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Theorem tglnunirn 25188
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 25186 . . . . . . 7 (𝐺 ∈ TarskiG → 𝐿 = (𝑥𝑃, 𝑦 ∈ (𝑃 ∖ {𝑥}) ↦ {𝑧𝑃 ∣ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))}))
54rneqd 5260 . . . . . 6 (𝐺 ∈ TarskiG → ran 𝐿 = ran (𝑥𝑃, 𝑦 ∈ (𝑃 ∖ {𝑥}) ↦ {𝑧𝑃 ∣ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))}))
65eleq2d 2672 . . . . 5 (𝐺 ∈ TarskiG → (𝑝 ∈ ran 𝐿𝑝 ∈ ran (𝑥𝑃, 𝑦 ∈ (𝑃 ∖ {𝑥}) ↦ {𝑧𝑃 ∣ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))})))
76biimpa 499 . . . 4 ((𝐺 ∈ TarskiG ∧ 𝑝 ∈ ran 𝐿) → 𝑝 ∈ ran (𝑥𝑃, 𝑦 ∈ (𝑃 ∖ {𝑥}) ↦ {𝑧𝑃 ∣ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))}))
8 eqid 2609 . . . . . 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 6097 . . . . . . . 8 (Base‘𝐺) ∈ V
101, 9eqeltri 2683 . . . . . . 7 𝑃 ∈ V
1110rabex 4734 . . . . . 6 {𝑧𝑃 ∣ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))} ∈ V
128, 11elrnmpt2 6648 . . . . 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 3649 . . . . . . . 8 {𝑧𝑃 ∣ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))} ⊆ 𝑃
14 sseq1 3588 . . . . . . . 8 (𝑝 = {𝑧𝑃 ∣ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))} → (𝑝𝑃 ↔ {𝑧𝑃 ∣ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))} ⊆ 𝑃))
1513, 14mpbiri 246 . . . . . . 7 (𝑝 = {𝑧𝑃 ∣ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))} → 𝑝𝑃)
1615rexlimivw 3010 . . . . . 6 (∃𝑦 ∈ (𝑃 ∖ {𝑥})𝑝 = {𝑧𝑃 ∣ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))} → 𝑝𝑃)
1716rexlimivw 3010 . . . . 5 (∃𝑥𝑃𝑦 ∈ (𝑃 ∖ {𝑥})𝑝 = {𝑧𝑃 ∣ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))} → 𝑝𝑃)
1812, 17sylbi 205 . . . 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 2948 . 2 (𝐺 ∈ TarskiG → ∀𝑝 ∈ ran 𝐿 𝑝𝑃)
21 unissb 4399 . 2 ( ran 𝐿𝑃 ↔ ∀𝑝 ∈ ran 𝐿 𝑝𝑃)
2220, 21sylibr 222 1 (𝐺 ∈ TarskiG → ran 𝐿𝑃)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382  w3o 1029   = wceq 1474  wcel 1976  wral 2895  wrex 2896  {crab 2899  Vcvv 3172  cdif 3536  wss 3539  {csn 4124   cuni 4366  ran crn 5028  cfv 5789  (class class class)co 6526  cmpt2 6528  Basecbs 15643  TarskiGcstrkg 25073  Itvcitv 25079  LineGclng 25080
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-sep 4703  ax-nul 4711  ax-pr 4827
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ral 2900  df-rex 2901  df-rab 2904  df-v 3174  df-sbc 3402  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-sn 4125  df-pr 4127  df-op 4131  df-uni 4367  df-br 4578  df-opab 4638  df-cnv 5035  df-dm 5037  df-rn 5038  df-iota 5753  df-fv 5797  df-ov 6529  df-oprab 6530  df-mpt2 6531  df-trkg 25096
This theorem is referenced by:  tglnpt  25189  tglineintmo  25282
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