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Theorem tglnunirn 25663
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 25661 . . . . . . 7 (𝐺 ∈ TarskiG → 𝐿 = (𝑥𝑃, 𝑦 ∈ (𝑃 ∖ {𝑥}) ↦ {𝑧𝑃 ∣ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))}))
54rneqd 5508 . . . . . 6 (𝐺 ∈ TarskiG → ran 𝐿 = ran (𝑥𝑃, 𝑦 ∈ (𝑃 ∖ {𝑥}) ↦ {𝑧𝑃 ∣ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))}))
65eleq2d 2825 . . . . 5 (𝐺 ∈ TarskiG → (𝑝 ∈ ran 𝐿𝑝 ∈ ran (𝑥𝑃, 𝑦 ∈ (𝑃 ∖ {𝑥}) ↦ {𝑧𝑃 ∣ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))})))
76biimpa 502 . . . 4 ((𝐺 ∈ TarskiG ∧ 𝑝 ∈ ran 𝐿) → 𝑝 ∈ ran (𝑥𝑃, 𝑦 ∈ (𝑃 ∖ {𝑥}) ↦ {𝑧𝑃 ∣ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))}))
8 eqid 2760 . . . . . 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 6363 . . . . . . . 8 (Base‘𝐺) ∈ V
101, 9eqeltri 2835 . . . . . . 7 𝑃 ∈ V
1110rabex 4964 . . . . . 6 {𝑧𝑃 ∣ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))} ∈ V
128, 11elrnmpt2 6939 . . . . 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 3828 . . . . . . . 8 {𝑧𝑃 ∣ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))} ⊆ 𝑃
14 sseq1 3767 . . . . . . . 8 (𝑝 = {𝑧𝑃 ∣ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))} → (𝑝𝑃 ↔ {𝑧𝑃 ∣ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))} ⊆ 𝑃))
1513, 14mpbiri 248 . . . . . . 7 (𝑝 = {𝑧𝑃 ∣ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))} → 𝑝𝑃)
1615rexlimivw 3167 . . . . . 6 (∃𝑦 ∈ (𝑃 ∖ {𝑥})𝑝 = {𝑧𝑃 ∣ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))} → 𝑝𝑃)
1716rexlimivw 3167 . . . . 5 (∃𝑥𝑃𝑦 ∈ (𝑃 ∖ {𝑥})𝑝 = {𝑧𝑃 ∣ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))} → 𝑝𝑃)
1812, 17sylbi 207 . . . 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 3104 . 2 (𝐺 ∈ TarskiG → ∀𝑝 ∈ ran 𝐿 𝑝𝑃)
21 unissb 4621 . 2 ( ran 𝐿𝑃 ↔ ∀𝑝 ∈ ran 𝐿 𝑝𝑃)
2220, 21sylibr 224 1 (𝐺 ∈ TarskiG → ran 𝐿𝑃)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  w3o 1071   = wceq 1632  wcel 2139  wral 3050  wrex 3051  {crab 3054  Vcvv 3340  cdif 3712  wss 3715  {csn 4321   cuni 4588  ran crn 5267  cfv 6049  (class class class)co 6814  cmpt2 6816  Basecbs 16079  TarskiGcstrkg 25549  Itvcitv 25555  LineGclng 25556
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pr 5055
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-opab 4865  df-cnv 5274  df-dm 5276  df-rn 5277  df-iota 6012  df-fv 6057  df-ov 6817  df-oprab 6818  df-mpt2 6819  df-trkg 25572
This theorem is referenced by:  tglnpt  25664  tglineintmo  25757
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