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Theorem ishpg 25451

Description: Value of the half-plane relation for a given line 𝐷. (Contributed by Thierry Arnoux, 4-Mar-2020.)

**Hypotheses**
Ref	Expression
ishpg.p	⊢ 𝑃 = (Base‘𝐺)
ishpg.i	⊢ 𝐼 = (Itv‘𝐺)
ishpg.l	⊢ 𝐿 = (LineG‘𝐺)
ishpg.o	⊢ 𝑂 = {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝑃 ∖ 𝐷) ∧ 𝑏 ∈ (𝑃 ∖ 𝐷)) ∧ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝑎𝐼𝑏))}
ishpg.g	⊢ (𝜑 → 𝐺 ∈ TarskiG)
ishpg.d	⊢ (𝜑 → 𝐷 ∈ ran 𝐿)

**Assertion**
Ref	Expression
ishpg	⊢ (𝜑 → ((hpG‘𝐺)‘𝐷) = {⟨𝑎, 𝑏⟩ ∣ ∃𝑐 ∈ 𝑃 (𝑎𝑂𝑐 ∧ 𝑏𝑂𝑐)})

Distinct variable groups: 𝐷,𝑎,𝑏,𝑐,𝑡 𝐺,𝑎,𝑏 𝐼,𝑎,𝑏,𝑐,𝑡 𝑃,𝑎,𝑏,𝑐,𝑡 Allowed substitution hints: 𝜑(𝑡,𝑎,𝑏,𝑐) 𝐺(𝑡,𝑐) 𝐿(𝑡,𝑎,𝑏,𝑐) 𝑂(𝑡,𝑎,𝑏,𝑐)

Proof of Theorem ishpg Dummy variables 𝑑 𝑒 𝑓 𝑔 𝑖 𝑝 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ishpg.g	. . . 4 ⊢ (𝜑 → 𝐺 ∈ TarskiG)
2		elex 3185	. . . 4 ⊢ (𝐺 ∈ TarskiG → 𝐺 ∈ V)
3		fveq2 6103	. . . . . . . 8 ⊢ (𝑔 = 𝐺 → (LineG‘𝑔) = (LineG‘𝐺))
4		ishpg.l	. . . . . . . 8 ⊢ 𝐿 = (LineG‘𝐺)
5	3, 4	syl6eqr 2662	. . . . . . 7 ⊢ (𝑔 = 𝐺 → (LineG‘𝑔) = 𝐿)
6	5	rneqd 5274	. . . . . 6 ⊢ (𝑔 = 𝐺 → ran (LineG‘𝑔) = ran 𝐿)
7		ishpg.p	. . . . . . . 8 ⊢ 𝑃 = (Base‘𝐺)
8		ishpg.i	. . . . . . . 8 ⊢ 𝐼 = (Itv‘𝐺)
9		simpl 472	. . . . . . . . . 10 ⊢ ((𝑝 = 𝑃 ∧ 𝑖 = 𝐼) → 𝑝 = 𝑃)
10	9	eqcomd 2616	. . . . . . . . 9 ⊢ ((𝑝 = 𝑃 ∧ 𝑖 = 𝐼) → 𝑃 = 𝑝)
11	10	difeq1d 3689	. . . . . . . . . . . . 13 ⊢ ((𝑝 = 𝑃 ∧ 𝑖 = 𝐼) → (𝑃 ∖ 𝑑) = (𝑝 ∖ 𝑑))
12	11	eleq2d 2673	. . . . . . . . . . . 12 ⊢ ((𝑝 = 𝑃 ∧ 𝑖 = 𝐼) → (𝑎 ∈ (𝑃 ∖ 𝑑) ↔ 𝑎 ∈ (𝑝 ∖ 𝑑)))
13	11	eleq2d 2673	. . . . . . . . . . . 12 ⊢ ((𝑝 = 𝑃 ∧ 𝑖 = 𝐼) → (𝑐 ∈ (𝑃 ∖ 𝑑) ↔ 𝑐 ∈ (𝑝 ∖ 𝑑)))
14	12, 13	anbi12d 743	. . . . . . . . . . 11 ⊢ ((𝑝 = 𝑃 ∧ 𝑖 = 𝐼) → ((𝑎 ∈ (𝑃 ∖ 𝑑) ∧ 𝑐 ∈ (𝑃 ∖ 𝑑)) ↔ (𝑎 ∈ (𝑝 ∖ 𝑑) ∧ 𝑐 ∈ (𝑝 ∖ 𝑑))))
15		simpr 476	. . . . . . . . . . . . . . 15 ⊢ ((𝑝 = 𝑃 ∧ 𝑖 = 𝐼) → 𝑖 = 𝐼)
16	15	eqcomd 2616	. . . . . . . . . . . . . 14 ⊢ ((𝑝 = 𝑃 ∧ 𝑖 = 𝐼) → 𝐼 = 𝑖)
17	16	oveqd 6566	. . . . . . . . . . . . 13 ⊢ ((𝑝 = 𝑃 ∧ 𝑖 = 𝐼) → (𝑎𝐼𝑐) = (𝑎𝑖𝑐))
18	17	eleq2d 2673	. . . . . . . . . . . 12 ⊢ ((𝑝 = 𝑃 ∧ 𝑖 = 𝐼) → (𝑡 ∈ (𝑎𝐼𝑐) ↔ 𝑡 ∈ (𝑎𝑖𝑐)))
19	18	rexbidv 3034	. . . . . . . . . . 11 ⊢ ((𝑝 = 𝑃 ∧ 𝑖 = 𝐼) → (∃𝑡 ∈ 𝑑 𝑡 ∈ (𝑎𝐼𝑐) ↔ ∃𝑡 ∈ 𝑑 𝑡 ∈ (𝑎𝑖𝑐)))
20	14, 19	anbi12d 743	. . . . . . . . . 10 ⊢ ((𝑝 = 𝑃 ∧ 𝑖 = 𝐼) → (((𝑎 ∈ (𝑃 ∖ 𝑑) ∧ 𝑐 ∈ (𝑃 ∖ 𝑑)) ∧ ∃𝑡 ∈ 𝑑 𝑡 ∈ (𝑎𝐼𝑐)) ↔ ((𝑎 ∈ (𝑝 ∖ 𝑑) ∧ 𝑐 ∈ (𝑝 ∖ 𝑑)) ∧ ∃𝑡 ∈ 𝑑 𝑡 ∈ (𝑎𝑖𝑐))))
21	11	eleq2d 2673	. . . . . . . . . . . 12 ⊢ ((𝑝 = 𝑃 ∧ 𝑖 = 𝐼) → (𝑏 ∈ (𝑃 ∖ 𝑑) ↔ 𝑏 ∈ (𝑝 ∖ 𝑑)))
22	21, 13	anbi12d 743	. . . . . . . . . . 11 ⊢ ((𝑝 = 𝑃 ∧ 𝑖 = 𝐼) → ((𝑏 ∈ (𝑃 ∖ 𝑑) ∧ 𝑐 ∈ (𝑃 ∖ 𝑑)) ↔ (𝑏 ∈ (𝑝 ∖ 𝑑) ∧ 𝑐 ∈ (𝑝 ∖ 𝑑))))
23	16	oveqd 6566	. . . . . . . . . . . . 13 ⊢ ((𝑝 = 𝑃 ∧ 𝑖 = 𝐼) → (𝑏𝐼𝑐) = (𝑏𝑖𝑐))
24	23	eleq2d 2673	. . . . . . . . . . . 12 ⊢ ((𝑝 = 𝑃 ∧ 𝑖 = 𝐼) → (𝑡 ∈ (𝑏𝐼𝑐) ↔ 𝑡 ∈ (𝑏𝑖𝑐)))
25	24	rexbidv 3034	. . . . . . . . . . 11 ⊢ ((𝑝 = 𝑃 ∧ 𝑖 = 𝐼) → (∃𝑡 ∈ 𝑑 𝑡 ∈ (𝑏𝐼𝑐) ↔ ∃𝑡 ∈ 𝑑 𝑡 ∈ (𝑏𝑖𝑐)))
26	22, 25	anbi12d 743	. . . . . . . . . 10 ⊢ ((𝑝 = 𝑃 ∧ 𝑖 = 𝐼) → (((𝑏 ∈ (𝑃 ∖ 𝑑) ∧ 𝑐 ∈ (𝑃 ∖ 𝑑)) ∧ ∃𝑡 ∈ 𝑑 𝑡 ∈ (𝑏𝐼𝑐)) ↔ ((𝑏 ∈ (𝑝 ∖ 𝑑) ∧ 𝑐 ∈ (𝑝 ∖ 𝑑)) ∧ ∃𝑡 ∈ 𝑑 𝑡 ∈ (𝑏𝑖𝑐))))
27	20, 26	anbi12d 743	. . . . . . . . 9 ⊢ ((𝑝 = 𝑃 ∧ 𝑖 = 𝐼) → ((((𝑎 ∈ (𝑃 ∖ 𝑑) ∧ 𝑐 ∈ (𝑃 ∖ 𝑑)) ∧ ∃𝑡 ∈ 𝑑 𝑡 ∈ (𝑎𝐼𝑐)) ∧ ((𝑏 ∈ (𝑃 ∖ 𝑑) ∧ 𝑐 ∈ (𝑃 ∖ 𝑑)) ∧ ∃𝑡 ∈ 𝑑 𝑡 ∈ (𝑏𝐼𝑐))) ↔ (((𝑎 ∈ (𝑝 ∖ 𝑑) ∧ 𝑐 ∈ (𝑝 ∖ 𝑑)) ∧ ∃𝑡 ∈ 𝑑 𝑡 ∈ (𝑎𝑖𝑐)) ∧ ((𝑏 ∈ (𝑝 ∖ 𝑑) ∧ 𝑐 ∈ (𝑝 ∖ 𝑑)) ∧ ∃𝑡 ∈ 𝑑 𝑡 ∈ (𝑏𝑖𝑐)))))
28	10, 27	rexeqbidv 3130	. . . . . . . 8 ⊢ ((𝑝 = 𝑃 ∧ 𝑖 = 𝐼) → (∃𝑐 ∈ 𝑃 (((𝑎 ∈ (𝑃 ∖ 𝑑) ∧ 𝑐 ∈ (𝑃 ∖ 𝑑)) ∧ ∃𝑡 ∈ 𝑑 𝑡 ∈ (𝑎𝐼𝑐)) ∧ ((𝑏 ∈ (𝑃 ∖ 𝑑) ∧ 𝑐 ∈ (𝑃 ∖ 𝑑)) ∧ ∃𝑡 ∈ 𝑑 𝑡 ∈ (𝑏𝐼𝑐))) ↔ ∃𝑐 ∈ 𝑝 (((𝑎 ∈ (𝑝 ∖ 𝑑) ∧ 𝑐 ∈ (𝑝 ∖ 𝑑)) ∧ ∃𝑡 ∈ 𝑑 𝑡 ∈ (𝑎𝑖𝑐)) ∧ ((𝑏 ∈ (𝑝 ∖ 𝑑) ∧ 𝑐 ∈ (𝑝 ∖ 𝑑)) ∧ ∃𝑡 ∈ 𝑑 𝑡 ∈ (𝑏𝑖𝑐)))))
29	7, 8, 28	sbcie2s 15744	. . . . . . 7 ⊢ (𝑔 = 𝐺 → ([(Base‘𝑔) / 𝑝][(Itv‘𝑔) / 𝑖]∃𝑐 ∈ 𝑝 (((𝑎 ∈ (𝑝 ∖ 𝑑) ∧ 𝑐 ∈ (𝑝 ∖ 𝑑)) ∧ ∃𝑡 ∈ 𝑑 𝑡 ∈ (𝑎𝑖𝑐)) ∧ ((𝑏 ∈ (𝑝 ∖ 𝑑) ∧ 𝑐 ∈ (𝑝 ∖ 𝑑)) ∧ ∃𝑡 ∈ 𝑑 𝑡 ∈ (𝑏𝑖𝑐))) ↔ ∃𝑐 ∈ 𝑃 (((𝑎 ∈ (𝑃 ∖ 𝑑) ∧ 𝑐 ∈ (𝑃 ∖ 𝑑)) ∧ ∃𝑡 ∈ 𝑑 𝑡 ∈ (𝑎𝐼𝑐)) ∧ ((𝑏 ∈ (𝑃 ∖ 𝑑) ∧ 𝑐 ∈ (𝑃 ∖ 𝑑)) ∧ ∃𝑡 ∈ 𝑑 𝑡 ∈ (𝑏𝐼𝑐)))))
30	29	opabbidv 4648	. . . . . 6 ⊢ (𝑔 = 𝐺 → {⟨𝑎, 𝑏⟩ ∣ [(Base‘𝑔) / 𝑝][(Itv‘𝑔) / 𝑖]∃𝑐 ∈ 𝑝 (((𝑎 ∈ (𝑝 ∖ 𝑑) ∧ 𝑐 ∈ (𝑝 ∖ 𝑑)) ∧ ∃𝑡 ∈ 𝑑 𝑡 ∈ (𝑎𝑖𝑐)) ∧ ((𝑏 ∈ (𝑝 ∖ 𝑑) ∧ 𝑐 ∈ (𝑝 ∖ 𝑑)) ∧ ∃𝑡 ∈ 𝑑 𝑡 ∈ (𝑏𝑖𝑐)))} = {⟨𝑎, 𝑏⟩ ∣ ∃𝑐 ∈ 𝑃 (((𝑎 ∈ (𝑃 ∖ 𝑑) ∧ 𝑐 ∈ (𝑃 ∖ 𝑑)) ∧ ∃𝑡 ∈ 𝑑 𝑡 ∈ (𝑎𝐼𝑐)) ∧ ((𝑏 ∈ (𝑃 ∖ 𝑑) ∧ 𝑐 ∈ (𝑃 ∖ 𝑑)) ∧ ∃𝑡 ∈ 𝑑 𝑡 ∈ (𝑏𝐼𝑐)))})
31	6, 30	mpteq12dv 4663	. . . . 5 ⊢ (𝑔 = 𝐺 → (𝑑 ∈ ran (LineG‘𝑔) ↦ {⟨𝑎, 𝑏⟩ ∣ [(Base‘𝑔) / 𝑝][(Itv‘𝑔) / 𝑖]∃𝑐 ∈ 𝑝 (((𝑎 ∈ (𝑝 ∖ 𝑑) ∧ 𝑐 ∈ (𝑝 ∖ 𝑑)) ∧ ∃𝑡 ∈ 𝑑 𝑡 ∈ (𝑎𝑖𝑐)) ∧ ((𝑏 ∈ (𝑝 ∖ 𝑑) ∧ 𝑐 ∈ (𝑝 ∖ 𝑑)) ∧ ∃𝑡 ∈ 𝑑 𝑡 ∈ (𝑏𝑖𝑐)))}) = (𝑑 ∈ ran 𝐿 ↦ {⟨𝑎, 𝑏⟩ ∣ ∃𝑐 ∈ 𝑃 (((𝑎 ∈ (𝑃 ∖ 𝑑) ∧ 𝑐 ∈ (𝑃 ∖ 𝑑)) ∧ ∃𝑡 ∈ 𝑑 𝑡 ∈ (𝑎𝐼𝑐)) ∧ ((𝑏 ∈ (𝑃 ∖ 𝑑) ∧ 𝑐 ∈ (𝑃 ∖ 𝑑)) ∧ ∃𝑡 ∈ 𝑑 𝑡 ∈ (𝑏𝐼𝑐)))}))
32		df-hpg 25450	. . . . 5 ⊢ hpG = (𝑔 ∈ V ↦ (𝑑 ∈ ran (LineG‘𝑔) ↦ {⟨𝑎, 𝑏⟩ ∣ [(Base‘𝑔) / 𝑝][(Itv‘𝑔) / 𝑖]∃𝑐 ∈ 𝑝 (((𝑎 ∈ (𝑝 ∖ 𝑑) ∧ 𝑐 ∈ (𝑝 ∖ 𝑑)) ∧ ∃𝑡 ∈ 𝑑 𝑡 ∈ (𝑎𝑖𝑐)) ∧ ((𝑏 ∈ (𝑝 ∖ 𝑑) ∧ 𝑐 ∈ (𝑝 ∖ 𝑑)) ∧ ∃𝑡 ∈ 𝑑 𝑡 ∈ (𝑏𝑖𝑐)))}))
33		fvex 6113	. . . . . . . 8 ⊢ (LineG‘𝐺) ∈ V
34	4, 33	eqeltri 2684	. . . . . . 7 ⊢ 𝐿 ∈ V
35	34	rnex 6992	. . . . . 6 ⊢ ran 𝐿 ∈ V
36	35	mptex 6390	. . . . 5 ⊢ (𝑑 ∈ ran 𝐿 ↦ {⟨𝑎, 𝑏⟩ ∣ ∃𝑐 ∈ 𝑃 (((𝑎 ∈ (𝑃 ∖ 𝑑) ∧ 𝑐 ∈ (𝑃 ∖ 𝑑)) ∧ ∃𝑡 ∈ 𝑑 𝑡 ∈ (𝑎𝐼𝑐)) ∧ ((𝑏 ∈ (𝑃 ∖ 𝑑) ∧ 𝑐 ∈ (𝑃 ∖ 𝑑)) ∧ ∃𝑡 ∈ 𝑑 𝑡 ∈ (𝑏𝐼𝑐)))}) ∈ V
37	31, 32, 36	fvmpt 6191	. . . 4 ⊢ (𝐺 ∈ V → (hpG‘𝐺) = (𝑑 ∈ ran 𝐿 ↦ {⟨𝑎, 𝑏⟩ ∣ ∃𝑐 ∈ 𝑃 (((𝑎 ∈ (𝑃 ∖ 𝑑) ∧ 𝑐 ∈ (𝑃 ∖ 𝑑)) ∧ ∃𝑡 ∈ 𝑑 𝑡 ∈ (𝑎𝐼𝑐)) ∧ ((𝑏 ∈ (𝑃 ∖ 𝑑) ∧ 𝑐 ∈ (𝑃 ∖ 𝑑)) ∧ ∃𝑡 ∈ 𝑑 𝑡 ∈ (𝑏𝐼𝑐)))}))
38	1, 2, 37	3syl 18	. . 3 ⊢ (𝜑 → (hpG‘𝐺) = (𝑑 ∈ ran 𝐿 ↦ {⟨𝑎, 𝑏⟩ ∣ ∃𝑐 ∈ 𝑃 (((𝑎 ∈ (𝑃 ∖ 𝑑) ∧ 𝑐 ∈ (𝑃 ∖ 𝑑)) ∧ ∃𝑡 ∈ 𝑑 𝑡 ∈ (𝑎𝐼𝑐)) ∧ ((𝑏 ∈ (𝑃 ∖ 𝑑) ∧ 𝑐 ∈ (𝑃 ∖ 𝑑)) ∧ ∃𝑡 ∈ 𝑑 𝑡 ∈ (𝑏𝐼𝑐)))}))
39		difeq2 3684	. . . . . . . . . 10 ⊢ (𝑑 = 𝐷 → (𝑃 ∖ 𝑑) = (𝑃 ∖ 𝐷))
40	39	eleq2d 2673	. . . . . . . . 9 ⊢ (𝑑 = 𝐷 → (𝑎 ∈ (𝑃 ∖ 𝑑) ↔ 𝑎 ∈ (𝑃 ∖ 𝐷)))
41	39	eleq2d 2673	. . . . . . . . 9 ⊢ (𝑑 = 𝐷 → (𝑐 ∈ (𝑃 ∖ 𝑑) ↔ 𝑐 ∈ (𝑃 ∖ 𝐷)))
42	40, 41	anbi12d 743	. . . . . . . 8 ⊢ (𝑑 = 𝐷 → ((𝑎 ∈ (𝑃 ∖ 𝑑) ∧ 𝑐 ∈ (𝑃 ∖ 𝑑)) ↔ (𝑎 ∈ (𝑃 ∖ 𝐷) ∧ 𝑐 ∈ (𝑃 ∖ 𝐷))))
43		id 22	. . . . . . . . 9 ⊢ (𝑑 = 𝐷 → 𝑑 = 𝐷)
44	43	rexeqdv 3122	. . . . . . . 8 ⊢ (𝑑 = 𝐷 → (∃𝑡 ∈ 𝑑 𝑡 ∈ (𝑎𝐼𝑐) ↔ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝑎𝐼𝑐)))
45	42, 44	anbi12d 743	. . . . . . 7 ⊢ (𝑑 = 𝐷 → (((𝑎 ∈ (𝑃 ∖ 𝑑) ∧ 𝑐 ∈ (𝑃 ∖ 𝑑)) ∧ ∃𝑡 ∈ 𝑑 𝑡 ∈ (𝑎𝐼𝑐)) ↔ ((𝑎 ∈ (𝑃 ∖ 𝐷) ∧ 𝑐 ∈ (𝑃 ∖ 𝐷)) ∧ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝑎𝐼𝑐))))
46	39	eleq2d 2673	. . . . . . . . 9 ⊢ (𝑑 = 𝐷 → (𝑏 ∈ (𝑃 ∖ 𝑑) ↔ 𝑏 ∈ (𝑃 ∖ 𝐷)))
47	46, 41	anbi12d 743	. . . . . . . 8 ⊢ (𝑑 = 𝐷 → ((𝑏 ∈ (𝑃 ∖ 𝑑) ∧ 𝑐 ∈ (𝑃 ∖ 𝑑)) ↔ (𝑏 ∈ (𝑃 ∖ 𝐷) ∧ 𝑐 ∈ (𝑃 ∖ 𝐷))))
48	43	rexeqdv 3122	. . . . . . . 8 ⊢ (𝑑 = 𝐷 → (∃𝑡 ∈ 𝑑 𝑡 ∈ (𝑏𝐼𝑐) ↔ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝑏𝐼𝑐)))
49	47, 48	anbi12d 743	. . . . . . 7 ⊢ (𝑑 = 𝐷 → (((𝑏 ∈ (𝑃 ∖ 𝑑) ∧ 𝑐 ∈ (𝑃 ∖ 𝑑)) ∧ ∃𝑡 ∈ 𝑑 𝑡 ∈ (𝑏𝐼𝑐)) ↔ ((𝑏 ∈ (𝑃 ∖ 𝐷) ∧ 𝑐 ∈ (𝑃 ∖ 𝐷)) ∧ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝑏𝐼𝑐))))
50	45, 49	anbi12d 743	. . . . . 6 ⊢ (𝑑 = 𝐷 → ((((𝑎 ∈ (𝑃 ∖ 𝑑) ∧ 𝑐 ∈ (𝑃 ∖ 𝑑)) ∧ ∃𝑡 ∈ 𝑑 𝑡 ∈ (𝑎𝐼𝑐)) ∧ ((𝑏 ∈ (𝑃 ∖ 𝑑) ∧ 𝑐 ∈ (𝑃 ∖ 𝑑)) ∧ ∃𝑡 ∈ 𝑑 𝑡 ∈ (𝑏𝐼𝑐))) ↔ (((𝑎 ∈ (𝑃 ∖ 𝐷) ∧ 𝑐 ∈ (𝑃 ∖ 𝐷)) ∧ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝑎𝐼𝑐)) ∧ ((𝑏 ∈ (𝑃 ∖ 𝐷) ∧ 𝑐 ∈ (𝑃 ∖ 𝐷)) ∧ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝑏𝐼𝑐)))))
51	50	rexbidv 3034	. . . . 5 ⊢ (𝑑 = 𝐷 → (∃𝑐 ∈ 𝑃 (((𝑎 ∈ (𝑃 ∖ 𝑑) ∧ 𝑐 ∈ (𝑃 ∖ 𝑑)) ∧ ∃𝑡 ∈ 𝑑 𝑡 ∈ (𝑎𝐼𝑐)) ∧ ((𝑏 ∈ (𝑃 ∖ 𝑑) ∧ 𝑐 ∈ (𝑃 ∖ 𝑑)) ∧ ∃𝑡 ∈ 𝑑 𝑡 ∈ (𝑏𝐼𝑐))) ↔ ∃𝑐 ∈ 𝑃 (((𝑎 ∈ (𝑃 ∖ 𝐷) ∧ 𝑐 ∈ (𝑃 ∖ 𝐷)) ∧ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝑎𝐼𝑐)) ∧ ((𝑏 ∈ (𝑃 ∖ 𝐷) ∧ 𝑐 ∈ (𝑃 ∖ 𝐷)) ∧ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝑏𝐼𝑐)))))
52	51	opabbidv 4648	. . . 4 ⊢ (𝑑 = 𝐷 → {⟨𝑎, 𝑏⟩ ∣ ∃𝑐 ∈ 𝑃 (((𝑎 ∈ (𝑃 ∖ 𝑑) ∧ 𝑐 ∈ (𝑃 ∖ 𝑑)) ∧ ∃𝑡 ∈ 𝑑 𝑡 ∈ (𝑎𝐼𝑐)) ∧ ((𝑏 ∈ (𝑃 ∖ 𝑑) ∧ 𝑐 ∈ (𝑃 ∖ 𝑑)) ∧ ∃𝑡 ∈ 𝑑 𝑡 ∈ (𝑏𝐼𝑐)))} = {⟨𝑎, 𝑏⟩ ∣ ∃𝑐 ∈ 𝑃 (((𝑎 ∈ (𝑃 ∖ 𝐷) ∧ 𝑐 ∈ (𝑃 ∖ 𝐷)) ∧ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝑎𝐼𝑐)) ∧ ((𝑏 ∈ (𝑃 ∖ 𝐷) ∧ 𝑐 ∈ (𝑃 ∖ 𝐷)) ∧ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝑏𝐼𝑐)))})
53	52	adantl 481	. . 3 ⊢ ((𝜑 ∧ 𝑑 = 𝐷) → {⟨𝑎, 𝑏⟩ ∣ ∃𝑐 ∈ 𝑃 (((𝑎 ∈ (𝑃 ∖ 𝑑) ∧ 𝑐 ∈ (𝑃 ∖ 𝑑)) ∧ ∃𝑡 ∈ 𝑑 𝑡 ∈ (𝑎𝐼𝑐)) ∧ ((𝑏 ∈ (𝑃 ∖ 𝑑) ∧ 𝑐 ∈ (𝑃 ∖ 𝑑)) ∧ ∃𝑡 ∈ 𝑑 𝑡 ∈ (𝑏𝐼𝑐)))} = {⟨𝑎, 𝑏⟩ ∣ ∃𝑐 ∈ 𝑃 (((𝑎 ∈ (𝑃 ∖ 𝐷) ∧ 𝑐 ∈ (𝑃 ∖ 𝐷)) ∧ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝑎𝐼𝑐)) ∧ ((𝑏 ∈ (𝑃 ∖ 𝐷) ∧ 𝑐 ∈ (𝑃 ∖ 𝐷)) ∧ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝑏𝐼𝑐)))})
54		ishpg.d	. . 3 ⊢ (𝜑 → 𝐷 ∈ ran 𝐿)
55		df-xp 5044	. . . . . 6 ⊢ (𝑃 × 𝑃) = {⟨𝑎, 𝑏⟩ ∣ (𝑎 ∈ 𝑃 ∧ 𝑏 ∈ 𝑃)}
56		fvex 6113	. . . . . . . 8 ⊢ (Base‘𝐺) ∈ V
57	7, 56	eqeltri 2684	. . . . . . 7 ⊢ 𝑃 ∈ V
58	57, 57	xpex 6860	. . . . . 6 ⊢ (𝑃 × 𝑃) ∈ V
59	55, 58	eqeltrri 2685	. . . . 5 ⊢ {⟨𝑎, 𝑏⟩ ∣ (𝑎 ∈ 𝑃 ∧ 𝑏 ∈ 𝑃)} ∈ V
60		eldifi 3694	. . . . . . . . . . . 12 ⊢ (𝑎 ∈ (𝑃 ∖ 𝐷) → 𝑎 ∈ 𝑃)
61		eldifi 3694	. . . . . . . . . . . 12 ⊢ (𝑏 ∈ (𝑃 ∖ 𝐷) → 𝑏 ∈ 𝑃)
62	60, 61	anim12i 588	. . . . . . . . . . 11 ⊢ ((𝑎 ∈ (𝑃 ∖ 𝐷) ∧ 𝑏 ∈ (𝑃 ∖ 𝐷)) → (𝑎 ∈ 𝑃 ∧ 𝑏 ∈ 𝑃))
63	62	adantrr 749	. . . . . . . . . 10 ⊢ ((𝑎 ∈ (𝑃 ∖ 𝐷) ∧ (𝑏 ∈ (𝑃 ∖ 𝐷) ∧ 𝑐 ∈ (𝑃 ∖ 𝐷))) → (𝑎 ∈ 𝑃 ∧ 𝑏 ∈ 𝑃))
64	63	adantlr 747	. . . . . . . . 9 ⊢ (((𝑎 ∈ (𝑃 ∖ 𝐷) ∧ 𝑐 ∈ (𝑃 ∖ 𝐷)) ∧ (𝑏 ∈ (𝑃 ∖ 𝐷) ∧ 𝑐 ∈ (𝑃 ∖ 𝐷))) → (𝑎 ∈ 𝑃 ∧ 𝑏 ∈ 𝑃))
65	64	adantlr 747	. . . . . . . 8 ⊢ ((((𝑎 ∈ (𝑃 ∖ 𝐷) ∧ 𝑐 ∈ (𝑃 ∖ 𝐷)) ∧ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝑎𝐼𝑐)) ∧ (𝑏 ∈ (𝑃 ∖ 𝐷) ∧ 𝑐 ∈ (𝑃 ∖ 𝐷))) → (𝑎 ∈ 𝑃 ∧ 𝑏 ∈ 𝑃))
66	65	adantrr 749	. . . . . . 7 ⊢ ((((𝑎 ∈ (𝑃 ∖ 𝐷) ∧ 𝑐 ∈ (𝑃 ∖ 𝐷)) ∧ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝑎𝐼𝑐)) ∧ ((𝑏 ∈ (𝑃 ∖ 𝐷) ∧ 𝑐 ∈ (𝑃 ∖ 𝐷)) ∧ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝑏𝐼𝑐))) → (𝑎 ∈ 𝑃 ∧ 𝑏 ∈ 𝑃))
67	66	rexlimivw 3011	. . . . . 6 ⊢ (∃𝑐 ∈ 𝑃 (((𝑎 ∈ (𝑃 ∖ 𝐷) ∧ 𝑐 ∈ (𝑃 ∖ 𝐷)) ∧ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝑎𝐼𝑐)) ∧ ((𝑏 ∈ (𝑃 ∖ 𝐷) ∧ 𝑐 ∈ (𝑃 ∖ 𝐷)) ∧ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝑏𝐼𝑐))) → (𝑎 ∈ 𝑃 ∧ 𝑏 ∈ 𝑃))
68	67	ssopab2i 4928	. . . . 5 ⊢ {⟨𝑎, 𝑏⟩ ∣ ∃𝑐 ∈ 𝑃 (((𝑎 ∈ (𝑃 ∖ 𝐷) ∧ 𝑐 ∈ (𝑃 ∖ 𝐷)) ∧ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝑎𝐼𝑐)) ∧ ((𝑏 ∈ (𝑃 ∖ 𝐷) ∧ 𝑐 ∈ (𝑃 ∖ 𝐷)) ∧ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝑏𝐼𝑐)))} ⊆ {⟨𝑎, 𝑏⟩ ∣ (𝑎 ∈ 𝑃 ∧ 𝑏 ∈ 𝑃)}
69	59, 68	ssexi 4731	. . . 4 ⊢ {⟨𝑎, 𝑏⟩ ∣ ∃𝑐 ∈ 𝑃 (((𝑎 ∈ (𝑃 ∖ 𝐷) ∧ 𝑐 ∈ (𝑃 ∖ 𝐷)) ∧ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝑎𝐼𝑐)) ∧ ((𝑏 ∈ (𝑃 ∖ 𝐷) ∧ 𝑐 ∈ (𝑃 ∖ 𝐷)) ∧ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝑏𝐼𝑐)))} ∈ V
70	69	a1i 11	. . 3 ⊢ (𝜑 → {⟨𝑎, 𝑏⟩ ∣ ∃𝑐 ∈ 𝑃 (((𝑎 ∈ (𝑃 ∖ 𝐷) ∧ 𝑐 ∈ (𝑃 ∖ 𝐷)) ∧ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝑎𝐼𝑐)) ∧ ((𝑏 ∈ (𝑃 ∖ 𝐷) ∧ 𝑐 ∈ (𝑃 ∖ 𝐷)) ∧ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝑏𝐼𝑐)))} ∈ V)
71	38, 53, 54, 70	fvmptd 6197	. 2 ⊢ (𝜑 → ((hpG‘𝐺)‘𝐷) = {⟨𝑎, 𝑏⟩ ∣ ∃𝑐 ∈ 𝑃 (((𝑎 ∈ (𝑃 ∖ 𝐷) ∧ 𝑐 ∈ (𝑃 ∖ 𝐷)) ∧ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝑎𝐼𝑐)) ∧ ((𝑏 ∈ (𝑃 ∖ 𝐷) ∧ 𝑐 ∈ (𝑃 ∖ 𝐷)) ∧ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝑏𝐼𝑐)))})
72		vex 3176	. . . . . . 7 ⊢ 𝑎 ∈ V
73		vex 3176	. . . . . . 7 ⊢ 𝑐 ∈ V
74		eleq1 2676	. . . . . . . . 9 ⊢ (𝑒 = 𝑎 → (𝑒 ∈ (𝑃 ∖ 𝐷) ↔ 𝑎 ∈ (𝑃 ∖ 𝐷)))
75	74	anbi1d 737	. . . . . . . 8 ⊢ (𝑒 = 𝑎 → ((𝑒 ∈ (𝑃 ∖ 𝐷) ∧ 𝑓 ∈ (𝑃 ∖ 𝐷)) ↔ (𝑎 ∈ (𝑃 ∖ 𝐷) ∧ 𝑓 ∈ (𝑃 ∖ 𝐷))))
76		oveq1 6556	. . . . . . . . . 10 ⊢ (𝑒 = 𝑎 → (𝑒𝐼𝑓) = (𝑎𝐼𝑓))
77	76	eleq2d 2673	. . . . . . . . 9 ⊢ (𝑒 = 𝑎 → (𝑡 ∈ (𝑒𝐼𝑓) ↔ 𝑡 ∈ (𝑎𝐼𝑓)))
78	77	rexbidv 3034	. . . . . . . 8 ⊢ (𝑒 = 𝑎 → (∃𝑡 ∈ 𝐷 𝑡 ∈ (𝑒𝐼𝑓) ↔ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝑎𝐼𝑓)))
79	75, 78	anbi12d 743	. . . . . . 7 ⊢ (𝑒 = 𝑎 → (((𝑒 ∈ (𝑃 ∖ 𝐷) ∧ 𝑓 ∈ (𝑃 ∖ 𝐷)) ∧ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝑒𝐼𝑓)) ↔ ((𝑎 ∈ (𝑃 ∖ 𝐷) ∧ 𝑓 ∈ (𝑃 ∖ 𝐷)) ∧ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝑎𝐼𝑓))))
80		eleq1 2676	. . . . . . . . 9 ⊢ (𝑓 = 𝑐 → (𝑓 ∈ (𝑃 ∖ 𝐷) ↔ 𝑐 ∈ (𝑃 ∖ 𝐷)))
81	80	anbi2d 736	. . . . . . . 8 ⊢ (𝑓 = 𝑐 → ((𝑎 ∈ (𝑃 ∖ 𝐷) ∧ 𝑓 ∈ (𝑃 ∖ 𝐷)) ↔ (𝑎 ∈ (𝑃 ∖ 𝐷) ∧ 𝑐 ∈ (𝑃 ∖ 𝐷))))
82		oveq2 6557	. . . . . . . . . 10 ⊢ (𝑓 = 𝑐 → (𝑎𝐼𝑓) = (𝑎𝐼𝑐))
83	82	eleq2d 2673	. . . . . . . . 9 ⊢ (𝑓 = 𝑐 → (𝑡 ∈ (𝑎𝐼𝑓) ↔ 𝑡 ∈ (𝑎𝐼𝑐)))
84	83	rexbidv 3034	. . . . . . . 8 ⊢ (𝑓 = 𝑐 → (∃𝑡 ∈ 𝐷 𝑡 ∈ (𝑎𝐼𝑓) ↔ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝑎𝐼𝑐)))
85	81, 84	anbi12d 743	. . . . . . 7 ⊢ (𝑓 = 𝑐 → (((𝑎 ∈ (𝑃 ∖ 𝐷) ∧ 𝑓 ∈ (𝑃 ∖ 𝐷)) ∧ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝑎𝐼𝑓)) ↔ ((𝑎 ∈ (𝑃 ∖ 𝐷) ∧ 𝑐 ∈ (𝑃 ∖ 𝐷)) ∧ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝑎𝐼𝑐))))
86		ishpg.o	. . . . . . . 8 ⊢ 𝑂 = {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝑃 ∖ 𝐷) ∧ 𝑏 ∈ (𝑃 ∖ 𝐷)) ∧ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝑎𝐼𝑏))}
87		simpl 472	. . . . . . . . . . . 12 ⊢ ((𝑎 = 𝑒 ∧ 𝑏 = 𝑓) → 𝑎 = 𝑒)
88	87	eleq1d 2672	. . . . . . . . . . 11 ⊢ ((𝑎 = 𝑒 ∧ 𝑏 = 𝑓) → (𝑎 ∈ (𝑃 ∖ 𝐷) ↔ 𝑒 ∈ (𝑃 ∖ 𝐷)))
89		simpr 476	. . . . . . . . . . . 12 ⊢ ((𝑎 = 𝑒 ∧ 𝑏 = 𝑓) → 𝑏 = 𝑓)
90	89	eleq1d 2672	. . . . . . . . . . 11 ⊢ ((𝑎 = 𝑒 ∧ 𝑏 = 𝑓) → (𝑏 ∈ (𝑃 ∖ 𝐷) ↔ 𝑓 ∈ (𝑃 ∖ 𝐷)))
91	88, 90	anbi12d 743	. . . . . . . . . 10 ⊢ ((𝑎 = 𝑒 ∧ 𝑏 = 𝑓) → ((𝑎 ∈ (𝑃 ∖ 𝐷) ∧ 𝑏 ∈ (𝑃 ∖ 𝐷)) ↔ (𝑒 ∈ (𝑃 ∖ 𝐷) ∧ 𝑓 ∈ (𝑃 ∖ 𝐷))))
92		oveq12 6558	. . . . . . . . . . . 12 ⊢ ((𝑎 = 𝑒 ∧ 𝑏 = 𝑓) → (𝑎𝐼𝑏) = (𝑒𝐼𝑓))
93	92	eleq2d 2673	. . . . . . . . . . 11 ⊢ ((𝑎 = 𝑒 ∧ 𝑏 = 𝑓) → (𝑡 ∈ (𝑎𝐼𝑏) ↔ 𝑡 ∈ (𝑒𝐼𝑓)))
94	93	rexbidv 3034	. . . . . . . . . 10 ⊢ ((𝑎 = 𝑒 ∧ 𝑏 = 𝑓) → (∃𝑡 ∈ 𝐷 𝑡 ∈ (𝑎𝐼𝑏) ↔ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝑒𝐼𝑓)))
95	91, 94	anbi12d 743	. . . . . . . . 9 ⊢ ((𝑎 = 𝑒 ∧ 𝑏 = 𝑓) → (((𝑎 ∈ (𝑃 ∖ 𝐷) ∧ 𝑏 ∈ (𝑃 ∖ 𝐷)) ∧ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝑎𝐼𝑏)) ↔ ((𝑒 ∈ (𝑃 ∖ 𝐷) ∧ 𝑓 ∈ (𝑃 ∖ 𝐷)) ∧ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝑒𝐼𝑓))))
96	95	cbvopabv 4654	. . . . . . . 8 ⊢ {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝑃 ∖ 𝐷) ∧ 𝑏 ∈ (𝑃 ∖ 𝐷)) ∧ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝑎𝐼𝑏))} = {⟨𝑒, 𝑓⟩ ∣ ((𝑒 ∈ (𝑃 ∖ 𝐷) ∧ 𝑓 ∈ (𝑃 ∖ 𝐷)) ∧ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝑒𝐼𝑓))}
97	86, 96	eqtri 2632	. . . . . . 7 ⊢ 𝑂 = {⟨𝑒, 𝑓⟩ ∣ ((𝑒 ∈ (𝑃 ∖ 𝐷) ∧ 𝑓 ∈ (𝑃 ∖ 𝐷)) ∧ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝑒𝐼𝑓))}
98	72, 73, 79, 85, 97	brab 4923	. . . . . 6 ⊢ (𝑎𝑂𝑐 ↔ ((𝑎 ∈ (𝑃 ∖ 𝐷) ∧ 𝑐 ∈ (𝑃 ∖ 𝐷)) ∧ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝑎𝐼𝑐)))
99		vex 3176	. . . . . . 7 ⊢ 𝑏 ∈ V
100		eleq1 2676	. . . . . . . . 9 ⊢ (𝑒 = 𝑏 → (𝑒 ∈ (𝑃 ∖ 𝐷) ↔ 𝑏 ∈ (𝑃 ∖ 𝐷)))
101	100	anbi1d 737	. . . . . . . 8 ⊢ (𝑒 = 𝑏 → ((𝑒 ∈ (𝑃 ∖ 𝐷) ∧ 𝑓 ∈ (𝑃 ∖ 𝐷)) ↔ (𝑏 ∈ (𝑃 ∖ 𝐷) ∧ 𝑓 ∈ (𝑃 ∖ 𝐷))))
102		oveq1 6556	. . . . . . . . . 10 ⊢ (𝑒 = 𝑏 → (𝑒𝐼𝑓) = (𝑏𝐼𝑓))
103	102	eleq2d 2673	. . . . . . . . 9 ⊢ (𝑒 = 𝑏 → (𝑡 ∈ (𝑒𝐼𝑓) ↔ 𝑡 ∈ (𝑏𝐼𝑓)))
104	103	rexbidv 3034	. . . . . . . 8 ⊢ (𝑒 = 𝑏 → (∃𝑡 ∈ 𝐷 𝑡 ∈ (𝑒𝐼𝑓) ↔ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝑏𝐼𝑓)))
105	101, 104	anbi12d 743	. . . . . . 7 ⊢ (𝑒 = 𝑏 → (((𝑒 ∈ (𝑃 ∖ 𝐷) ∧ 𝑓 ∈ (𝑃 ∖ 𝐷)) ∧ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝑒𝐼𝑓)) ↔ ((𝑏 ∈ (𝑃 ∖ 𝐷) ∧ 𝑓 ∈ (𝑃 ∖ 𝐷)) ∧ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝑏𝐼𝑓))))
106	80	anbi2d 736	. . . . . . . 8 ⊢ (𝑓 = 𝑐 → ((𝑏 ∈ (𝑃 ∖ 𝐷) ∧ 𝑓 ∈ (𝑃 ∖ 𝐷)) ↔ (𝑏 ∈ (𝑃 ∖ 𝐷) ∧ 𝑐 ∈ (𝑃 ∖ 𝐷))))
107		oveq2 6557	. . . . . . . . . 10 ⊢ (𝑓 = 𝑐 → (𝑏𝐼𝑓) = (𝑏𝐼𝑐))
108	107	eleq2d 2673	. . . . . . . . 9 ⊢ (𝑓 = 𝑐 → (𝑡 ∈ (𝑏𝐼𝑓) ↔ 𝑡 ∈ (𝑏𝐼𝑐)))
109	108	rexbidv 3034	. . . . . . . 8 ⊢ (𝑓 = 𝑐 → (∃𝑡 ∈ 𝐷 𝑡 ∈ (𝑏𝐼𝑓) ↔ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝑏𝐼𝑐)))
110	106, 109	anbi12d 743	. . . . . . 7 ⊢ (𝑓 = 𝑐 → (((𝑏 ∈ (𝑃 ∖ 𝐷) ∧ 𝑓 ∈ (𝑃 ∖ 𝐷)) ∧ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝑏𝐼𝑓)) ↔ ((𝑏 ∈ (𝑃 ∖ 𝐷) ∧ 𝑐 ∈ (𝑃 ∖ 𝐷)) ∧ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝑏𝐼𝑐))))
111	99, 73, 105, 110, 97	brab 4923	. . . . . 6 ⊢ (𝑏𝑂𝑐 ↔ ((𝑏 ∈ (𝑃 ∖ 𝐷) ∧ 𝑐 ∈ (𝑃 ∖ 𝐷)) ∧ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝑏𝐼𝑐)))
112	98, 111	anbi12i 729	. . . . 5 ⊢ ((𝑎𝑂𝑐 ∧ 𝑏𝑂𝑐) ↔ (((𝑎 ∈ (𝑃 ∖ 𝐷) ∧ 𝑐 ∈ (𝑃 ∖ 𝐷)) ∧ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝑎𝐼𝑐)) ∧ ((𝑏 ∈ (𝑃 ∖ 𝐷) ∧ 𝑐 ∈ (𝑃 ∖ 𝐷)) ∧ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝑏𝐼𝑐))))
113	112	rexbii 3023	. . . 4 ⊢ (∃𝑐 ∈ 𝑃 (𝑎𝑂𝑐 ∧ 𝑏𝑂𝑐) ↔ ∃𝑐 ∈ 𝑃 (((𝑎 ∈ (𝑃 ∖ 𝐷) ∧ 𝑐 ∈ (𝑃 ∖ 𝐷)) ∧ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝑎𝐼𝑐)) ∧ ((𝑏 ∈ (𝑃 ∖ 𝐷) ∧ 𝑐 ∈ (𝑃 ∖ 𝐷)) ∧ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝑏𝐼𝑐))))
114	113	opabbii 4649	. . 3 ⊢ {⟨𝑎, 𝑏⟩ ∣ ∃𝑐 ∈ 𝑃 (𝑎𝑂𝑐 ∧ 𝑏𝑂𝑐)} = {⟨𝑎, 𝑏⟩ ∣ ∃𝑐 ∈ 𝑃 (((𝑎 ∈ (𝑃 ∖ 𝐷) ∧ 𝑐 ∈ (𝑃 ∖ 𝐷)) ∧ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝑎𝐼𝑐)) ∧ ((𝑏 ∈ (𝑃 ∖ 𝐷) ∧ 𝑐 ∈ (𝑃 ∖ 𝐷)) ∧ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝑏𝐼𝑐)))}
115	114	a1i 11	. 2 ⊢ (𝜑 → {⟨𝑎, 𝑏⟩ ∣ ∃𝑐 ∈ 𝑃 (𝑎𝑂𝑐 ∧ 𝑏𝑂𝑐)} = {⟨𝑎, 𝑏⟩ ∣ ∃𝑐 ∈ 𝑃 (((𝑎 ∈ (𝑃 ∖ 𝐷) ∧ 𝑐 ∈ (𝑃 ∖ 𝐷)) ∧ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝑎𝐼𝑐)) ∧ ((𝑏 ∈ (𝑃 ∖ 𝐷) ∧ 𝑐 ∈ (𝑃 ∖ 𝐷)) ∧ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝑏𝐼𝑐)))})
116	71, 115	eqtr4d 2647	1 ⊢ (𝜑 → ((hpG‘𝐺)‘𝐷) = {⟨𝑎, 𝑏⟩ ∣ ∃𝑐 ∈ 𝑃 (𝑎𝑂𝑐 ∧ 𝑏𝑂𝑐)})

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ∃wrex 2897 Vcvv 3173 [wsbc 3402 ∖ cdif 3537 class class class wbr 4583 {copab 4642 ↦ cmpt 4643 × cxp 5036 ran crn 5039 ‘cfv 5804 (class class class)co 6549 Basecbs 15695 TarskiGcstrkg 25129 Itvcitv 25135 LineGclng 25136 hpGchpg 25449

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847

This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-ov 6552 df-hpg 25450

This theorem is referenced by: hpgbr 25452

W3C validator