Theorem tglnfn 26341

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 6659	. . . . . . 7 ⊢ 𝑃 ∈ V
3	2	rabex 5199	. . . . . 6 ⊢ {𝑧 ∈ 𝑃 ∣ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))} ∈ V
4	3	rgen2w 3119	. . . . 5 ⊢ ∀𝑥 ∈ 𝑃 ∀𝑦 ∈ (𝑃 ∖ {𝑥}){𝑧 ∈ 𝑃 ∣ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))} ∈ V
5		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))})
6	5	fmpox 7747	. . . . 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 233	. . . 4 ⊢ (𝑥 ∈ 𝑃, 𝑦 ∈ (𝑃 ∖ {𝑥}) ↦ {𝑧 ∈ 𝑃 ∣ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))}):∪ 𝑥 ∈ 𝑃 ({𝑥} × (𝑃 ∖ {𝑥}))⟶V
8		ffn 6487	. . . 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 5992	. . . 4 ⊢ ∪ 𝑥 ∈ 𝑃 ({𝑥} × (𝑃 ∖ {𝑥})) = ((𝑃 × 𝑃) ∖ I )
11	10	fneq2i 6421	. . 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 233	. 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 26340	. . 3 ⊢ (𝐺 ∈ TarskiG → 𝐿 = (𝑥 ∈ 𝑃, 𝑦 ∈ (𝑃 ∖ {𝑥}) ↦ {𝑧 ∈ 𝑃 ∣ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))}))
16	15	fneq1d 6416	. 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 261	1 ⊢ (𝐺 ∈ TarskiG → 𝐿 Fn ((𝑃 × 𝑃) ∖ I ))

Syntax hints:   → wi 4   ∨ w3o 1083   = wceq 1538   ∈ wcel 2111  ∀wral 3106  {crab 3110  Vcvv 3441   ∖ cdif 3878  {csn 4525  ∪ ciun 4881   I cid 5424   × cxp 5517   Fn wfn 6319  ⟶wf 6320  ‘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-pow 5231  ax-pr 5295  ax-un 7441

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-ne 2988  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  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-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-fv 6332  df-ov 7138  df-oprab 7139  df-mpo 7140  df-1st 7671  df-2nd 7672  df-trkg 26247

This theorem is referenced by:  tglngne  26344  tgelrnln  26424