Theorem tglnunirn 28527

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 28525	. . . . . . 7 ⊢ (𝐺 ∈ TarskiG → 𝐿 = (𝑥 ∈ 𝑃, 𝑦 ∈ (𝑃 ∖ {𝑥}) ↦ {𝑧 ∈ 𝑃 ∣ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))}))
5	4	rneqd 5918	. . . . . 6 ⊢ (𝐺 ∈ TarskiG → ran 𝐿 = ran (𝑥 ∈ 𝑃, 𝑦 ∈ (𝑃 ∖ {𝑥}) ↦ {𝑧 ∈ 𝑃 ∣ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))}))
6	5	eleq2d 2820	. . . . 5 ⊢ (𝐺 ∈ TarskiG → (𝑝 ∈ ran 𝐿 ↔ 𝑝 ∈ ran (𝑥 ∈ 𝑃, 𝑦 ∈ (𝑃 ∖ {𝑥}) ↦ {𝑧 ∈ 𝑃 ∣ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))})))
7	6	biimpa 476	. . . 4 ⊢ ((𝐺 ∈ TarskiG ∧ 𝑝 ∈ ran 𝐿) → 𝑝 ∈ ran (𝑥 ∈ 𝑃, 𝑦 ∈ (𝑃 ∖ {𝑥}) ↦ {𝑧 ∈ 𝑃 ∣ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))}))
8		eqid 2735	. . . . . 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 6890	. . . . . . 7 ⊢ 𝑃 ∈ V
10	9	rabex 5309	. . . . . 6 ⊢ {𝑧 ∈ 𝑃 ∣ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))} ∈ V
11	8, 10	elrnmpo 7543	. . . . 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 4055	. . . . . . . 8 ⊢ {𝑧 ∈ 𝑃 ∣ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))} ⊆ 𝑃
13		sseq1 3984	. . . . . . . 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 258	. . . . . . 7 ⊢ (𝑝 = {𝑧 ∈ 𝑃 ∣ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))} → 𝑝 ⊆ 𝑃)
15	14	rexlimivw 3137	. . . . . 6 ⊢ (∃𝑦 ∈ (𝑃 ∖ {𝑥})𝑝 = {𝑧 ∈ 𝑃 ∣ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))} → 𝑝 ⊆ 𝑃)
16	15	rexlimivw 3137	. . . . 5 ⊢ (∃𝑥 ∈ 𝑃 ∃𝑦 ∈ (𝑃 ∖ {𝑥})𝑝 = {𝑧 ∈ 𝑃 ∣ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))} → 𝑝 ⊆ 𝑃)
17	11, 16	sylbi 217	. . . 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 3132	. 2 ⊢ (𝐺 ∈ TarskiG → ∀𝑝 ∈ ran 𝐿 𝑝 ⊆ 𝑃)
20		unissb 4915	. 2 ⊢ (∪ ran 𝐿 ⊆ 𝑃 ↔ ∀𝑝 ∈ ran 𝐿 𝑝 ⊆ 𝑃)
21	19, 20	sylibr 234	1 ⊢ (𝐺 ∈ TarskiG → ∪ ran 𝐿 ⊆ 𝑃)

Syntax hints:   → wi 4   ∧ wa 395   ∨ w3o 1085   = wceq 1540   ∈ wcel 2108  ∀wral 3051  ∃wrex 3060  {crab 3415   ∖ cdif 3923   ⊆ wss 3926  {csn 4601  ∪ cuni 4883  ran crn 5655  ‘cfv 6531  (class class class)co 7405   ∈ cmpo 7407  Basecbs 17228  TarskiGcstrkg 28406  Itvcitv 28412  LineGclng 28413

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-sep 5266  ax-nul 5276  ax-pr 5402

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rab 3416  df-v 3461  df-sbc 3766  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-nul 4309  df-if 4501  df-pw 4577  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-br 5120  df-opab 5182  df-cnv 5662  df-dm 5664  df-rn 5665  df-iota 6484  df-fv 6539  df-ov 7408  df-oprab 7409  df-mpo 7410  df-trkg 28432

This theorem is referenced by:  tglnpt  28528  tglineintmo  28621