Theorem fvline 36167

Description: Calculate the value of the Line function. (Contributed by Scott Fenton, 25-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)

Assertion

Ref	Expression
fvline	⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐴 ≠ 𝐵)) → (𝐴Line𝐵) = {𝑥 ∣ 𝑥 Colinear ⟨𝐴, 𝐵⟩})

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Allowed substitution hint:   𝑁(𝑥)

Proof of Theorem fvline

Dummy variables 𝑎 𝑏 𝑙 𝑛 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2736	. . . . 5 ⊢ [⟨𝐴, 𝐵⟩]◡ Colinear = [⟨𝐴, 𝐵⟩]◡ Colinear
2		fveq2 6881	. . . . . . . . 9 ⊢ (𝑛 = 𝑁 → (𝔼‘𝑛) = (𝔼‘𝑁))
3	2	eleq2d 2821	. . . . . . . 8 ⊢ (𝑛 = 𝑁 → (𝐴 ∈ (𝔼‘𝑛) ↔ 𝐴 ∈ (𝔼‘𝑁)))
4	2	eleq2d 2821	. . . . . . . 8 ⊢ (𝑛 = 𝑁 → (𝐵 ∈ (𝔼‘𝑛) ↔ 𝐵 ∈ (𝔼‘𝑁)))
5	3, 4	3anbi12d 1439	. . . . . . 7 ⊢ (𝑛 = 𝑁 → ((𝐴 ∈ (𝔼‘𝑛) ∧ 𝐵 ∈ (𝔼‘𝑛) ∧ 𝐴 ≠ 𝐵) ↔ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐴 ≠ 𝐵)))
6	5	anbi1d 631	. . . . . 6 ⊢ (𝑛 = 𝑁 → (((𝐴 ∈ (𝔼‘𝑛) ∧ 𝐵 ∈ (𝔼‘𝑛) ∧ 𝐴 ≠ 𝐵) ∧ [⟨𝐴, 𝐵⟩]◡ Colinear = [⟨𝐴, 𝐵⟩]◡ Colinear ) ↔ ((𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐴 ≠ 𝐵) ∧ [⟨𝐴, 𝐵⟩]◡ Colinear = [⟨𝐴, 𝐵⟩]◡ Colinear )))
7	6	rspcev 3606	. . . . 5 ⊢ ((𝑁 ∈ ℕ ∧ ((𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐴 ≠ 𝐵) ∧ [⟨𝐴, 𝐵⟩]◡ Colinear = [⟨𝐴, 𝐵⟩]◡ Colinear )) → ∃𝑛 ∈ ℕ ((𝐴 ∈ (𝔼‘𝑛) ∧ 𝐵 ∈ (𝔼‘𝑛) ∧ 𝐴 ≠ 𝐵) ∧ [⟨𝐴, 𝐵⟩]◡ Colinear = [⟨𝐴, 𝐵⟩]◡ Colinear ))
8	1, 7	mpanr2 704	. . . 4 ⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐴 ≠ 𝐵)) → ∃𝑛 ∈ ℕ ((𝐴 ∈ (𝔼‘𝑛) ∧ 𝐵 ∈ (𝔼‘𝑛) ∧ 𝐴 ≠ 𝐵) ∧ [⟨𝐴, 𝐵⟩]◡ Colinear = [⟨𝐴, 𝐵⟩]◡ Colinear ))
9		simpr1 1195	. . . . 5 ⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐴 ≠ 𝐵)) → 𝐴 ∈ (𝔼‘𝑁))
10		simpr2 1196	. . . . 5 ⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐴 ≠ 𝐵)) → 𝐵 ∈ (𝔼‘𝑁))
11		colinearex 36083	. . . . . . . 8 ⊢ Colinear ∈ V
12	11	cnvex 7926	. . . . . . 7 ⊢ ◡ Colinear ∈ V
13		ecexg 8728	. . . . . . 7 ⊢ (◡ Colinear ∈ V → [⟨𝐴, 𝐵⟩]◡ Colinear ∈ V)
14	12, 13	ax-mp 5	. . . . . 6 ⊢ [⟨𝐴, 𝐵⟩]◡ Colinear ∈ V
15		eleq1 2823	. . . . . . . . . 10 ⊢ (𝑎 = 𝐴 → (𝑎 ∈ (𝔼‘𝑛) ↔ 𝐴 ∈ (𝔼‘𝑛)))
16		neeq1 2995	. . . . . . . . . 10 ⊢ (𝑎 = 𝐴 → (𝑎 ≠ 𝑏 ↔ 𝐴 ≠ 𝑏))
17	15, 16	3anbi13d 1440	. . . . . . . . 9 ⊢ (𝑎 = 𝐴 → ((𝑎 ∈ (𝔼‘𝑛) ∧ 𝑏 ∈ (𝔼‘𝑛) ∧ 𝑎 ≠ 𝑏) ↔ (𝐴 ∈ (𝔼‘𝑛) ∧ 𝑏 ∈ (𝔼‘𝑛) ∧ 𝐴 ≠ 𝑏)))
18		opeq1 4854	. . . . . . . . . . 11 ⊢ (𝑎 = 𝐴 → ⟨𝑎, 𝑏⟩ = ⟨𝐴, 𝑏⟩)
19	18	eceq1d 8764	. . . . . . . . . 10 ⊢ (𝑎 = 𝐴 → [⟨𝑎, 𝑏⟩]◡ Colinear = [⟨𝐴, 𝑏⟩]◡ Colinear )
20	19	eqeq2d 2747	. . . . . . . . 9 ⊢ (𝑎 = 𝐴 → (𝑙 = [⟨𝑎, 𝑏⟩]◡ Colinear ↔ 𝑙 = [⟨𝐴, 𝑏⟩]◡ Colinear ))
21	17, 20	anbi12d 632	. . . . . . . 8 ⊢ (𝑎 = 𝐴 → (((𝑎 ∈ (𝔼‘𝑛) ∧ 𝑏 ∈ (𝔼‘𝑛) ∧ 𝑎 ≠ 𝑏) ∧ 𝑙 = [⟨𝑎, 𝑏⟩]◡ Colinear ) ↔ ((𝐴 ∈ (𝔼‘𝑛) ∧ 𝑏 ∈ (𝔼‘𝑛) ∧ 𝐴 ≠ 𝑏) ∧ 𝑙 = [⟨𝐴, 𝑏⟩]◡ Colinear )))
22	21	rexbidv 3165	. . . . . . 7 ⊢ (𝑎 = 𝐴 → (∃𝑛 ∈ ℕ ((𝑎 ∈ (𝔼‘𝑛) ∧ 𝑏 ∈ (𝔼‘𝑛) ∧ 𝑎 ≠ 𝑏) ∧ 𝑙 = [⟨𝑎, 𝑏⟩]◡ Colinear ) ↔ ∃𝑛 ∈ ℕ ((𝐴 ∈ (𝔼‘𝑛) ∧ 𝑏 ∈ (𝔼‘𝑛) ∧ 𝐴 ≠ 𝑏) ∧ 𝑙 = [⟨𝐴, 𝑏⟩]◡ Colinear )))
23		eleq1 2823	. . . . . . . . . 10 ⊢ (𝑏 = 𝐵 → (𝑏 ∈ (𝔼‘𝑛) ↔ 𝐵 ∈ (𝔼‘𝑛)))
24		neeq2 2996	. . . . . . . . . 10 ⊢ (𝑏 = 𝐵 → (𝐴 ≠ 𝑏 ↔ 𝐴 ≠ 𝐵))
25	23, 24	3anbi23d 1441	. . . . . . . . 9 ⊢ (𝑏 = 𝐵 → ((𝐴 ∈ (𝔼‘𝑛) ∧ 𝑏 ∈ (𝔼‘𝑛) ∧ 𝐴 ≠ 𝑏) ↔ (𝐴 ∈ (𝔼‘𝑛) ∧ 𝐵 ∈ (𝔼‘𝑛) ∧ 𝐴 ≠ 𝐵)))
26		opeq2 4855	. . . . . . . . . . 11 ⊢ (𝑏 = 𝐵 → ⟨𝐴, 𝑏⟩ = ⟨𝐴, 𝐵⟩)
27	26	eceq1d 8764	. . . . . . . . . 10 ⊢ (𝑏 = 𝐵 → [⟨𝐴, 𝑏⟩]◡ Colinear = [⟨𝐴, 𝐵⟩]◡ Colinear )
28	27	eqeq2d 2747	. . . . . . . . 9 ⊢ (𝑏 = 𝐵 → (𝑙 = [⟨𝐴, 𝑏⟩]◡ Colinear ↔ 𝑙 = [⟨𝐴, 𝐵⟩]◡ Colinear ))
29	25, 28	anbi12d 632	. . . . . . . 8 ⊢ (𝑏 = 𝐵 → (((𝐴 ∈ (𝔼‘𝑛) ∧ 𝑏 ∈ (𝔼‘𝑛) ∧ 𝐴 ≠ 𝑏) ∧ 𝑙 = [⟨𝐴, 𝑏⟩]◡ Colinear ) ↔ ((𝐴 ∈ (𝔼‘𝑛) ∧ 𝐵 ∈ (𝔼‘𝑛) ∧ 𝐴 ≠ 𝐵) ∧ 𝑙 = [⟨𝐴, 𝐵⟩]◡ Colinear )))
30	29	rexbidv 3165	. . . . . . 7 ⊢ (𝑏 = 𝐵 → (∃𝑛 ∈ ℕ ((𝐴 ∈ (𝔼‘𝑛) ∧ 𝑏 ∈ (𝔼‘𝑛) ∧ 𝐴 ≠ 𝑏) ∧ 𝑙 = [⟨𝐴, 𝑏⟩]◡ Colinear ) ↔ ∃𝑛 ∈ ℕ ((𝐴 ∈ (𝔼‘𝑛) ∧ 𝐵 ∈ (𝔼‘𝑛) ∧ 𝐴 ≠ 𝐵) ∧ 𝑙 = [⟨𝐴, 𝐵⟩]◡ Colinear )))
31		eqeq1 2740	. . . . . . . . 9 ⊢ (𝑙 = [⟨𝐴, 𝐵⟩]◡ Colinear → (𝑙 = [⟨𝐴, 𝐵⟩]◡ Colinear ↔ [⟨𝐴, 𝐵⟩]◡ Colinear = [⟨𝐴, 𝐵⟩]◡ Colinear ))
32	31	anbi2d 630	. . . . . . . 8 ⊢ (𝑙 = [⟨𝐴, 𝐵⟩]◡ Colinear → (((𝐴 ∈ (𝔼‘𝑛) ∧ 𝐵 ∈ (𝔼‘𝑛) ∧ 𝐴 ≠ 𝐵) ∧ 𝑙 = [⟨𝐴, 𝐵⟩]◡ Colinear ) ↔ ((𝐴 ∈ (𝔼‘𝑛) ∧ 𝐵 ∈ (𝔼‘𝑛) ∧ 𝐴 ≠ 𝐵) ∧ [⟨𝐴, 𝐵⟩]◡ Colinear = [⟨𝐴, 𝐵⟩]◡ Colinear )))
33	32	rexbidv 3165	. . . . . . 7 ⊢ (𝑙 = [⟨𝐴, 𝐵⟩]◡ Colinear → (∃𝑛 ∈ ℕ ((𝐴 ∈ (𝔼‘𝑛) ∧ 𝐵 ∈ (𝔼‘𝑛) ∧ 𝐴 ≠ 𝐵) ∧ 𝑙 = [⟨𝐴, 𝐵⟩]◡ Colinear ) ↔ ∃𝑛 ∈ ℕ ((𝐴 ∈ (𝔼‘𝑛) ∧ 𝐵 ∈ (𝔼‘𝑛) ∧ 𝐴 ≠ 𝐵) ∧ [⟨𝐴, 𝐵⟩]◡ Colinear = [⟨𝐴, 𝐵⟩]◡ Colinear )))
34	22, 30, 33	eloprabg 7522	. . . . . 6 ⊢ ((𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ [⟨𝐴, 𝐵⟩]◡ Colinear ∈ V) → (⟨⟨𝐴, 𝐵⟩, [⟨𝐴, 𝐵⟩]◡ Colinear ⟩ ∈ {⟨⟨𝑎, 𝑏⟩, 𝑙⟩ ∣ ∃𝑛 ∈ ℕ ((𝑎 ∈ (𝔼‘𝑛) ∧ 𝑏 ∈ (𝔼‘𝑛) ∧ 𝑎 ≠ 𝑏) ∧ 𝑙 = [⟨𝑎, 𝑏⟩]◡ Colinear )} ↔ ∃𝑛 ∈ ℕ ((𝐴 ∈ (𝔼‘𝑛) ∧ 𝐵 ∈ (𝔼‘𝑛) ∧ 𝐴 ≠ 𝐵) ∧ [⟨𝐴, 𝐵⟩]◡ Colinear = [⟨𝐴, 𝐵⟩]◡ Colinear )))
35	14, 34	mp3an3 1452	. . . . 5 ⊢ ((𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) → (⟨⟨𝐴, 𝐵⟩, [⟨𝐴, 𝐵⟩]◡ Colinear ⟩ ∈ {⟨⟨𝑎, 𝑏⟩, 𝑙⟩ ∣ ∃𝑛 ∈ ℕ ((𝑎 ∈ (𝔼‘𝑛) ∧ 𝑏 ∈ (𝔼‘𝑛) ∧ 𝑎 ≠ 𝑏) ∧ 𝑙 = [⟨𝑎, 𝑏⟩]◡ Colinear )} ↔ ∃𝑛 ∈ ℕ ((𝐴 ∈ (𝔼‘𝑛) ∧ 𝐵 ∈ (𝔼‘𝑛) ∧ 𝐴 ≠ 𝐵) ∧ [⟨𝐴, 𝐵⟩]◡ Colinear = [⟨𝐴, 𝐵⟩]◡ Colinear )))
36	9, 10, 35	syl2anc 584	. . . 4 ⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐴 ≠ 𝐵)) → (⟨⟨𝐴, 𝐵⟩, [⟨𝐴, 𝐵⟩]◡ Colinear ⟩ ∈ {⟨⟨𝑎, 𝑏⟩, 𝑙⟩ ∣ ∃𝑛 ∈ ℕ ((𝑎 ∈ (𝔼‘𝑛) ∧ 𝑏 ∈ (𝔼‘𝑛) ∧ 𝑎 ≠ 𝑏) ∧ 𝑙 = [⟨𝑎, 𝑏⟩]◡ Colinear )} ↔ ∃𝑛 ∈ ℕ ((𝐴 ∈ (𝔼‘𝑛) ∧ 𝐵 ∈ (𝔼‘𝑛) ∧ 𝐴 ≠ 𝐵) ∧ [⟨𝐴, 𝐵⟩]◡ Colinear = [⟨𝐴, 𝐵⟩]◡ Colinear )))
37	8, 36	mpbird 257	. . 3 ⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐴 ≠ 𝐵)) → ⟨⟨𝐴, 𝐵⟩, [⟨𝐴, 𝐵⟩]◡ Colinear ⟩ ∈ {⟨⟨𝑎, 𝑏⟩, 𝑙⟩ ∣ ∃𝑛 ∈ ℕ ((𝑎 ∈ (𝔼‘𝑛) ∧ 𝑏 ∈ (𝔼‘𝑛) ∧ 𝑎 ≠ 𝑏) ∧ 𝑙 = [⟨𝑎, 𝑏⟩]◡ Colinear )})
38		df-ov 7413	. . . 4 ⊢ (𝐴Line𝐵) = (Line‘⟨𝐴, 𝐵⟩)
39		df-br 5125	. . . . . 6 ⊢ (⟨𝐴, 𝐵⟩Line[⟨𝐴, 𝐵⟩]◡ Colinear ↔ ⟨⟨𝐴, 𝐵⟩, [⟨𝐴, 𝐵⟩]◡ Colinear ⟩ ∈ Line)
40		df-line2 36160	. . . . . . 7 ⊢ Line = {⟨⟨𝑎, 𝑏⟩, 𝑙⟩ ∣ ∃𝑛 ∈ ℕ ((𝑎 ∈ (𝔼‘𝑛) ∧ 𝑏 ∈ (𝔼‘𝑛) ∧ 𝑎 ≠ 𝑏) ∧ 𝑙 = [⟨𝑎, 𝑏⟩]◡ Colinear )}
41	40	eleq2i 2827	. . . . . 6 ⊢ (⟨⟨𝐴, 𝐵⟩, [⟨𝐴, 𝐵⟩]◡ Colinear ⟩ ∈ Line ↔ ⟨⟨𝐴, 𝐵⟩, [⟨𝐴, 𝐵⟩]◡ Colinear ⟩ ∈ {⟨⟨𝑎, 𝑏⟩, 𝑙⟩ ∣ ∃𝑛 ∈ ℕ ((𝑎 ∈ (𝔼‘𝑛) ∧ 𝑏 ∈ (𝔼‘𝑛) ∧ 𝑎 ≠ 𝑏) ∧ 𝑙 = [⟨𝑎, 𝑏⟩]◡ Colinear )})
42	39, 41	bitri 275	. . . . 5 ⊢ (⟨𝐴, 𝐵⟩Line[⟨𝐴, 𝐵⟩]◡ Colinear ↔ ⟨⟨𝐴, 𝐵⟩, [⟨𝐴, 𝐵⟩]◡ Colinear ⟩ ∈ {⟨⟨𝑎, 𝑏⟩, 𝑙⟩ ∣ ∃𝑛 ∈ ℕ ((𝑎 ∈ (𝔼‘𝑛) ∧ 𝑏 ∈ (𝔼‘𝑛) ∧ 𝑎 ≠ 𝑏) ∧ 𝑙 = [⟨𝑎, 𝑏⟩]◡ Colinear )})
43		funline 36165	. . . . . 6 ⊢ Fun Line
44		funbrfv 6932	. . . . . 6 ⊢ (Fun Line → (⟨𝐴, 𝐵⟩Line[⟨𝐴, 𝐵⟩]◡ Colinear → (Line‘⟨𝐴, 𝐵⟩) = [⟨𝐴, 𝐵⟩]◡ Colinear ))
45	43, 44	ax-mp 5	. . . . 5 ⊢ (⟨𝐴, 𝐵⟩Line[⟨𝐴, 𝐵⟩]◡ Colinear → (Line‘⟨𝐴, 𝐵⟩) = [⟨𝐴, 𝐵⟩]◡ Colinear )
46	42, 45	sylbir 235	. . . 4 ⊢ (⟨⟨𝐴, 𝐵⟩, [⟨𝐴, 𝐵⟩]◡ Colinear ⟩ ∈ {⟨⟨𝑎, 𝑏⟩, 𝑙⟩ ∣ ∃𝑛 ∈ ℕ ((𝑎 ∈ (𝔼‘𝑛) ∧ 𝑏 ∈ (𝔼‘𝑛) ∧ 𝑎 ≠ 𝑏) ∧ 𝑙 = [⟨𝑎, 𝑏⟩]◡ Colinear )} → (Line‘⟨𝐴, 𝐵⟩) = [⟨𝐴, 𝐵⟩]◡ Colinear )
47	38, 46	eqtrid 2783	. . 3 ⊢ (⟨⟨𝐴, 𝐵⟩, [⟨𝐴, 𝐵⟩]◡ Colinear ⟩ ∈ {⟨⟨𝑎, 𝑏⟩, 𝑙⟩ ∣ ∃𝑛 ∈ ℕ ((𝑎 ∈ (𝔼‘𝑛) ∧ 𝑏 ∈ (𝔼‘𝑛) ∧ 𝑎 ≠ 𝑏) ∧ 𝑙 = [⟨𝑎, 𝑏⟩]◡ Colinear )} → (𝐴Line𝐵) = [⟨𝐴, 𝐵⟩]◡ Colinear )
48	37, 47	syl 17	. 2 ⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐴 ≠ 𝐵)) → (𝐴Line𝐵) = [⟨𝐴, 𝐵⟩]◡ Colinear )
49		opex 5444	. . . 4 ⊢ ⟨𝐴, 𝐵⟩ ∈ V
50		dfec2 8727	. . . 4 ⊢ (⟨𝐴, 𝐵⟩ ∈ V → [⟨𝐴, 𝐵⟩]◡ Colinear = {𝑥 ∣ ⟨𝐴, 𝐵⟩◡ Colinear 𝑥})
51	49, 50	ax-mp 5	. . 3 ⊢ [⟨𝐴, 𝐵⟩]◡ Colinear = {𝑥 ∣ ⟨𝐴, 𝐵⟩◡ Colinear 𝑥}
52		vex 3468	. . . . 5 ⊢ 𝑥 ∈ V
53	49, 52	brcnv 5867	. . . 4 ⊢ (⟨𝐴, 𝐵⟩◡ Colinear 𝑥 ↔ 𝑥 Colinear ⟨𝐴, 𝐵⟩)
54	53	abbii 2803	. . 3 ⊢ {𝑥 ∣ ⟨𝐴, 𝐵⟩◡ Colinear 𝑥} = {𝑥 ∣ 𝑥 Colinear ⟨𝐴, 𝐵⟩}
55	51, 54	eqtri 2759	. 2 ⊢ [⟨𝐴, 𝐵⟩]◡ Colinear = {𝑥 ∣ 𝑥 Colinear ⟨𝐴, 𝐵⟩}
56	48, 55	eqtrdi 2787	1 ⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐴 ≠ 𝐵)) → (𝐴Line𝐵) = {𝑥 ∣ 𝑥 Colinear ⟨𝐴, 𝐵⟩})

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109  {cab 2714   ≠ wne 2933  ∃wrex 3061  Vcvv 3464  ⟨cop 4612   class class class wbr 5124  ^◡ccnv 5658  Fun wfun 6530  ‘cfv 6536  (class class class)co 7410  {coprab 7411  [cec 8722  ℕcn 12245  𝔼cee 28872   Colinear ccolin 36060  Linecline2 36157

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2708  ax-rep 5254  ax-sep 5271  ax-nul 5281  ax-pow 5340  ax-pr 5407  ax-un 7734  ax-cnex 11190  ax-1cn 11192  ax-addcl 11194

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2810  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3062  df-reu 3365  df-rab 3421  df-v 3466  df-sbc 3771  df-csb 3880  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-pss 3951  df-nul 4314  df-if 4506  df-pw 4582  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4889  df-iun 4974  df-br 5125  df-opab 5187  df-mpt 5207  df-tr 5235  df-id 5553  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-we 5613  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-pred 6295  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6489  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-ov 7413  df-oprab 7414  df-om 7867  df-2nd 7994  df-frecs 8285  df-wrecs 8316  df-recs 8390  df-rdg 8429  df-ec 8726  df-nn 12246  df-colinear 36062  df-line2 36160

This theorem is referenced by:  liness  36168  fvline2  36169  ellines  36175