Theorem tgplnfn 29014

Description: The plane generating function as a function. (Contributed by Thierry Arnoux, 17-Jun-2026.)

Hypotheses

Ref	Expression
tgplnfn.p	⊢ 𝑃 = (Base‘𝐺)
tgplnfn.l	⊢ 𝐿 = (LineG‘𝐺)
tgplnfn.i	⊢ 𝐸 = (hlG‘𝐺)
tgplnfn.1	⊢ (𝜑 → 𝐺 ∈ 𝑉)

Assertion

Ref	Expression
tgplnfn	⊢ (𝜑 → 𝐸 Fn ((ran 𝐿 × 𝑃) ∖ ◡ E ))

Proof of Theorem tgplnfn

Dummy variables 𝑎 𝑟 𝑥 𝑡 𝑔 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		tgplnfn.p	. . . . . . . 8 ⊢ 𝑃 = (Base‘𝐺)
2	1	fvexi 6896	. . . . . . 7 ⊢ 𝑃 ∈ V
3	2	rabex 5310	. . . . . 6 ⊢ {𝑥 ∈ 𝑃 ∣ (𝑥 ∈ 𝑎 ∨ 𝑥((hpG‘𝐺)‘𝑎)𝑟 ∨ ∃𝑡 ∈ 𝑎 𝑡 ∈ (𝑥(Itv‘𝐺)𝑟))} ∈ V
4	3	rgen2w 3090	. . . . 5 ⊢ ∀𝑎 ∈ ran 𝐿∀𝑟 ∈ (𝑃 ∖ 𝑎){𝑥 ∈ 𝑃 ∣ (𝑥 ∈ 𝑎 ∨ 𝑥((hpG‘𝐺)‘𝑎)𝑟 ∨ ∃𝑡 ∈ 𝑎 𝑡 ∈ (𝑥(Itv‘𝐺)𝑟))} ∈ V
5		eqid 2769	. . . . . 6 ⊢ (𝑎 ∈ ran 𝐿, 𝑟 ∈ (𝑃 ∖ 𝑎) ↦ {𝑥 ∈ 𝑃 ∣ (𝑥 ∈ 𝑎 ∨ 𝑥((hpG‘𝐺)‘𝑎)𝑟 ∨ ∃𝑡 ∈ 𝑎 𝑡 ∈ (𝑥(Itv‘𝐺)𝑟))}) = (𝑎 ∈ ran 𝐿, 𝑟 ∈ (𝑃 ∖ 𝑎) ↦ {𝑥 ∈ 𝑃 ∣ (𝑥 ∈ 𝑎 ∨ 𝑥((hpG‘𝐺)‘𝑎)𝑟 ∨ ∃𝑡 ∈ 𝑎 𝑡 ∈ (𝑥(Itv‘𝐺)𝑟))})
6	5	fmpox 8063	. . . . 5 ⊢ (∀𝑎 ∈ ran 𝐿∀𝑟 ∈ (𝑃 ∖ 𝑎){𝑥 ∈ 𝑃 ∣ (𝑥 ∈ 𝑎 ∨ 𝑥((hpG‘𝐺)‘𝑎)𝑟 ∨ ∃𝑡 ∈ 𝑎 𝑡 ∈ (𝑥(Itv‘𝐺)𝑟))} ∈ V ↔ (𝑎 ∈ ran 𝐿, 𝑟 ∈ (𝑃 ∖ 𝑎) ↦ {𝑥 ∈ 𝑃 ∣ (𝑥 ∈ 𝑎 ∨ 𝑥((hpG‘𝐺)‘𝑎)𝑟 ∨ ∃𝑡 ∈ 𝑎 𝑡 ∈ (𝑥(Itv‘𝐺)𝑟))}):∪ 𝑎 ∈ ran 𝐿({𝑎} × (𝑃 ∖ 𝑎))⟶V)
7	4, 6	mpbi 233	. . . 4 ⊢ (𝑎 ∈ ran 𝐿, 𝑟 ∈ (𝑃 ∖ 𝑎) ↦ {𝑥 ∈ 𝑃 ∣ (𝑥 ∈ 𝑎 ∨ 𝑥((hpG‘𝐺)‘𝑎)𝑟 ∨ ∃𝑡 ∈ 𝑎 𝑡 ∈ (𝑥(Itv‘𝐺)𝑟))}):∪ 𝑎 ∈ ran 𝐿({𝑎} × (𝑃 ∖ 𝑎))⟶V
8		ffn 6706	. . . 4 ⊢ ((𝑎 ∈ ran 𝐿, 𝑟 ∈ (𝑃 ∖ 𝑎) ↦ {𝑥 ∈ 𝑃 ∣ (𝑥 ∈ 𝑎 ∨ 𝑥((hpG‘𝐺)‘𝑎)𝑟 ∨ ∃𝑡 ∈ 𝑎 𝑡 ∈ (𝑥(Itv‘𝐺)𝑟))}):∪ 𝑎 ∈ ran 𝐿({𝑎} × (𝑃 ∖ 𝑎))⟶V → (𝑎 ∈ ran 𝐿, 𝑟 ∈ (𝑃 ∖ 𝑎) ↦ {𝑥 ∈ 𝑃 ∣ (𝑥 ∈ 𝑎 ∨ 𝑥((hpG‘𝐺)‘𝑎)𝑟 ∨ ∃𝑡 ∈ 𝑎 𝑡 ∈ (𝑥(Itv‘𝐺)𝑟))}) Fn ∪ 𝑎 ∈ ran 𝐿({𝑎} × (𝑃 ∖ 𝑎)))
9	7, 8	ax-mp 5	. . 3 ⊢ (𝑎 ∈ ran 𝐿, 𝑟 ∈ (𝑃 ∖ 𝑎) ↦ {𝑥 ∈ 𝑃 ∣ (𝑥 ∈ 𝑎 ∨ 𝑥((hpG‘𝐺)‘𝑎)𝑟 ∨ ∃𝑡 ∈ 𝑎 𝑡 ∈ (𝑥(Itv‘𝐺)𝑟))}) Fn ∪ 𝑎 ∈ ran 𝐿({𝑎} × (𝑃 ∖ 𝑎))
10		xpdifcnvepel 6167	. . . 4 ⊢ ∪ 𝑎 ∈ ran 𝐿({𝑎} × (𝑃 ∖ 𝑎)) = ((ran 𝐿 × 𝑃) ∖ ◡ E )
11	10	fneq2i 6634	. . 3 ⊢ ((𝑎 ∈ ran 𝐿, 𝑟 ∈ (𝑃 ∖ 𝑎) ↦ {𝑥 ∈ 𝑃 ∣ (𝑥 ∈ 𝑎 ∨ 𝑥((hpG‘𝐺)‘𝑎)𝑟 ∨ ∃𝑡 ∈ 𝑎 𝑡 ∈ (𝑥(Itv‘𝐺)𝑟))}) Fn ∪ 𝑎 ∈ ran 𝐿({𝑎} × (𝑃 ∖ 𝑎)) ↔ (𝑎 ∈ ran 𝐿, 𝑟 ∈ (𝑃 ∖ 𝑎) ↦ {𝑥 ∈ 𝑃 ∣ (𝑥 ∈ 𝑎 ∨ 𝑥((hpG‘𝐺)‘𝑎)𝑟 ∨ ∃𝑡 ∈ 𝑎 𝑡 ∈ (𝑥(Itv‘𝐺)𝑟))}) Fn ((ran 𝐿 × 𝑃) ∖ ◡ E ))
12	9, 11	mpbi 233	. 2 ⊢ (𝑎 ∈ ran 𝐿, 𝑟 ∈ (𝑃 ∖ 𝑎) ↦ {𝑥 ∈ 𝑃 ∣ (𝑥 ∈ 𝑎 ∨ 𝑥((hpG‘𝐺)‘𝑎)𝑟 ∨ ∃𝑡 ∈ 𝑎 𝑡 ∈ (𝑥(Itv‘𝐺)𝑟))}) Fn ((ran 𝐿 × 𝑃) ∖ ◡ E )
13		tgplnfn.i	. . . 4 ⊢ 𝐸 = (hlG‘𝐺)
14		df-plng 29013	. . . . 5 ⊢ hlG = (𝑔 ∈ V ↦ (𝑎 ∈ ran (LineG‘𝑔), 𝑟 ∈ ((Base‘𝑔) ∖ 𝑎) ↦ {𝑥 ∈ (Base‘𝑔) ∣ (𝑥 ∈ 𝑎 ∨ 𝑥((hpG‘𝑔)‘𝑎)𝑟 ∨ ∃𝑡 ∈ 𝑎 𝑡 ∈ (𝑥(Itv‘𝑔)𝑟))}))
15		fveq2 6882	. . . . . . . 8 ⊢ (𝑔 = 𝐺 → (LineG‘𝑔) = (LineG‘𝐺))
16		tgplnfn.l	. . . . . . . 8 ⊢ 𝐿 = (LineG‘𝐺)
17	15, 16	eqtr4di 2822	. . . . . . 7 ⊢ (𝑔 = 𝐺 → (LineG‘𝑔) = 𝐿)
18	17	rneqd 5929	. . . . . 6 ⊢ (𝑔 = 𝐺 → ran (LineG‘𝑔) = ran 𝐿)
19		fveq2 6882	. . . . . . . 8 ⊢ (𝑔 = 𝐺 → (Base‘𝑔) = (Base‘𝐺))
20	19, 1	eqtr4di 2822	. . . . . . 7 ⊢ (𝑔 = 𝐺 → (Base‘𝑔) = 𝑃)
21	20	difeq1d 4088	. . . . . 6 ⊢ (𝑔 = 𝐺 → ((Base‘𝑔) ∖ 𝑎) = (𝑃 ∖ 𝑎))
22		biidd 265	. . . . . . . 8 ⊢ (𝑔 = 𝐺 → (𝑥 ∈ 𝑎 ↔ 𝑥 ∈ 𝑎))
23		fveq2 6882	. . . . . . . . . 10 ⊢ (𝑔 = 𝐺 → (hpG‘𝑔) = (hpG‘𝐺))
24	23	fveq1d 6884	. . . . . . . . 9 ⊢ (𝑔 = 𝐺 → ((hpG‘𝑔)‘𝑎) = ((hpG‘𝐺)‘𝑎))
25	24	breqd 5124	. . . . . . . 8 ⊢ (𝑔 = 𝐺 → (𝑥((hpG‘𝑔)‘𝑎)𝑟 ↔ 𝑥((hpG‘𝐺)‘𝑎)𝑟))
26		fveq2 6882	. . . . . . . . . . 11 ⊢ (𝑔 = 𝐺 → (Itv‘𝑔) = (Itv‘𝐺))
27	26	oveqd 7428	. . . . . . . . . 10 ⊢ (𝑔 = 𝐺 → (𝑥(Itv‘𝑔)𝑟) = (𝑥(Itv‘𝐺)𝑟))
28	27	eleq2d 2855	. . . . . . . . 9 ⊢ (𝑔 = 𝐺 → (𝑡 ∈ (𝑥(Itv‘𝑔)𝑟) ↔ 𝑡 ∈ (𝑥(Itv‘𝐺)𝑟)))
29	28	rexbidv 3195	. . . . . . . 8 ⊢ (𝑔 = 𝐺 → (∃𝑡 ∈ 𝑎 𝑡 ∈ (𝑥(Itv‘𝑔)𝑟) ↔ ∃𝑡 ∈ 𝑎 𝑡 ∈ (𝑥(Itv‘𝐺)𝑟)))
30	22, 25, 29	3orbi123d 1461	. . . . . . 7 ⊢ (𝑔 = 𝐺 → ((𝑥 ∈ 𝑎 ∨ 𝑥((hpG‘𝑔)‘𝑎)𝑟 ∨ ∃𝑡 ∈ 𝑎 𝑡 ∈ (𝑥(Itv‘𝑔)𝑟)) ↔ (𝑥 ∈ 𝑎 ∨ 𝑥((hpG‘𝐺)‘𝑎)𝑟 ∨ ∃𝑡 ∈ 𝑎 𝑡 ∈ (𝑥(Itv‘𝐺)𝑟))))
31	20, 30	rabeqbidv 3441	. . . . . 6 ⊢ (𝑔 = 𝐺 → {𝑥 ∈ (Base‘𝑔) ∣ (𝑥 ∈ 𝑎 ∨ 𝑥((hpG‘𝑔)‘𝑎)𝑟 ∨ ∃𝑡 ∈ 𝑎 𝑡 ∈ (𝑥(Itv‘𝑔)𝑟))} = {𝑥 ∈ 𝑃 ∣ (𝑥 ∈ 𝑎 ∨ 𝑥((hpG‘𝐺)‘𝑎)𝑟 ∨ ∃𝑡 ∈ 𝑎 𝑡 ∈ (𝑥(Itv‘𝐺)𝑟))})
32	18, 21, 31	mpoeq123dv 7486	. . . . 5 ⊢ (𝑔 = 𝐺 → (𝑎 ∈ ran (LineG‘𝑔), 𝑟 ∈ ((Base‘𝑔) ∖ 𝑎) ↦ {𝑥 ∈ (Base‘𝑔) ∣ (𝑥 ∈ 𝑎 ∨ 𝑥((hpG‘𝑔)‘𝑎)𝑟 ∨ ∃𝑡 ∈ 𝑎 𝑡 ∈ (𝑥(Itv‘𝑔)𝑟))}) = (𝑎 ∈ ran 𝐿, 𝑟 ∈ (𝑃 ∖ 𝑎) ↦ {𝑥 ∈ 𝑃 ∣ (𝑥 ∈ 𝑎 ∨ 𝑥((hpG‘𝐺)‘𝑎)𝑟 ∨ ∃𝑡 ∈ 𝑎 𝑡 ∈ (𝑥(Itv‘𝐺)𝑟))}))
33		tgplnfn.1	. . . . . 6 ⊢ (𝜑 → 𝐺 ∈ 𝑉)
34	33	elexd 3486	. . . . 5 ⊢ (𝜑 → 𝐺 ∈ V)
35	16	fvexi 6896	. . . . . . . 8 ⊢ 𝐿 ∈ V
36	35	rnex 7906	. . . . . . 7 ⊢ ran 𝐿 ∈ V
37	36	a1i 11	. . . . . 6 ⊢ (𝜑 → ran 𝐿 ∈ V)
38	2	difexi 5301	. . . . . . 7 ⊢ (𝑃 ∖ 𝑎) ∈ V
39	38	a1i 11	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ ran 𝐿) → (𝑃 ∖ 𝑎) ∈ V)
40	37, 39	mpoexd 8076	. . . . 5 ⊢ (𝜑 → (𝑎 ∈ ran 𝐿, 𝑟 ∈ (𝑃 ∖ 𝑎) ↦ {𝑥 ∈ 𝑃 ∣ (𝑥 ∈ 𝑎 ∨ 𝑥((hpG‘𝐺)‘𝑎)𝑟 ∨ ∃𝑡 ∈ 𝑎 𝑡 ∈ (𝑥(Itv‘𝐺)𝑟))}) ∈ V)
41	14, 32, 34, 40	fvmptd3 7014	. . . 4 ⊢ (𝜑 → (hlG‘𝐺) = (𝑎 ∈ ran 𝐿, 𝑟 ∈ (𝑃 ∖ 𝑎) ↦ {𝑥 ∈ 𝑃 ∣ (𝑥 ∈ 𝑎 ∨ 𝑥((hpG‘𝐺)‘𝑎)𝑟 ∨ ∃𝑡 ∈ 𝑎 𝑡 ∈ (𝑥(Itv‘𝐺)𝑟))}))
42	13, 41	eqtrid 2816	. . 3 ⊢ (𝜑 → 𝐸 = (𝑎 ∈ ran 𝐿, 𝑟 ∈ (𝑃 ∖ 𝑎) ↦ {𝑥 ∈ 𝑃 ∣ (𝑥 ∈ 𝑎 ∨ 𝑥((hpG‘𝐺)‘𝑎)𝑟 ∨ ∃𝑡 ∈ 𝑎 𝑡 ∈ (𝑥(Itv‘𝐺)𝑟))}))
43	42	fneq1d 6629	. 2 ⊢ (𝜑 → (𝐸 Fn ((ran 𝐿 × 𝑃) ∖ ◡ E ) ↔ (𝑎 ∈ ran 𝐿, 𝑟 ∈ (𝑃 ∖ 𝑎) ↦ {𝑥 ∈ 𝑃 ∣ (𝑥 ∈ 𝑎 ∨ 𝑥((hpG‘𝐺)‘𝑎)𝑟 ∨ ∃𝑡 ∈ 𝑎 𝑡 ∈ (𝑥(Itv‘𝐺)𝑟))}) Fn ((ran 𝐿 × 𝑃) ∖ ◡ E )))
44	12, 43	mpbiri 261	1 ⊢ (𝜑 → 𝐸 Fn ((ran 𝐿 × 𝑃) ∖ ◡ E ))

Syntax hints:   → wi 4   ∧ wa 400   ∨ w3o 1100   = wceq 1567   ∈ wcel 2149  ∀wral 3085  ∃wrex 3095  {crab 3423  Vcvv 3463   ∖ cdif 3910  {csn 4594  ∪ ciun 4960   class class class wbr 5113   E cep 5561   × cxp 5660  ^◡ccnv 5661  ran crn 5663   Fn wfn 6532  ⟶wf 6533  ‘cfv 6537  (class class class)co 7411   ∈ cmpo 7413  Basecbs 17268  Itvcitv 28667  LineGclng 28668  hpGchpg 28997  hlGcplng 29012

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-rep 5242  ax-sep 5261  ax-nul 5271  ax-pow 5337  ax-pr 5405  ax-un 7733

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-reu 3377  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-iun 4962  df-br 5114  df-opab 5178  df-mpt 5197  df-id 5557  df-eprel 5562  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-ov 7414  df-oprab 7415  df-mpo 7416  df-1st 7985  df-2nd 7986  df-plng 29013

This theorem is referenced by:  tgelrnpln  29015