Theorem brfs 36061

Description: Binary relation form of the general five segment predicate. (Contributed by Scott Fenton, 5-Oct-2013.)

Assertion

Ref	Expression
brfs	⊢ (((𝑁 ∈ ℕ ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁)) ∧ (𝐹 ∈ (𝔼‘𝑁) ∧ 𝐺 ∈ (𝔼‘𝑁) ∧ 𝐻 ∈ (𝔼‘𝑁))) → (⟨⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩⟩ FiveSeg ⟨⟨𝐸, 𝐹⟩, ⟨𝐺, 𝐻⟩⟩ ↔ (𝐴 Colinear ⟨𝐵, 𝐶⟩ ∧ ⟨𝐴, ⟨𝐵, 𝐶⟩⟩Cgr3⟨𝐸, ⟨𝐹, 𝐺⟩⟩ ∧ (⟨𝐴, 𝐷⟩Cgr⟨𝐸, 𝐻⟩ ∧ ⟨𝐵, 𝐷⟩Cgr⟨𝐹, 𝐻⟩))))

Proof of Theorem brfs

Dummy variables 𝑎 𝑏 𝑐 𝑑 𝑒 𝑓 𝑔 ℎ 𝑝 𝑞 𝑛 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		breq1 5151	. . 3 ⊢ (𝑎 = 𝐴 → (𝑎 Colinear ⟨𝑏, 𝑐⟩ ↔ 𝐴 Colinear ⟨𝑏, 𝑐⟩))
2		opeq1 4878	. . . 4 ⊢ (𝑎 = 𝐴 → ⟨𝑎, ⟨𝑏, 𝑐⟩⟩ = ⟨𝐴, ⟨𝑏, 𝑐⟩⟩)
3	2	breq1d 5158	. . 3 ⊢ (𝑎 = 𝐴 → (⟨𝑎, ⟨𝑏, 𝑐⟩⟩Cgr3⟨𝑒, ⟨𝑓, 𝑔⟩⟩ ↔ ⟨𝐴, ⟨𝑏, 𝑐⟩⟩Cgr3⟨𝑒, ⟨𝑓, 𝑔⟩⟩))
4		opeq1 4878	. . . . 5 ⊢ (𝑎 = 𝐴 → ⟨𝑎, 𝑑⟩ = ⟨𝐴, 𝑑⟩)
5	4	breq1d 5158	. . . 4 ⊢ (𝑎 = 𝐴 → (⟨𝑎, 𝑑⟩Cgr⟨𝑒, ℎ⟩ ↔ ⟨𝐴, 𝑑⟩Cgr⟨𝑒, ℎ⟩))
6	5	anbi1d 631	. . 3 ⊢ (𝑎 = 𝐴 → ((⟨𝑎, 𝑑⟩Cgr⟨𝑒, ℎ⟩ ∧ ⟨𝑏, 𝑑⟩Cgr⟨𝑓, ℎ⟩) ↔ (⟨𝐴, 𝑑⟩Cgr⟨𝑒, ℎ⟩ ∧ ⟨𝑏, 𝑑⟩Cgr⟨𝑓, ℎ⟩)))
7	1, 3, 6	3anbi123d 1435	. 2 ⊢ (𝑎 = 𝐴 → ((𝑎 Colinear ⟨𝑏, 𝑐⟩ ∧ ⟨𝑎, ⟨𝑏, 𝑐⟩⟩Cgr3⟨𝑒, ⟨𝑓, 𝑔⟩⟩ ∧ (⟨𝑎, 𝑑⟩Cgr⟨𝑒, ℎ⟩ ∧ ⟨𝑏, 𝑑⟩Cgr⟨𝑓, ℎ⟩)) ↔ (𝐴 Colinear ⟨𝑏, 𝑐⟩ ∧ ⟨𝐴, ⟨𝑏, 𝑐⟩⟩Cgr3⟨𝑒, ⟨𝑓, 𝑔⟩⟩ ∧ (⟨𝐴, 𝑑⟩Cgr⟨𝑒, ℎ⟩ ∧ ⟨𝑏, 𝑑⟩Cgr⟨𝑓, ℎ⟩))))
8		opeq1 4878	. . . 4 ⊢ (𝑏 = 𝐵 → ⟨𝑏, 𝑐⟩ = ⟨𝐵, 𝑐⟩)
9	8	breq2d 5160	. . 3 ⊢ (𝑏 = 𝐵 → (𝐴 Colinear ⟨𝑏, 𝑐⟩ ↔ 𝐴 Colinear ⟨𝐵, 𝑐⟩))
10	8	opeq2d 4885	. . . 4 ⊢ (𝑏 = 𝐵 → ⟨𝐴, ⟨𝑏, 𝑐⟩⟩ = ⟨𝐴, ⟨𝐵, 𝑐⟩⟩)
11	10	breq1d 5158	. . 3 ⊢ (𝑏 = 𝐵 → (⟨𝐴, ⟨𝑏, 𝑐⟩⟩Cgr3⟨𝑒, ⟨𝑓, 𝑔⟩⟩ ↔ ⟨𝐴, ⟨𝐵, 𝑐⟩⟩Cgr3⟨𝑒, ⟨𝑓, 𝑔⟩⟩))
12		opeq1 4878	. . . . 5 ⊢ (𝑏 = 𝐵 → ⟨𝑏, 𝑑⟩ = ⟨𝐵, 𝑑⟩)
13	12	breq1d 5158	. . . 4 ⊢ (𝑏 = 𝐵 → (⟨𝑏, 𝑑⟩Cgr⟨𝑓, ℎ⟩ ↔ ⟨𝐵, 𝑑⟩Cgr⟨𝑓, ℎ⟩))
14	13	anbi2d 630	. . 3 ⊢ (𝑏 = 𝐵 → ((⟨𝐴, 𝑑⟩Cgr⟨𝑒, ℎ⟩ ∧ ⟨𝑏, 𝑑⟩Cgr⟨𝑓, ℎ⟩) ↔ (⟨𝐴, 𝑑⟩Cgr⟨𝑒, ℎ⟩ ∧ ⟨𝐵, 𝑑⟩Cgr⟨𝑓, ℎ⟩)))
15	9, 11, 14	3anbi123d 1435	. 2 ⊢ (𝑏 = 𝐵 → ((𝐴 Colinear ⟨𝑏, 𝑐⟩ ∧ ⟨𝐴, ⟨𝑏, 𝑐⟩⟩Cgr3⟨𝑒, ⟨𝑓, 𝑔⟩⟩ ∧ (⟨𝐴, 𝑑⟩Cgr⟨𝑒, ℎ⟩ ∧ ⟨𝑏, 𝑑⟩Cgr⟨𝑓, ℎ⟩)) ↔ (𝐴 Colinear ⟨𝐵, 𝑐⟩ ∧ ⟨𝐴, ⟨𝐵, 𝑐⟩⟩Cgr3⟨𝑒, ⟨𝑓, 𝑔⟩⟩ ∧ (⟨𝐴, 𝑑⟩Cgr⟨𝑒, ℎ⟩ ∧ ⟨𝐵, 𝑑⟩Cgr⟨𝑓, ℎ⟩))))
16		opeq2 4879	. . . 4 ⊢ (𝑐 = 𝐶 → ⟨𝐵, 𝑐⟩ = ⟨𝐵, 𝐶⟩)
17	16	breq2d 5160	. . 3 ⊢ (𝑐 = 𝐶 → (𝐴 Colinear ⟨𝐵, 𝑐⟩ ↔ 𝐴 Colinear ⟨𝐵, 𝐶⟩))
18	16	opeq2d 4885	. . . 4 ⊢ (𝑐 = 𝐶 → ⟨𝐴, ⟨𝐵, 𝑐⟩⟩ = ⟨𝐴, ⟨𝐵, 𝐶⟩⟩)
19	18	breq1d 5158	. . 3 ⊢ (𝑐 = 𝐶 → (⟨𝐴, ⟨𝐵, 𝑐⟩⟩Cgr3⟨𝑒, ⟨𝑓, 𝑔⟩⟩ ↔ ⟨𝐴, ⟨𝐵, 𝐶⟩⟩Cgr3⟨𝑒, ⟨𝑓, 𝑔⟩⟩))
20	17, 19	3anbi12d 1436	. 2 ⊢ (𝑐 = 𝐶 → ((𝐴 Colinear ⟨𝐵, 𝑐⟩ ∧ ⟨𝐴, ⟨𝐵, 𝑐⟩⟩Cgr3⟨𝑒, ⟨𝑓, 𝑔⟩⟩ ∧ (⟨𝐴, 𝑑⟩Cgr⟨𝑒, ℎ⟩ ∧ ⟨𝐵, 𝑑⟩Cgr⟨𝑓, ℎ⟩)) ↔ (𝐴 Colinear ⟨𝐵, 𝐶⟩ ∧ ⟨𝐴, ⟨𝐵, 𝐶⟩⟩Cgr3⟨𝑒, ⟨𝑓, 𝑔⟩⟩ ∧ (⟨𝐴, 𝑑⟩Cgr⟨𝑒, ℎ⟩ ∧ ⟨𝐵, 𝑑⟩Cgr⟨𝑓, ℎ⟩))))
21		opeq2 4879	. . . . 5 ⊢ (𝑑 = 𝐷 → ⟨𝐴, 𝑑⟩ = ⟨𝐴, 𝐷⟩)
22	21	breq1d 5158	. . . 4 ⊢ (𝑑 = 𝐷 → (⟨𝐴, 𝑑⟩Cgr⟨𝑒, ℎ⟩ ↔ ⟨𝐴, 𝐷⟩Cgr⟨𝑒, ℎ⟩))
23		opeq2 4879	. . . . 5 ⊢ (𝑑 = 𝐷 → ⟨𝐵, 𝑑⟩ = ⟨𝐵, 𝐷⟩)
24	23	breq1d 5158	. . . 4 ⊢ (𝑑 = 𝐷 → (⟨𝐵, 𝑑⟩Cgr⟨𝑓, ℎ⟩ ↔ ⟨𝐵, 𝐷⟩Cgr⟨𝑓, ℎ⟩))
25	22, 24	anbi12d 632	. . 3 ⊢ (𝑑 = 𝐷 → ((⟨𝐴, 𝑑⟩Cgr⟨𝑒, ℎ⟩ ∧ ⟨𝐵, 𝑑⟩Cgr⟨𝑓, ℎ⟩) ↔ (⟨𝐴, 𝐷⟩Cgr⟨𝑒, ℎ⟩ ∧ ⟨𝐵, 𝐷⟩Cgr⟨𝑓, ℎ⟩)))
26	25	3anbi3d 1441	. 2 ⊢ (𝑑 = 𝐷 → ((𝐴 Colinear ⟨𝐵, 𝐶⟩ ∧ ⟨𝐴, ⟨𝐵, 𝐶⟩⟩Cgr3⟨𝑒, ⟨𝑓, 𝑔⟩⟩ ∧ (⟨𝐴, 𝑑⟩Cgr⟨𝑒, ℎ⟩ ∧ ⟨𝐵, 𝑑⟩Cgr⟨𝑓, ℎ⟩)) ↔ (𝐴 Colinear ⟨𝐵, 𝐶⟩ ∧ ⟨𝐴, ⟨𝐵, 𝐶⟩⟩Cgr3⟨𝑒, ⟨𝑓, 𝑔⟩⟩ ∧ (⟨𝐴, 𝐷⟩Cgr⟨𝑒, ℎ⟩ ∧ ⟨𝐵, 𝐷⟩Cgr⟨𝑓, ℎ⟩))))
27		opeq1 4878	. . . 4 ⊢ (𝑒 = 𝐸 → ⟨𝑒, ⟨𝑓, 𝑔⟩⟩ = ⟨𝐸, ⟨𝑓, 𝑔⟩⟩)
28	27	breq2d 5160	. . 3 ⊢ (𝑒 = 𝐸 → (⟨𝐴, ⟨𝐵, 𝐶⟩⟩Cgr3⟨𝑒, ⟨𝑓, 𝑔⟩⟩ ↔ ⟨𝐴, ⟨𝐵, 𝐶⟩⟩Cgr3⟨𝐸, ⟨𝑓, 𝑔⟩⟩))
29		opeq1 4878	. . . . 5 ⊢ (𝑒 = 𝐸 → ⟨𝑒, ℎ⟩ = ⟨𝐸, ℎ⟩)
30	29	breq2d 5160	. . . 4 ⊢ (𝑒 = 𝐸 → (⟨𝐴, 𝐷⟩Cgr⟨𝑒, ℎ⟩ ↔ ⟨𝐴, 𝐷⟩Cgr⟨𝐸, ℎ⟩))
31	30	anbi1d 631	. . 3 ⊢ (𝑒 = 𝐸 → ((⟨𝐴, 𝐷⟩Cgr⟨𝑒, ℎ⟩ ∧ ⟨𝐵, 𝐷⟩Cgr⟨𝑓, ℎ⟩) ↔ (⟨𝐴, 𝐷⟩Cgr⟨𝐸, ℎ⟩ ∧ ⟨𝐵, 𝐷⟩Cgr⟨𝑓, ℎ⟩)))
32	28, 31	3anbi23d 1438	. 2 ⊢ (𝑒 = 𝐸 → ((𝐴 Colinear ⟨𝐵, 𝐶⟩ ∧ ⟨𝐴, ⟨𝐵, 𝐶⟩⟩Cgr3⟨𝑒, ⟨𝑓, 𝑔⟩⟩ ∧ (⟨𝐴, 𝐷⟩Cgr⟨𝑒, ℎ⟩ ∧ ⟨𝐵, 𝐷⟩Cgr⟨𝑓, ℎ⟩)) ↔ (𝐴 Colinear ⟨𝐵, 𝐶⟩ ∧ ⟨𝐴, ⟨𝐵, 𝐶⟩⟩Cgr3⟨𝐸, ⟨𝑓, 𝑔⟩⟩ ∧ (⟨𝐴, 𝐷⟩Cgr⟨𝐸, ℎ⟩ ∧ ⟨𝐵, 𝐷⟩Cgr⟨𝑓, ℎ⟩))))
33		opeq1 4878	. . . . 5 ⊢ (𝑓 = 𝐹 → ⟨𝑓, 𝑔⟩ = ⟨𝐹, 𝑔⟩)
34	33	opeq2d 4885	. . . 4 ⊢ (𝑓 = 𝐹 → ⟨𝐸, ⟨𝑓, 𝑔⟩⟩ = ⟨𝐸, ⟨𝐹, 𝑔⟩⟩)
35	34	breq2d 5160	. . 3 ⊢ (𝑓 = 𝐹 → (⟨𝐴, ⟨𝐵, 𝐶⟩⟩Cgr3⟨𝐸, ⟨𝑓, 𝑔⟩⟩ ↔ ⟨𝐴, ⟨𝐵, 𝐶⟩⟩Cgr3⟨𝐸, ⟨𝐹, 𝑔⟩⟩))
36		opeq1 4878	. . . . 5 ⊢ (𝑓 = 𝐹 → ⟨𝑓, ℎ⟩ = ⟨𝐹, ℎ⟩)
37	36	breq2d 5160	. . . 4 ⊢ (𝑓 = 𝐹 → (⟨𝐵, 𝐷⟩Cgr⟨𝑓, ℎ⟩ ↔ ⟨𝐵, 𝐷⟩Cgr⟨𝐹, ℎ⟩))
38	37	anbi2d 630	. . 3 ⊢ (𝑓 = 𝐹 → ((⟨𝐴, 𝐷⟩Cgr⟨𝐸, ℎ⟩ ∧ ⟨𝐵, 𝐷⟩Cgr⟨𝑓, ℎ⟩) ↔ (⟨𝐴, 𝐷⟩Cgr⟨𝐸, ℎ⟩ ∧ ⟨𝐵, 𝐷⟩Cgr⟨𝐹, ℎ⟩)))
39	35, 38	3anbi23d 1438	. 2 ⊢ (𝑓 = 𝐹 → ((𝐴 Colinear ⟨𝐵, 𝐶⟩ ∧ ⟨𝐴, ⟨𝐵, 𝐶⟩⟩Cgr3⟨𝐸, ⟨𝑓, 𝑔⟩⟩ ∧ (⟨𝐴, 𝐷⟩Cgr⟨𝐸, ℎ⟩ ∧ ⟨𝐵, 𝐷⟩Cgr⟨𝑓, ℎ⟩)) ↔ (𝐴 Colinear ⟨𝐵, 𝐶⟩ ∧ ⟨𝐴, ⟨𝐵, 𝐶⟩⟩Cgr3⟨𝐸, ⟨𝐹, 𝑔⟩⟩ ∧ (⟨𝐴, 𝐷⟩Cgr⟨𝐸, ℎ⟩ ∧ ⟨𝐵, 𝐷⟩Cgr⟨𝐹, ℎ⟩))))
40		opeq2 4879	. . . . 5 ⊢ (𝑔 = 𝐺 → ⟨𝐹, 𝑔⟩ = ⟨𝐹, 𝐺⟩)
41	40	opeq2d 4885	. . . 4 ⊢ (𝑔 = 𝐺 → ⟨𝐸, ⟨𝐹, 𝑔⟩⟩ = ⟨𝐸, ⟨𝐹, 𝐺⟩⟩)
42	41	breq2d 5160	. . 3 ⊢ (𝑔 = 𝐺 → (⟨𝐴, ⟨𝐵, 𝐶⟩⟩Cgr3⟨𝐸, ⟨𝐹, 𝑔⟩⟩ ↔ ⟨𝐴, ⟨𝐵, 𝐶⟩⟩Cgr3⟨𝐸, ⟨𝐹, 𝐺⟩⟩))
43	42	3anbi2d 1440	. 2 ⊢ (𝑔 = 𝐺 → ((𝐴 Colinear ⟨𝐵, 𝐶⟩ ∧ ⟨𝐴, ⟨𝐵, 𝐶⟩⟩Cgr3⟨𝐸, ⟨𝐹, 𝑔⟩⟩ ∧ (⟨𝐴, 𝐷⟩Cgr⟨𝐸, ℎ⟩ ∧ ⟨𝐵, 𝐷⟩Cgr⟨𝐹, ℎ⟩)) ↔ (𝐴 Colinear ⟨𝐵, 𝐶⟩ ∧ ⟨𝐴, ⟨𝐵, 𝐶⟩⟩Cgr3⟨𝐸, ⟨𝐹, 𝐺⟩⟩ ∧ (⟨𝐴, 𝐷⟩Cgr⟨𝐸, ℎ⟩ ∧ ⟨𝐵, 𝐷⟩Cgr⟨𝐹, ℎ⟩))))
44		opeq2 4879	. . . . 5 ⊢ (ℎ = 𝐻 → ⟨𝐸, ℎ⟩ = ⟨𝐸, 𝐻⟩)
45	44	breq2d 5160	. . . 4 ⊢ (ℎ = 𝐻 → (⟨𝐴, 𝐷⟩Cgr⟨𝐸, ℎ⟩ ↔ ⟨𝐴, 𝐷⟩Cgr⟨𝐸, 𝐻⟩))
46		opeq2 4879	. . . . 5 ⊢ (ℎ = 𝐻 → ⟨𝐹, ℎ⟩ = ⟨𝐹, 𝐻⟩)
47	46	breq2d 5160	. . . 4 ⊢ (ℎ = 𝐻 → (⟨𝐵, 𝐷⟩Cgr⟨𝐹, ℎ⟩ ↔ ⟨𝐵, 𝐷⟩Cgr⟨𝐹, 𝐻⟩))
48	45, 47	anbi12d 632	. . 3 ⊢ (ℎ = 𝐻 → ((⟨𝐴, 𝐷⟩Cgr⟨𝐸, ℎ⟩ ∧ ⟨𝐵, 𝐷⟩Cgr⟨𝐹, ℎ⟩) ↔ (⟨𝐴, 𝐷⟩Cgr⟨𝐸, 𝐻⟩ ∧ ⟨𝐵, 𝐷⟩Cgr⟨𝐹, 𝐻⟩)))
49	48	3anbi3d 1441	. 2 ⊢ (ℎ = 𝐻 → ((𝐴 Colinear ⟨𝐵, 𝐶⟩ ∧ ⟨𝐴, ⟨𝐵, 𝐶⟩⟩Cgr3⟨𝐸, ⟨𝐹, 𝐺⟩⟩ ∧ (⟨𝐴, 𝐷⟩Cgr⟨𝐸, ℎ⟩ ∧ ⟨𝐵, 𝐷⟩Cgr⟨𝐹, ℎ⟩)) ↔ (𝐴 Colinear ⟨𝐵, 𝐶⟩ ∧ ⟨𝐴, ⟨𝐵, 𝐶⟩⟩Cgr3⟨𝐸, ⟨𝐹, 𝐺⟩⟩ ∧ (⟨𝐴, 𝐷⟩Cgr⟨𝐸, 𝐻⟩ ∧ ⟨𝐵, 𝐷⟩Cgr⟨𝐹, 𝐻⟩))))
50		fveq2 6907	. 2 ⊢ (𝑛 = 𝑁 → (𝔼‘𝑛) = (𝔼‘𝑁))
51		df-fs 36024	. 2 ⊢ FiveSeg = {⟨𝑝, 𝑞⟩ ∣ ∃𝑛 ∈ ℕ ∃𝑎 ∈ (𝔼‘𝑛)∃𝑏 ∈ (𝔼‘𝑛)∃𝑐 ∈ (𝔼‘𝑛)∃𝑑 ∈ (𝔼‘𝑛)∃𝑒 ∈ (𝔼‘𝑛)∃𝑓 ∈ (𝔼‘𝑛)∃𝑔 ∈ (𝔼‘𝑛)∃ℎ ∈ (𝔼‘𝑛)(𝑝 = ⟨⟨𝑎, 𝑏⟩, ⟨𝑐, 𝑑⟩⟩ ∧ 𝑞 = ⟨⟨𝑒, 𝑓⟩, ⟨𝑔, ℎ⟩⟩ ∧ (𝑎 Colinear ⟨𝑏, 𝑐⟩ ∧ ⟨𝑎, ⟨𝑏, 𝑐⟩⟩Cgr3⟨𝑒, ⟨𝑓, 𝑔⟩⟩ ∧ (⟨𝑎, 𝑑⟩Cgr⟨𝑒, ℎ⟩ ∧ ⟨𝑏, 𝑑⟩Cgr⟨𝑓, ℎ⟩)))}
52	7, 15, 20, 26, 32, 39, 43, 49, 50, 51	br8 35736	1 ⊢ (((𝑁 ∈ ℕ ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁)) ∧ (𝐹 ∈ (𝔼‘𝑁) ∧ 𝐺 ∈ (𝔼‘𝑁) ∧ 𝐻 ∈ (𝔼‘𝑁))) → (⟨⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩⟩ FiveSeg ⟨⟨𝐸, 𝐹⟩, ⟨𝐺, 𝐻⟩⟩ ↔ (𝐴 Colinear ⟨𝐵, 𝐶⟩ ∧ ⟨𝐴, ⟨𝐵, 𝐶⟩⟩Cgr3⟨𝐸, ⟨𝐹, 𝐺⟩⟩ ∧ (⟨𝐴, 𝐷⟩Cgr⟨𝐸, 𝐻⟩ ∧ ⟨𝐵, 𝐷⟩Cgr⟨𝐹, 𝐻⟩))))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1537   ∈ wcel 2106  ⟨cop 4637   class class class wbr 5148  ‘cfv 6563  ℕcn 12264  𝔼cee 28918  Cgrccgr 28920  Cgr3ccgr3 36018   Colinear ccolin 36019   FiveSeg cfs 36020

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-iota 6516  df-fv 6571  df-fs 36024

This theorem is referenced by:  fscgr  36062  linecgr  36063