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Theorem tglnunirn 27539
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 27537 . . . . . . 7 (𝐺 ∈ TarskiG β†’ 𝐿 = (π‘₯ ∈ 𝑃, 𝑦 ∈ (𝑃 βˆ– {π‘₯}) ↦ {𝑧 ∈ 𝑃 ∣ (𝑧 ∈ (π‘₯𝐼𝑦) ∨ π‘₯ ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (π‘₯𝐼𝑧))}))
54rneqd 5897 . . . . . 6 (𝐺 ∈ TarskiG β†’ ran 𝐿 = ran (π‘₯ ∈ 𝑃, 𝑦 ∈ (𝑃 βˆ– {π‘₯}) ↦ {𝑧 ∈ 𝑃 ∣ (𝑧 ∈ (π‘₯𝐼𝑦) ∨ π‘₯ ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (π‘₯𝐼𝑧))}))
65eleq2d 2820 . . . . 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 478 . . . 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 2733 . . . . . 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 6860 . . . . . . 7 𝑃 ∈ V
109rabex 5293 . . . . . 6 {𝑧 ∈ 𝑃 ∣ (𝑧 ∈ (π‘₯𝐼𝑦) ∨ π‘₯ ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (π‘₯𝐼𝑧))} ∈ V
118, 10elrnmpo 7496 . . . . 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 4041 . . . . . . . 8 {𝑧 ∈ 𝑃 ∣ (𝑧 ∈ (π‘₯𝐼𝑦) ∨ π‘₯ ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (π‘₯𝐼𝑧))} βŠ† 𝑃
13 sseq1 3973 . . . . . . . 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 258 . . . . . . 7 (𝑝 = {𝑧 ∈ 𝑃 ∣ (𝑧 ∈ (π‘₯𝐼𝑦) ∨ π‘₯ ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (π‘₯𝐼𝑧))} β†’ 𝑝 βŠ† 𝑃)
1514rexlimivw 3145 . . . . . 6 (βˆƒπ‘¦ ∈ (𝑃 βˆ– {π‘₯})𝑝 = {𝑧 ∈ 𝑃 ∣ (𝑧 ∈ (π‘₯𝐼𝑦) ∨ π‘₯ ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (π‘₯𝐼𝑧))} β†’ 𝑝 βŠ† 𝑃)
1615rexlimivw 3145 . . . . 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 3140 . 2 (𝐺 ∈ TarskiG β†’ βˆ€π‘ ∈ ran 𝐿 𝑝 βŠ† 𝑃)
20 unissb 4904 . 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 397   ∨ w3o 1087   = wceq 1542   ∈ wcel 2107  βˆ€wral 3061  βˆƒwrex 3070  {crab 3406   βˆ– cdif 3911   βŠ† wss 3914  {csn 4590  βˆͺ cuni 4869  ran crn 5638  β€˜cfv 6500  (class class class)co 7361   ∈ cmpo 7363  Basecbs 17091  TarskiGcstrkg 27418  Itvcitv 27424  LineGclng 27425
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5260  ax-nul 5267  ax-pr 5388
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3449  df-sbc 3744  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-br 5110  df-opab 5172  df-cnv 5645  df-dm 5647  df-rn 5648  df-iota 6452  df-fv 6508  df-ov 7364  df-oprab 7365  df-mpo 7366  df-trkg 27444
This theorem is referenced by:  tglnpt  27540  tglineintmo  27633
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