Theorem tgplnfn 29262

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 6904	. . . . . . 7 ⊢ 𝑃 ∈ V
3	2	rabex 5300	. . . . . 6 ⊢ {𝑥 ∈ 𝑃 ∣ (𝑥 ∈ 𝑎 ∨ 𝑥((hpG‘𝐺)‘𝑎)𝑟 ∨ ∃𝑡 ∈ 𝑎 𝑡 ∈ (𝑥(Itv‘𝐺)𝑟))} ∈ V
4	3	rgen2w 3081	. . . . 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 2760	. . . . . 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 8082	. . . . 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 6714	. . . 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 6162	. . . 4 ⊢ ∪ 𝑎 ∈ ran 𝐿({𝑎} × (𝑃 ∖ 𝑎)) = ((ran 𝐿 × 𝑃) ∖ ◡ E )
11	10	fneq2i 6642	. . 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 29261	. . . . 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 6890	. . . . . . . 8 ⊢ (𝑔 = 𝐺 → (LineG‘𝑔) = (LineG‘𝐺))
16		tgplnfn.l	. . . . . . . 8 ⊢ 𝐿 = (LineG‘𝐺)
17	15, 16	eqtr4di 2813	. . . . . . 7 ⊢ (𝑔 = 𝐺 → (LineG‘𝑔) = 𝐿)
18	17	rneqd 5921	. . . . . 6 ⊢ (𝑔 = 𝐺 → ran (LineG‘𝑔) = ran 𝐿)
19		fveq2 6890	. . . . . . . 8 ⊢ (𝑔 = 𝐺 → (Base‘𝑔) = (Base‘𝐺))
20	19, 1	eqtr4di 2813	. . . . . . 7 ⊢ (𝑔 = 𝐺 → (Base‘𝑔) = 𝑃)
21	20	difeq1d 4061	. . . . . 6 ⊢ (𝑔 = 𝐺 → ((Base‘𝑔) ∖ 𝑎) = (𝑃 ∖ 𝑎))
22		biidd 265	. . . . . . . 8 ⊢ (𝑔 = 𝐺 → (𝑥 ∈ 𝑎 ↔ 𝑥 ∈ 𝑎))
23		fveq2 6890	. . . . . . . . . 10 ⊢ (𝑔 = 𝐺 → (hpG‘𝑔) = (hpG‘𝐺))
24	23	fveq1d 6892	. . . . . . . . 9 ⊢ (𝑔 = 𝐺 → ((hpG‘𝑔)‘𝑎) = ((hpG‘𝐺)‘𝑎))
25	24	breqd 5113	. . . . . . . 8 ⊢ (𝑔 = 𝐺 → (𝑥((hpG‘𝑔)‘𝑎)𝑟 ↔ 𝑥((hpG‘𝐺)‘𝑎)𝑟))
26		fveq2 6890	. . . . . . . . . . 11 ⊢ (𝑔 = 𝐺 → (Itv‘𝑔) = (Itv‘𝐺))
27	26	oveqd 7441	. . . . . . . . . 10 ⊢ (𝑔 = 𝐺 → (𝑥(Itv‘𝑔)𝑟) = (𝑥(Itv‘𝐺)𝑟))
28	27	eleq2d 2846	. . . . . . . . 9 ⊢ (𝑔 = 𝐺 → (𝑡 ∈ (𝑥(Itv‘𝑔)𝑟) ↔ 𝑡 ∈ (𝑥(Itv‘𝐺)𝑟)))
29	28	rexbidv 3186	. . . . . . . 8 ⊢ (𝑔 = 𝐺 → (∃𝑡 ∈ 𝑎 𝑡 ∈ (𝑥(Itv‘𝑔)𝑟) ↔ ∃𝑡 ∈ 𝑎 𝑡 ∈ (𝑥(Itv‘𝐺)𝑟)))
30	22, 25, 29	3orbi123d 1463	. . . . . . 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 3429	. . . . . 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 7499	. . . . 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 3473	. . . . 5 ⊢ (𝜑 → 𝐺 ∈ V)
35	16	fvexi 6904	. . . . . . . 8 ⊢ 𝐿 ∈ V
36	35	rnex 7926	. . . . . . 7 ⊢ ran 𝐿 ∈ V
37	36	a1i 11	. . . . . 6 ⊢ (𝜑 → ran 𝐿 ∈ V)
38	2	difexi 5292	. . . . . . 7 ⊢ (𝑃 ∖ 𝑎) ∈ V
39	38	a1i 11	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ ran 𝐿) → (𝑃 ∖ 𝑎) ∈ V)
40	37, 39	mpoexd 8097	. . . . 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 7022	. . . 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 2807	. . 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 6637	. 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 1569   ∈ wcel 2144  ∀wral 3076  ∃wrex 3086  {crab 3412  Vcvv 3450   ∖ cdif 4044  {csn 4584  ∪ ciun 4950   class class class wbr 5102   E cep 5550   × cxp 5649  ^◡ccnv 5650  ran crn 5652   Fn wfn 6538  ⟶wf 6539  ‘cfv 6543  (class class class)co 7424   ∈ cmpo 7426  Basecbs 17396  Itvcitv 28904  LineGclng 28905  hpGchpg 29244  hlGcplng 29260

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5327  ax-pr 5391  ax-un 7755

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3848  df-ss 3897  df-dif 4046  df-in 4090  df-un 4150  df-nul 4278  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5546  df-eprel 5551  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-iota 6497  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7427  df-oprab 7428  df-mpo 7429  df-1st 8005  df-2nd 8006  df-plng 29261

This theorem is used by:  tgelrnpln  29263