 (βˆͺ ran 𝐿 βŠ† 𝑃 ↔ βˆ€π‘ ∈ ran 𝐿 𝑝 βŠ† 𝑃)
2119, 20sylibr 233 1 (𝐺 ∈ TarskiG β†’ βˆͺ ran 𝐿 βŠ† 𝑃)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 396   ∨ w3o 1086   = wceq 1541   ∈ wcel 2106  βˆ€wral 3061  βˆƒwrex 3070  {crab 3432   βˆ– cdif 3945   βŠ† wss 3948  {csn 4628  βˆͺ cuni 4908  ran crn 5677  β€˜cfv 6543  (class class class)co 7408   ∈ cmpo 7410  Basecbs 17143  TarskiGcstrkg 27675  Itvcitv 27681  LineGclng 27682
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pr 5427
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-sbc 3778  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-cnv 5684  df-dm 5686  df-rn 5687  df-iota 6495  df-fv 6551  df-ov 7411  df-oprab 7412  df-mpo 7413  df-trkg 27701
This theorem is referenced by:  tglnpt  27797  tglineintmo  27890
  Copyright terms: Public domain W3C validator