Theorem tgplnfn 29310

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 6930	. . . . . . 7 ⊢ 𝑃 ∈ V
3	2	rabex 5307	. . . . . 6 ⊢ {𝑥 ∈ 𝑃 ∣ (𝑥 ∈ 𝑎 ∨ 𝑥((hpG‘𝐺)‘𝑎)𝑟 ∨ ∃𝑡 ∈ 𝑎 𝑡 ∈ (𝑥(Itv‘𝐺)𝑟))} ∈ V
4	3	rgen2w 3085	. . . . 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 2763	. . . . . 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 8113	. . . . 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 6739	. . . 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 6178	. . . 4 ⊢ ∪ 𝑎 ∈ ran 𝐿({𝑎} × (𝑃 ∖ 𝑎)) = ((ran 𝐿 × 𝑃) ∖ ◡ E )
11	10	fneq2i 6667	. . 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 29309	. . . . 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 6916	. . . . . . . 8 ⊢ (𝑔 = 𝐺 → (LineG‘𝑔) = (LineG‘𝐺))
16		tgplnfn.l	. . . . . . . 8 ⊢ 𝐿 = (LineG‘𝐺)
17	15, 16	eqtr4di 2816	. . . . . . 7 ⊢ (𝑔 = 𝐺 → (LineG‘𝑔) = 𝐿)
18	17	rneqd 5932	. . . . . 6 ⊢ (𝑔 = 𝐺 → ran (LineG‘𝑔) = ran 𝐿)
19		fveq2 6916	. . . . . . . 8 ⊢ (𝑔 = 𝐺 → (Base‘𝑔) = (Base‘𝐺))
20	19, 1	eqtr4di 2816	. . . . . . 7 ⊢ (𝑔 = 𝐺 → (Base‘𝑔) = 𝑃)
21	20	difeq1d 4065	. . . . . 6 ⊢ (𝑔 = 𝐺 → ((Base‘𝑔) ∖ 𝑎) = (𝑃 ∖ 𝑎))
22		biidd 265	. . . . . . . 8 ⊢ (𝑔 = 𝐺 → (𝑥 ∈ 𝑎 ↔ 𝑥 ∈ 𝑎))
23		fveq2 6916	. . . . . . . . . 10 ⊢ (𝑔 = 𝐺 → (hpG‘𝑔) = (hpG‘𝐺))
24	23	fveq1d 6918	. . . . . . . . 9 ⊢ (𝑔 = 𝐺 → ((hpG‘𝑔)‘𝑎) = ((hpG‘𝐺)‘𝑎))
25	24	breqd 5119	. . . . . . . 8 ⊢ (𝑔 = 𝐺 → (𝑥((hpG‘𝑔)‘𝑎)𝑟 ↔ 𝑥((hpG‘𝐺)‘𝑎)𝑟))
26		fveq2 6916	. . . . . . . . . . 11 ⊢ (𝑔 = 𝐺 → (Itv‘𝑔) = (Itv‘𝐺))
27	26	oveqd 7468	. . . . . . . . . 10 ⊢ (𝑔 = 𝐺 → (𝑥(Itv‘𝑔)𝑟) = (𝑥(Itv‘𝐺)𝑟))
28	27	eleq2d 2850	. . . . . . . . 9 ⊢ (𝑔 = 𝐺 → (𝑡 ∈ (𝑥(Itv‘𝑔)𝑟) ↔ 𝑡 ∈ (𝑥(Itv‘𝐺)𝑟)))
29	28	rexbidv 3190	. . . . . . . 8 ⊢ (𝑔 = 𝐺 → (∃𝑡 ∈ 𝑎 𝑡 ∈ (𝑥(Itv‘𝑔)𝑟) ↔ ∃𝑡 ∈ 𝑎 𝑡 ∈ (𝑥(Itv‘𝐺)𝑟)))
30	22, 25, 29	3orbi123d 1464	. . . . . . 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 3433	. . . . . 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 7526	. . . . 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 3477	. . . . 5 ⊢ (𝜑 → 𝐺 ∈ V)
35	16	fvexi 6930	. . . . . . . 8 ⊢ 𝐿 ∈ V
36	35	rnex 7956	. . . . . . 7 ⊢ ran 𝐿 ∈ V
37	36	a1i 11	. . . . . 6 ⊢ (𝜑 → ran 𝐿 ∈ V)
38	2	difexi 5299	. . . . . . 7 ⊢ (𝑃 ∖ 𝑎) ∈ V
39	38	a1i 11	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ ran 𝐿) → (𝑃 ∖ 𝑎) ∈ V)
40	37, 39	mpoexd 8128	. . . . 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 7048	. . . 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 2810	. . 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 6662	. 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 ))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∨ w3o 1102   = wceq 1570   ∈ wcel 2147  ∀wral 3080  ∃wrex 3090  {crab 3416  Vcvv 3454   ∖ cdif 4048  {csn 4588  ∪ ciun 4956   class class class wbr 5108   E cep 5557   × cxp 5656  ^◡ccnv 5657  ran crn 5659   Fn wfn 6561  ⟶wf 6562  ‘cfv 6566  (class class class)co 7451   ∈ cmpo 7453  Basecbs 17431  Itvcitv 28952  LineGclng 28953  hpGchpg 29292  hlGcplng 29308

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5238  ax-sep 5255  ax-nul 5266  ax-pow 5334  ax-pr 5398  ax-un 7783

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-iun 4958  df-br 5109  df-opab 5174  df-mpt 5193  df-id 5553  df-eprel 5558  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-iota 6520  df-fun 6568  df-fn 6569  df-f 6570  df-f1 6571  df-fo 6572  df-f1o 6573  df-fv 6574  df-ov 7454  df-oprab 7455  df-mpo 7456  df-1st 8035  df-2nd 8036  df-plng 29309

This theorem is used by:  tgelrnpln  29311