Theorem tglnunirn 26342

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 𝐿 ⊆ 𝑃)

Proof of Theorem tglnunirn

Dummy variables 𝑝 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		tglng.p	. . . . . . . 8 ⊢ 𝑃 = (Base‘𝐺)
2		tglng.l	. . . . . . . 8 ⊢ 𝐿 = (LineG‘𝐺)
3		tglng.i	. . . . . . . 8 ⊢ 𝐼 = (Itv‘𝐺)
4	1, 2, 3	tglng 26340	. . . . . . 7 ⊢ (𝐺 ∈ TarskiG → 𝐿 = (𝑥 ∈ 𝑃, 𝑦 ∈ (𝑃 ∖ {𝑥}) ↦ {𝑧 ∈ 𝑃 ∣ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))}))
5	4	rneqd 5772	. . . . . 6 ⊢ (𝐺 ∈ TarskiG → ran 𝐿 = ran (𝑥 ∈ 𝑃, 𝑦 ∈ (𝑃 ∖ {𝑥}) ↦ {𝑧 ∈ 𝑃 ∣ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))}))
6	5	eleq2d 2875	. . . . 5 ⊢ (𝐺 ∈ TarskiG → (𝑝 ∈ ran 𝐿 ↔ 𝑝 ∈ ran (𝑥 ∈ 𝑃, 𝑦 ∈ (𝑃 ∖ {𝑥}) ↦ {𝑧 ∈ 𝑃 ∣ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))})))
7	6	biimpa 480	. . . 4 ⊢ ((𝐺 ∈ TarskiG ∧ 𝑝 ∈ ran 𝐿) → 𝑝 ∈ ran (𝑥 ∈ 𝑃, 𝑦 ∈ (𝑃 ∖ {𝑥}) ↦ {𝑧 ∈ 𝑃 ∣ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))}))
8		eqid 2798	. . . . . 6 ⊢ (𝑥 ∈ 𝑃, 𝑦 ∈ (𝑃 ∖ {𝑥}) ↦ {𝑧 ∈ 𝑃 ∣ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))}) = (𝑥 ∈ 𝑃, 𝑦 ∈ (𝑃 ∖ {𝑥}) ↦ {𝑧 ∈ 𝑃 ∣ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))})
9	1	fvexi 6659	. . . . . . 7 ⊢ 𝑃 ∈ V
10	9	rabex 5199	. . . . . 6 ⊢ {𝑧 ∈ 𝑃 ∣ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))} ∈ V
11	8, 10	elrnmpo 7266	. . . . 5 ⊢ (𝑝 ∈ ran (𝑥 ∈ 𝑃, 𝑦 ∈ (𝑃 ∖ {𝑥}) ↦ {𝑧 ∈ 𝑃 ∣ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))}) ↔ ∃𝑥 ∈ 𝑃 ∃𝑦 ∈ (𝑃 ∖ {𝑥})𝑝 = {𝑧 ∈ 𝑃 ∣ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))})
12		ssrab2 4007	. . . . . . . 8 ⊢ {𝑧 ∈ 𝑃 ∣ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))} ⊆ 𝑃
13		sseq1 3940	. . . . . . . 8 ⊢ (𝑝 = {𝑧 ∈ 𝑃 ∣ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))} → (𝑝 ⊆ 𝑃 ↔ {𝑧 ∈ 𝑃 ∣ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))} ⊆ 𝑃))
14	12, 13	mpbiri 261	. . . . . . 7 ⊢ (𝑝 = {𝑧 ∈ 𝑃 ∣ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))} → 𝑝 ⊆ 𝑃)
15	14	rexlimivw 3241	. . . . . 6 ⊢ (∃𝑦 ∈ (𝑃 ∖ {𝑥})𝑝 = {𝑧 ∈ 𝑃 ∣ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))} → 𝑝 ⊆ 𝑃)
16	15	rexlimivw 3241	. . . . 5 ⊢ (∃𝑥 ∈ 𝑃 ∃𝑦 ∈ (𝑃 ∖ {𝑥})𝑝 = {𝑧 ∈ 𝑃 ∣ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))} → 𝑝 ⊆ 𝑃)
17	11, 16	sylbi 220	. . . 4 ⊢ (𝑝 ∈ ran (𝑥 ∈ 𝑃, 𝑦 ∈ (𝑃 ∖ {𝑥}) ↦ {𝑧 ∈ 𝑃 ∣ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))}) → 𝑝 ⊆ 𝑃)
18	7, 17	syl 17	. . 3 ⊢ ((𝐺 ∈ TarskiG ∧ 𝑝 ∈ ran 𝐿) → 𝑝 ⊆ 𝑃)
19	18	ralrimiva 3149	. 2 ⊢ (𝐺 ∈ TarskiG → ∀𝑝 ∈ ran 𝐿 𝑝 ⊆ 𝑃)
20		unissb 4832	. 2 ⊢ (∪ ran 𝐿 ⊆ 𝑃 ↔ ∀𝑝 ∈ ran 𝐿 𝑝 ⊆ 𝑃)
21	19, 20	sylibr 237	1 ⊢ (𝐺 ∈ TarskiG → ∪ ran 𝐿 ⊆ 𝑃)

Syntax hints:   → wi 4   ∧ wa 399   ∨ w3o 1083   = wceq 1538   ∈ wcel 2111  ∀wral 3106  ∃wrex 3107  {crab 3110   ∖ cdif 3878   ⊆ wss 3881  {csn 4525  ∪ cuni 4800  ran crn 5520  ‘cfv 6324  (class class class)co 7135   ∈ cmpo 7137  Basecbs 16475  TarskiGcstrkg 26224  Itvcitv 26230  LineGclng 26231

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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pr 5295

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-br 5031  df-opab 5093  df-cnv 5527  df-dm 5529  df-rn 5530  df-iota 6283  df-fv 6332  df-ov 7138  df-oprab 7139  df-mpo 7140  df-trkg 26247

This theorem is referenced by:  tglnpt  26343  tglineintmo  26436