Theorem tgplnfn 29005

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 6885	. . . . . . 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 3084	. . . . 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 2765	. . . . . 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 8052	. . . . 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 6695	. . . 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 6158	. . . 4 ⊢ ∪ 𝑎 ∈ ran 𝐿({𝑎} × (𝑃 ∖ 𝑎)) = ((ran 𝐿 × 𝑃) ∖ ◡ E )
11	10	fneq2i 6623	. . 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 29004	. . . . 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 6871	. . . . . . . 8 ⊢ (𝑔 = 𝐺 → (LineG‘𝑔) = (LineG‘𝐺))
16		tgplnfn.l	. . . . . . . 8 ⊢ 𝐿 = (LineG‘𝐺)
17	15, 16	eqtr4di 2818	. . . . . . 7 ⊢ (𝑔 = 𝐺 → (LineG‘𝑔) = 𝐿)
18	17	rneqd 5919	. . . . . 6 ⊢ (𝑔 = 𝐺 → ran (LineG‘𝑔) = ran 𝐿)
19		fveq2 6871	. . . . . . . 8 ⊢ (𝑔 = 𝐺 → (Base‘𝑔) = (Base‘𝐺))
20	19, 1	eqtr4di 2818	. . . . . . 7 ⊢ (𝑔 = 𝐺 → (Base‘𝑔) = 𝑃)
21	20	difeq1d 4082	. . . . . 6 ⊢ (𝑔 = 𝐺 → ((Base‘𝑔) ∖ 𝑎) = (𝑃 ∖ 𝑎))
22		biidd 265	. . . . . . . 8 ⊢ (𝑔 = 𝐺 → (𝑥 ∈ 𝑎 ↔ 𝑥 ∈ 𝑎))
23		fveq2 6871	. . . . . . . . . 10 ⊢ (𝑔 = 𝐺 → (hpG‘𝑔) = (hpG‘𝐺))
24	23	fveq1d 6873	. . . . . . . . 9 ⊢ (𝑔 = 𝐺 → ((hpG‘𝑔)‘𝑎) = ((hpG‘𝐺)‘𝑎))
25	24	breqd 5116	. . . . . . . 8 ⊢ (𝑔 = 𝐺 → (𝑥((hpG‘𝑔)‘𝑎)𝑟 ↔ 𝑥((hpG‘𝐺)‘𝑎)𝑟))
26		fveq2 6871	. . . . . . . . . . 11 ⊢ (𝑔 = 𝐺 → (Itv‘𝑔) = (Itv‘𝐺))
27	26	oveqd 7417	. . . . . . . . . 10 ⊢ (𝑔 = 𝐺 → (𝑥(Itv‘𝑔)𝑟) = (𝑥(Itv‘𝐺)𝑟))
28	27	eleq2d 2851	. . . . . . . . 9 ⊢ (𝑔 = 𝐺 → (𝑡 ∈ (𝑥(Itv‘𝑔)𝑟) ↔ 𝑡 ∈ (𝑥(Itv‘𝐺)𝑟)))
29	28	rexbidv 3189	. . . . . . . 8 ⊢ (𝑔 = 𝐺 → (∃𝑡 ∈ 𝑎 𝑡 ∈ (𝑥(Itv‘𝑔)𝑟) ↔ ∃𝑡 ∈ 𝑎 𝑡 ∈ (𝑥(Itv‘𝐺)𝑟)))
30	22, 25, 29	3orbi123d 1459	. . . . . . 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 3435	. . . . . 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 7475	. . . . 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 3480	. . . . 5 ⊢ (𝜑 → 𝐺 ∈ V)
35	16	fvexi 6885	. . . . . . . 8 ⊢ 𝐿 ∈ V
36	35	rnex 7895	. . . . . . 7 ⊢ ran 𝐿 ∈ V
37	36	a1i 11	. . . . . 6 ⊢ (𝜑 → ran 𝐿 ∈ V)
38	2	difexi 5291	. . . . . . 7 ⊢ (𝑃 ∖ 𝑎) ∈ V
39	38	a1i 11	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ ran 𝐿) → (𝑃 ∖ 𝑎) ∈ V)
40	37, 39	mpoexd 8065	. . . . 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 7003	. . . 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 2812	. . 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 6618	. 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 1563   ∈ wcel 2145  ∀wral 3079  ∃wrex 3089  {crab 3417  Vcvv 3457   ∖ cdif 3904  {csn 4585  ∪ ciun 4952   class class class wbr 5105   E cep 5551   × cxp 5650  ^◡ccnv 5651  ran crn 5653   Fn wfn 6520  ⟶wf 6521  ‘cfv 6525  (class class class)co 7400   ∈ cmpo 7402  Basecbs 17259  Itvcitv 28660  LineGclng 28661  hpGchpg 28988  hlGcplng 29003

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737  ax-rep 5232  ax-sep 5251  ax-nul 5261  ax-pow 5327  ax-pr 5395  ax-un 7722

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3080  df-rex 3090  df-reu 3371  df-rab 3418  df-v 3459  df-sbc 3748  df-csb 3856  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-nul 4289  df-if 4484  df-pw 4560  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4869  df-iun 4954  df-br 5106  df-opab 5168  df-mpt 5187  df-id 5547  df-eprel 5552  df-xp 5658  df-rel 5659  df-cnv 5660  df-co 5661  df-dm 5662  df-rn 5663  df-res 5664  df-ima 5665  df-iota 6481  df-fun 6527  df-fn 6528  df-f 6529  df-f1 6530  df-fo 6531  df-f1o 6532  df-fv 6533  df-ov 7403  df-oprab 7404  df-mpo 7405  df-1st 7974  df-2nd 7975  df-plng 29004

This theorem is referenced by:  tgelrnpln  29006