Theorem tglnfn 28570

Description: Lines as functions. (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
tglnfn	⊢ (𝐺 ∈ TarskiG → 𝐿 Fn ((𝑃 × 𝑃) ∖ I ))

Proof of Theorem tglnfn

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

Step	Hyp	Ref	Expression
1		tglng.p	. . . . . . . 8 ⊢ 𝑃 = (Base‘𝐺)
2	1	fvexi 6921	. . . . . . 7 ⊢ 𝑃 ∈ V
3	2	rabex 5345	. . . . . 6 ⊢ {𝑧 ∈ 𝑃 ∣ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))} ∈ V
4	3	rgen2w 3064	. . . . 5 ⊢ ∀𝑥 ∈ 𝑃 ∀𝑦 ∈ (𝑃 ∖ {𝑥}){𝑧 ∈ 𝑃 ∣ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))} ∈ V
5		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))})
6	5	fmpox 8091	. . . . 5 ⊢ (∀𝑥 ∈ 𝑃 ∀𝑦 ∈ (𝑃 ∖ {𝑥}){𝑧 ∈ 𝑃 ∣ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))} ∈ V ↔ (𝑥 ∈ 𝑃, 𝑦 ∈ (𝑃 ∖ {𝑥}) ↦ {𝑧 ∈ 𝑃 ∣ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))}):∪ 𝑥 ∈ 𝑃 ({𝑥} × (𝑃 ∖ {𝑥}))⟶V)
7	4, 6	mpbi 230	. . . 4 ⊢ (𝑥 ∈ 𝑃, 𝑦 ∈ (𝑃 ∖ {𝑥}) ↦ {𝑧 ∈ 𝑃 ∣ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))}):∪ 𝑥 ∈ 𝑃 ({𝑥} × (𝑃 ∖ {𝑥}))⟶V
8		ffn 6737	. . . 4 ⊢ ((𝑥 ∈ 𝑃, 𝑦 ∈ (𝑃 ∖ {𝑥}) ↦ {𝑧 ∈ 𝑃 ∣ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))}):∪ 𝑥 ∈ 𝑃 ({𝑥} × (𝑃 ∖ {𝑥}))⟶V → (𝑥 ∈ 𝑃, 𝑦 ∈ (𝑃 ∖ {𝑥}) ↦ {𝑧 ∈ 𝑃 ∣ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))}) Fn ∪ 𝑥 ∈ 𝑃 ({𝑥} × (𝑃 ∖ {𝑥})))
9	7, 8	ax-mp 5	. . 3 ⊢ (𝑥 ∈ 𝑃, 𝑦 ∈ (𝑃 ∖ {𝑥}) ↦ {𝑧 ∈ 𝑃 ∣ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))}) Fn ∪ 𝑥 ∈ 𝑃 ({𝑥} × (𝑃 ∖ {𝑥}))
10		xpdifid 6190	. . . 4 ⊢ ∪ 𝑥 ∈ 𝑃 ({𝑥} × (𝑃 ∖ {𝑥})) = ((𝑃 × 𝑃) ∖ I )
11	10	fneq2i 6667	. . 3 ⊢ ((𝑥 ∈ 𝑃, 𝑦 ∈ (𝑃 ∖ {𝑥}) ↦ {𝑧 ∈ 𝑃 ∣ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))}) Fn ∪ 𝑥 ∈ 𝑃 ({𝑥} × (𝑃 ∖ {𝑥})) ↔ (𝑥 ∈ 𝑃, 𝑦 ∈ (𝑃 ∖ {𝑥}) ↦ {𝑧 ∈ 𝑃 ∣ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))}) Fn ((𝑃 × 𝑃) ∖ I ))
12	9, 11	mpbi 230	. 2 ⊢ (𝑥 ∈ 𝑃, 𝑦 ∈ (𝑃 ∖ {𝑥}) ↦ {𝑧 ∈ 𝑃 ∣ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))}) Fn ((𝑃 × 𝑃) ∖ I )
13		tglng.l	. . . 4 ⊢ 𝐿 = (LineG‘𝐺)
14		tglng.i	. . . 4 ⊢ 𝐼 = (Itv‘𝐺)
15	1, 13, 14	tglng 28569	. . 3 ⊢ (𝐺 ∈ TarskiG → 𝐿 = (𝑥 ∈ 𝑃, 𝑦 ∈ (𝑃 ∖ {𝑥}) ↦ {𝑧 ∈ 𝑃 ∣ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))}))
16	15	fneq1d 6662	. 2 ⊢ (𝐺 ∈ TarskiG → (𝐿 Fn ((𝑃 × 𝑃) ∖ I ) ↔ (𝑥 ∈ 𝑃, 𝑦 ∈ (𝑃 ∖ {𝑥}) ↦ {𝑧 ∈ 𝑃 ∣ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))}) Fn ((𝑃 × 𝑃) ∖ I )))
17	12, 16	mpbiri 258	1 ⊢ (𝐺 ∈ TarskiG → 𝐿 Fn ((𝑃 × 𝑃) ∖ I ))

Syntax hints:   → wi 4   ∨ w3o 1085   = wceq 1537   ∈ wcel 2106  ∀wral 3059  {crab 3433  Vcvv 3478   ∖ cdif 3960  {csn 4631  ∪ ciun 4996   I cid 5582   × cxp 5687   Fn wfn 6558  ⟶wf 6559  ‘cfv 6563  (class class class)co 7431   ∈ cmpo 7433  Basecbs 17245  TarskiGcstrkg 28450  Itvcitv 28456  LineGclng 28457

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-un 7754

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8013  df-2nd 8014  df-trkg 28476

This theorem is referenced by:  tglngne  28573  tgelrnln  28653