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Theorem tglnpt 26327
 Description: Lines are sets of points. (Contributed by Thierry Arnoux, 17-Oct-2019.)
Hypotheses
Ref Expression
tglng.p 𝑃 = (Base‘𝐺)
tglng.l 𝐿 = (LineG‘𝐺)
tglng.i 𝐼 = (Itv‘𝐺)
tglnpt.g (𝜑𝐺 ∈ TarskiG)
tglnpt.a (𝜑𝐴 ∈ ran 𝐿)
tglnpt.x (𝜑𝑋𝐴)
Assertion
Ref Expression
tglnpt (𝜑𝑋𝑃)

Proof of Theorem tglnpt
StepHypRef Expression
1 tglnpt.g . . 3 (𝜑𝐺 ∈ TarskiG)
2 tglng.p . . . 4 𝑃 = (Base‘𝐺)
3 tglng.l . . . 4 𝐿 = (LineG‘𝐺)
4 tglng.i . . . 4 𝐼 = (Itv‘𝐺)
52, 3, 4tglnunirn 26326 . . 3 (𝐺 ∈ TarskiG → ran 𝐿𝑃)
61, 5syl 17 . 2 (𝜑 ran 𝐿𝑃)
7 tglnpt.a . . . 4 (𝜑𝐴 ∈ ran 𝐿)
8 elssuni 4859 . . . 4 (𝐴 ∈ ran 𝐿𝐴 ran 𝐿)
97, 8syl 17 . . 3 (𝜑𝐴 ran 𝐿)
10 tglnpt.x . . 3 (𝜑𝑋𝐴)
119, 10sseldd 3966 . 2 (𝜑𝑋 ran 𝐿)
126, 11sseldd 3966 1 (𝜑𝑋𝑃)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1531   ∈ wcel 2108   ⊆ wss 3934  ∪ cuni 4830  ran crn 5549  ‘cfv 6348  Basecbs 16475  TarskiGcstrkg 26208  Itvcitv 26214  LineGclng 26215 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1905  ax-6 1964  ax-7 2009  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2154  ax-12 2170  ax-ext 2791  ax-sep 5194  ax-nul 5201  ax-pr 5320 This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1083  df-3an 1084  df-tru 1534  df-ex 1775  df-nf 1779  df-sb 2064  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ral 3141  df-rex 3142  df-rab 3145  df-v 3495  df-sbc 3771  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-nul 4290  df-if 4466  df-sn 4560  df-pr 4562  df-op 4566  df-uni 4831  df-br 5058  df-opab 5120  df-cnv 5556  df-dm 5558  df-rn 5559  df-iota 6307  df-fv 6356  df-ov 7151  df-oprab 7152  df-mpo 7153  df-trkg 26231 This theorem is referenced by:  mirln  26454  mirln2  26455  perpcom  26491  perpneq  26492  ragperp  26495  foot  26500  footne  26501  footeq  26502  hlperpnel  26503  perprag  26504  perpdragALT  26505  perpdrag  26506  colperpexlem3  26510  oppne3  26521  oppcom  26522  oppnid  26524  opphllem1  26525  opphllem2  26526  opphllem3  26527  opphllem4  26528  opphllem5  26529  opphllem6  26530  oppperpex  26531  opphl  26532  outpasch  26533  lnopp2hpgb  26541  hpgerlem  26543  colopp  26547  colhp  26548  lmieu  26562  lmimid  26572  lnperpex  26581  trgcopy  26582  trgcopyeulem  26583
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