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Theorem efgval2 19633

Description: Value of the free group construction. (Contributed by Mario Carneiro, 27-Sep-2015.)

**Hypotheses**
Ref	Expression
efgval.w	⊢ 𝑊 = ( I ‘Word (𝐼 × 2_o))
efgval.r	⊢ ∼ = ( ~_FG ‘𝐼)
efgval2.m	⊢ 𝑀 = (𝑦 ∈ 𝐼, 𝑧 ∈ 2_o ↦ ⟨𝑦, (1_o ∖ 𝑧)⟩)
efgval2.t	⊢ 𝑇 = (𝑣 ∈ 𝑊 ↦ (𝑛 ∈ (0...(♯‘𝑣)), 𝑤 ∈ (𝐼 × 2_o) ↦ (𝑣 splice ⟨𝑛, 𝑛, ⟨“𝑤(𝑀‘𝑤)”⟩⟩)))

**Assertion**
Ref	Expression
efgval2	⊢ ∼ = ∩ {𝑟 ∣ (𝑟 Er 𝑊 ∧ ∀𝑥 ∈ 𝑊 ran (𝑇‘𝑥) ⊆ [𝑥]𝑟)}

Distinct variable groups: 𝑦,𝑟,𝑧 𝑣,𝑛,𝑤,𝑦,𝑧,𝑟,𝑥 𝑛,𝑀 𝑣,𝑟,𝑤,𝑥,𝑀 𝑇,𝑟,𝑥 𝑛,𝑊,𝑟,𝑣,𝑤 𝑥,𝑦,𝑧,𝑊 ∼ ,𝑟,𝑥,𝑦,𝑧 𝑛,𝐼,𝑟,𝑣,𝑤,𝑥,𝑦,𝑧 Allowed substitution hints: ∼ (𝑤,𝑣,𝑛) 𝑇(𝑦,𝑧,𝑤,𝑣,𝑛) 𝑀(𝑦,𝑧)

Proof of Theorem efgval2 Dummy variables 𝑎 𝑏 𝑢 𝑚 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		efgval.w	. . 3 ⊢ 𝑊 = ( I ‘Word (𝐼 × 2_o))
2		efgval.r	. . 3 ⊢ ∼ = ( ~_FG ‘𝐼)
3	1, 2	efgval 19626	. 2 ⊢ ∼ = ∩ {𝑟 ∣ (𝑟 Er 𝑊 ∧ ∀𝑥 ∈ 𝑊 ∀𝑚 ∈ (0...(♯‘𝑥))∀𝑎 ∈ 𝐼 ∀𝑏 ∈ 2_o 𝑥𝑟(𝑥 splice ⟨𝑚, 𝑚, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1_o ∖ 𝑏)⟩”⟩⟩))}
4		efgval2.m	. . . . . . . . . . 11 ⊢ 𝑀 = (𝑦 ∈ 𝐼, 𝑧 ∈ 2_o ↦ ⟨𝑦, (1_o ∖ 𝑧)⟩)
5		efgval2.t	. . . . . . . . . . 11 ⊢ 𝑇 = (𝑣 ∈ 𝑊 ↦ (𝑛 ∈ (0...(♯‘𝑣)), 𝑤 ∈ (𝐼 × 2_o) ↦ (𝑣 splice ⟨𝑛, 𝑛, ⟨“𝑤(𝑀‘𝑤)”⟩⟩)))
6	1, 2, 4, 5	efgtf 19631	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝑊 → ((𝑇‘𝑥) = (𝑚 ∈ (0...(♯‘𝑥)), 𝑢 ∈ (𝐼 × 2_o) ↦ (𝑥 splice ⟨𝑚, 𝑚, ⟨“𝑢(𝑀‘𝑢)”⟩⟩)) ∧ (𝑇‘𝑥):((0...(♯‘𝑥)) × (𝐼 × 2_o))⟶𝑊))
7	6	simpld 493	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝑊 → (𝑇‘𝑥) = (𝑚 ∈ (0...(♯‘𝑥)), 𝑢 ∈ (𝐼 × 2_o) ↦ (𝑥 splice ⟨𝑚, 𝑚, ⟨“𝑢(𝑀‘𝑢)”⟩⟩)))
8	7	rneqd 5936	. . . . . . . 8 ⊢ (𝑥 ∈ 𝑊 → ran (𝑇‘𝑥) = ran (𝑚 ∈ (0...(♯‘𝑥)), 𝑢 ∈ (𝐼 × 2_o) ↦ (𝑥 splice ⟨𝑚, 𝑚, ⟨“𝑢(𝑀‘𝑢)”⟩⟩)))
9	8	sseq1d 4012	. . . . . . 7 ⊢ (𝑥 ∈ 𝑊 → (ran (𝑇‘𝑥) ⊆ [𝑥]𝑟 ↔ ran (𝑚 ∈ (0...(♯‘𝑥)), 𝑢 ∈ (𝐼 × 2_o) ↦ (𝑥 splice ⟨𝑚, 𝑚, ⟨“𝑢(𝑀‘𝑢)”⟩⟩)) ⊆ [𝑥]𝑟))
10		dfss3 3969	. . . . . . . 8 ⊢ (ran (𝑚 ∈ (0...(♯‘𝑥)), 𝑢 ∈ (𝐼 × 2_o) ↦ (𝑥 splice ⟨𝑚, 𝑚, ⟨“𝑢(𝑀‘𝑢)”⟩⟩)) ⊆ [𝑥]𝑟 ↔ ∀𝑎 ∈ ran (𝑚 ∈ (0...(♯‘𝑥)), 𝑢 ∈ (𝐼 × 2_o) ↦ (𝑥 splice ⟨𝑚, 𝑚, ⟨“𝑢(𝑀‘𝑢)”⟩⟩))𝑎 ∈ [𝑥]𝑟)
11		ovex 7444	. . . . . . . . . . 11 ⊢ (𝑥 splice ⟨𝑚, 𝑚, ⟨“𝑢(𝑀‘𝑢)”⟩⟩) ∈ V
12	11	rgen2w 3064	. . . . . . . . . 10 ⊢ ∀𝑚 ∈ (0...(♯‘𝑥))∀𝑢 ∈ (𝐼 × 2_o)(𝑥 splice ⟨𝑚, 𝑚, ⟨“𝑢(𝑀‘𝑢)”⟩⟩) ∈ V
13		eqid 2730	. . . . . . . . . . 11 ⊢ (𝑚 ∈ (0...(♯‘𝑥)), 𝑢 ∈ (𝐼 × 2_o) ↦ (𝑥 splice ⟨𝑚, 𝑚, ⟨“𝑢(𝑀‘𝑢)”⟩⟩)) = (𝑚 ∈ (0...(♯‘𝑥)), 𝑢 ∈ (𝐼 × 2_o) ↦ (𝑥 splice ⟨𝑚, 𝑚, ⟨“𝑢(𝑀‘𝑢)”⟩⟩))
14		vex 3476	. . . . . . . . . . . . 13 ⊢ 𝑎 ∈ V
15		vex 3476	. . . . . . . . . . . . 13 ⊢ 𝑥 ∈ V
16	14, 15	elec 8749	. . . . . . . . . . . 12 ⊢ (𝑎 ∈ [𝑥]𝑟 ↔ 𝑥𝑟𝑎)
17		breq2 5151	. . . . . . . . . . . 12 ⊢ (𝑎 = (𝑥 splice ⟨𝑚, 𝑚, ⟨“𝑢(𝑀‘𝑢)”⟩⟩) → (𝑥𝑟𝑎 ↔ 𝑥𝑟(𝑥 splice ⟨𝑚, 𝑚, ⟨“𝑢(𝑀‘𝑢)”⟩⟩)))
18	16, 17	bitrid 282	. . . . . . . . . . 11 ⊢ (𝑎 = (𝑥 splice ⟨𝑚, 𝑚, ⟨“𝑢(𝑀‘𝑢)”⟩⟩) → (𝑎 ∈ [𝑥]𝑟 ↔ 𝑥𝑟(𝑥 splice ⟨𝑚, 𝑚, ⟨“𝑢(𝑀‘𝑢)”⟩⟩)))
19	13, 18	ralrnmpo 7549	. . . . . . . . . 10 ⊢ (∀𝑚 ∈ (0...(♯‘𝑥))∀𝑢 ∈ (𝐼 × 2_o)(𝑥 splice ⟨𝑚, 𝑚, ⟨“𝑢(𝑀‘𝑢)”⟩⟩) ∈ V → (∀𝑎 ∈ ran (𝑚 ∈ (0...(♯‘𝑥)), 𝑢 ∈ (𝐼 × 2_o) ↦ (𝑥 splice ⟨𝑚, 𝑚, ⟨“𝑢(𝑀‘𝑢)”⟩⟩))𝑎 ∈ [𝑥]𝑟 ↔ ∀𝑚 ∈ (0...(♯‘𝑥))∀𝑢 ∈ (𝐼 × 2_o)𝑥𝑟(𝑥 splice ⟨𝑚, 𝑚, ⟨“𝑢(𝑀‘𝑢)”⟩⟩)))
20	12, 19	ax-mp 5	. . . . . . . . 9 ⊢ (∀𝑎 ∈ ran (𝑚 ∈ (0...(♯‘𝑥)), 𝑢 ∈ (𝐼 × 2_o) ↦ (𝑥 splice ⟨𝑚, 𝑚, ⟨“𝑢(𝑀‘𝑢)”⟩⟩))𝑎 ∈ [𝑥]𝑟 ↔ ∀𝑚 ∈ (0...(♯‘𝑥))∀𝑢 ∈ (𝐼 × 2_o)𝑥𝑟(𝑥 splice ⟨𝑚, 𝑚, ⟨“𝑢(𝑀‘𝑢)”⟩⟩))
21		id 22	. . . . . . . . . . . . . . . 16 ⊢ (𝑢 = ⟨𝑎, 𝑏⟩ → 𝑢 = ⟨𝑎, 𝑏⟩)
22		fveq2 6890	. . . . . . . . . . . . . . . . 17 ⊢ (𝑢 = ⟨𝑎, 𝑏⟩ → (𝑀‘𝑢) = (𝑀‘⟨𝑎, 𝑏⟩))
23		df-ov 7414	. . . . . . . . . . . . . . . . 17 ⊢ (𝑎𝑀𝑏) = (𝑀‘⟨𝑎, 𝑏⟩)
24	22, 23	eqtr4di 2788	. . . . . . . . . . . . . . . 16 ⊢ (𝑢 = ⟨𝑎, 𝑏⟩ → (𝑀‘𝑢) = (𝑎𝑀𝑏))
25	21, 24	s2eqd 14818	. . . . . . . . . . . . . . 15 ⊢ (𝑢 = ⟨𝑎, 𝑏⟩ → ⟨“𝑢(𝑀‘𝑢)”⟩ = ⟨“⟨𝑎, 𝑏⟩(𝑎𝑀𝑏)”⟩)
26	25	oteq3d 4886	. . . . . . . . . . . . . 14 ⊢ (𝑢 = ⟨𝑎, 𝑏⟩ → ⟨𝑚, 𝑚, ⟨“𝑢(𝑀‘𝑢)”⟩⟩ = ⟨𝑚, 𝑚, ⟨“⟨𝑎, 𝑏⟩(𝑎𝑀𝑏)”⟩⟩)
27	26	oveq2d 7427	. . . . . . . . . . . . 13 ⊢ (𝑢 = ⟨𝑎, 𝑏⟩ → (𝑥 splice ⟨𝑚, 𝑚, ⟨“𝑢(𝑀‘𝑢)”⟩⟩) = (𝑥 splice ⟨𝑚, 𝑚, ⟨“⟨𝑎, 𝑏⟩(𝑎𝑀𝑏)”⟩⟩))
28	27	breq2d 5159	. . . . . . . . . . . 12 ⊢ (𝑢 = ⟨𝑎, 𝑏⟩ → (𝑥𝑟(𝑥 splice ⟨𝑚, 𝑚, ⟨“𝑢(𝑀‘𝑢)”⟩⟩) ↔ 𝑥𝑟(𝑥 splice ⟨𝑚, 𝑚, ⟨“⟨𝑎, 𝑏⟩(𝑎𝑀𝑏)”⟩⟩)))
29	28	ralxp 5840	. . . . . . . . . . 11 ⊢ (∀𝑢 ∈ (𝐼 × 2_o)𝑥𝑟(𝑥 splice ⟨𝑚, 𝑚, ⟨“𝑢(𝑀‘𝑢)”⟩⟩) ↔ ∀𝑎 ∈ 𝐼 ∀𝑏 ∈ 2_o 𝑥𝑟(𝑥 splice ⟨𝑚, 𝑚, ⟨“⟨𝑎, 𝑏⟩(𝑎𝑀𝑏)”⟩⟩))
30		eqidd 2731	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑎 ∈ 𝐼 ∧ 𝑏 ∈ 2_o) → ⟨𝑎, 𝑏⟩ = ⟨𝑎, 𝑏⟩)
31	4	efgmval 19621	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑎 ∈ 𝐼 ∧ 𝑏 ∈ 2_o) → (𝑎𝑀𝑏) = ⟨𝑎, (1_o ∖ 𝑏)⟩)
32	30, 31	s2eqd 14818	. . . . . . . . . . . . . . . 16 ⊢ ((𝑎 ∈ 𝐼 ∧ 𝑏 ∈ 2_o) → ⟨“⟨𝑎, 𝑏⟩(𝑎𝑀𝑏)”⟩ = ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1_o ∖ 𝑏)⟩”⟩)
33	32	oteq3d 4886	. . . . . . . . . . . . . . 15 ⊢ ((𝑎 ∈ 𝐼 ∧ 𝑏 ∈ 2_o) → ⟨𝑚, 𝑚, ⟨“⟨𝑎, 𝑏⟩(𝑎𝑀𝑏)”⟩⟩ = ⟨𝑚, 𝑚, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1_o ∖ 𝑏)⟩”⟩⟩)
34	33	oveq2d 7427	. . . . . . . . . . . . . 14 ⊢ ((𝑎 ∈ 𝐼 ∧ 𝑏 ∈ 2_o) → (𝑥 splice ⟨𝑚, 𝑚, ⟨“⟨𝑎, 𝑏⟩(𝑎𝑀𝑏)”⟩⟩) = (𝑥 splice ⟨𝑚, 𝑚, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1_o ∖ 𝑏)⟩”⟩⟩))
35	34	breq2d 5159	. . . . . . . . . . . . 13 ⊢ ((𝑎 ∈ 𝐼 ∧ 𝑏 ∈ 2_o) → (𝑥𝑟(𝑥 splice ⟨𝑚, 𝑚, ⟨“⟨𝑎, 𝑏⟩(𝑎𝑀𝑏)”⟩⟩) ↔ 𝑥𝑟(𝑥 splice ⟨𝑚, 𝑚, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1_o ∖ 𝑏)⟩”⟩⟩)))
36	35	ralbidva 3173	. . . . . . . . . . . 12 ⊢ (𝑎 ∈ 𝐼 → (∀𝑏 ∈ 2_o 𝑥𝑟(𝑥 splice ⟨𝑚, 𝑚, ⟨“⟨𝑎, 𝑏⟩(𝑎𝑀𝑏)”⟩⟩) ↔ ∀𝑏 ∈ 2_o 𝑥𝑟(𝑥 splice ⟨𝑚, 𝑚, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1_o ∖ 𝑏)⟩”⟩⟩)))
37	36	ralbiia 3089	. . . . . . . . . . 11 ⊢ (∀𝑎 ∈ 𝐼 ∀𝑏 ∈ 2_o 𝑥𝑟(𝑥 splice ⟨𝑚, 𝑚, ⟨“⟨𝑎, 𝑏⟩(𝑎𝑀𝑏)”⟩⟩) ↔ ∀𝑎 ∈ 𝐼 ∀𝑏 ∈ 2_o 𝑥𝑟(𝑥 splice ⟨𝑚, 𝑚, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1_o ∖ 𝑏)⟩”⟩⟩))
38	29, 37	bitri 274	. . . . . . . . . 10 ⊢ (∀𝑢 ∈ (𝐼 × 2_o)𝑥𝑟(𝑥 splice ⟨𝑚, 𝑚, ⟨“𝑢(𝑀‘𝑢)”⟩⟩) ↔ ∀𝑎 ∈ 𝐼 ∀𝑏 ∈ 2_o 𝑥𝑟(𝑥 splice ⟨𝑚, 𝑚, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1_o ∖ 𝑏)⟩”⟩⟩))
39	38	ralbii 3091	. . . . . . . . 9 ⊢ (∀𝑚 ∈ (0...(♯‘𝑥))∀𝑢 ∈ (𝐼 × 2_o)𝑥𝑟(𝑥 splice ⟨𝑚, 𝑚, ⟨“𝑢(𝑀‘𝑢)”⟩⟩) ↔ ∀𝑚 ∈ (0...(♯‘𝑥))∀𝑎 ∈ 𝐼 ∀𝑏 ∈ 2_o 𝑥𝑟(𝑥 splice ⟨𝑚, 𝑚, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1_o ∖ 𝑏)⟩”⟩⟩))
40	20, 39	bitri 274	. . . . . . . 8 ⊢ (∀𝑎 ∈ ran (𝑚 ∈ (0...(♯‘𝑥)), 𝑢 ∈ (𝐼 × 2_o) ↦ (𝑥 splice ⟨𝑚, 𝑚, ⟨“𝑢(𝑀‘𝑢)”⟩⟩))𝑎 ∈ [𝑥]𝑟 ↔ ∀𝑚 ∈ (0...(♯‘𝑥))∀𝑎 ∈ 𝐼 ∀𝑏 ∈ 2_o 𝑥𝑟(𝑥 splice ⟨𝑚, 𝑚, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1_o ∖ 𝑏)⟩”⟩⟩))
41	10, 40	bitri 274	. . . . . . 7 ⊢ (ran (𝑚 ∈ (0...(♯‘𝑥)), 𝑢 ∈ (𝐼 × 2_o) ↦ (𝑥 splice ⟨𝑚, 𝑚, ⟨“𝑢(𝑀‘𝑢)”⟩⟩)) ⊆ [𝑥]𝑟 ↔ ∀𝑚 ∈ (0...(♯‘𝑥))∀𝑎 ∈ 𝐼 ∀𝑏 ∈ 2_o 𝑥𝑟(𝑥 splice ⟨𝑚, 𝑚, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1_o ∖ 𝑏)⟩”⟩⟩))
42	9, 41	bitrdi 286	. . . . . 6 ⊢ (𝑥 ∈ 𝑊 → (ran (𝑇‘𝑥) ⊆ [𝑥]𝑟 ↔ ∀𝑚 ∈ (0...(♯‘𝑥))∀𝑎 ∈ 𝐼 ∀𝑏 ∈ 2_o 𝑥𝑟(𝑥 splice ⟨𝑚, 𝑚, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1_o ∖ 𝑏)⟩”⟩⟩)))
43	42	ralbiia 3089	. . . . 5 ⊢ (∀𝑥 ∈ 𝑊 ran (𝑇‘𝑥) ⊆ [𝑥]𝑟 ↔ ∀𝑥 ∈ 𝑊 ∀𝑚 ∈ (0...(♯‘𝑥))∀𝑎 ∈ 𝐼 ∀𝑏 ∈ 2_o 𝑥𝑟(𝑥 splice ⟨𝑚, 𝑚, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1_o ∖ 𝑏)⟩”⟩⟩))
44	43	anbi2i 621	. . . 4 ⊢ ((𝑟 Er 𝑊 ∧ ∀𝑥 ∈ 𝑊 ran (𝑇‘𝑥) ⊆ [𝑥]𝑟) ↔ (𝑟 Er 𝑊 ∧ ∀𝑥 ∈ 𝑊 ∀𝑚 ∈ (0...(♯‘𝑥))∀𝑎 ∈ 𝐼 ∀𝑏 ∈ 2_o 𝑥𝑟(𝑥 splice ⟨𝑚, 𝑚, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1_o ∖ 𝑏)⟩”⟩⟩)))
45	44	abbii 2800	. . 3 ⊢ {𝑟 ∣ (𝑟 Er 𝑊 ∧ ∀𝑥 ∈ 𝑊 ran (𝑇‘𝑥) ⊆ [𝑥]𝑟)} = {𝑟 ∣ (𝑟 Er 𝑊 ∧ ∀𝑥 ∈ 𝑊 ∀𝑚 ∈ (0...(♯‘𝑥))∀𝑎 ∈ 𝐼 ∀𝑏 ∈ 2_o 𝑥𝑟(𝑥 splice ⟨𝑚, 𝑚, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1_o ∖ 𝑏)⟩”⟩⟩))}
46	45	inteqi 4953	. 2 ⊢ ∩ {𝑟 ∣ (𝑟 Er 𝑊 ∧ ∀𝑥 ∈ 𝑊 ran (𝑇‘𝑥) ⊆ [𝑥]𝑟)} = ∩ {𝑟 ∣ (𝑟 Er 𝑊 ∧ ∀𝑥 ∈ 𝑊 ∀𝑚 ∈ (0...(♯‘𝑥))∀𝑎 ∈ 𝐼 ∀𝑏 ∈ 2_o 𝑥𝑟(𝑥 splice ⟨𝑚, 𝑚, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1_o ∖ 𝑏)⟩”⟩⟩))}
47	3, 46	eqtr4i 2761	1 ⊢ ∼ = ∩ {𝑟 ∣ (𝑟 Er 𝑊 ∧ ∀𝑥 ∈ 𝑊 ran (𝑇‘𝑥) ⊆ [𝑥]𝑟)}

Colors of variables: wff setvar class

Syntax hints: ↔ wb 205 ∧ wa 394 = wceq 1539 ∈ wcel 2104 {cab 2707 ∀wral 3059 Vcvv 3472 ∖ cdif 3944 ⊆ wss 3947 ⟨cop 4633 ⟨cotp 4635 ∩ cint 4949 class class class wbr 5147 ↦ cmpt 5230 I cid 5572 × cxp 5673 ran crn 5676 ⟶wf 6538 ‘cfv 6542 (class class class)co 7411 ∈ cmpo 7413 1_oc1o 8461 2_oc2o 8462 Er wer 8702 [cec 8703 0cc0 11112 ...cfz 13488 ♯chash 14294 Word cword 14468 splice csplice 14703 ⟨“cs2 14796 ~_FG cefg 19615

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 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2701 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7727 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189

This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2532 df-eu 2561 df-clab 2708 df-cleq 2722 df-clel 2808 df-nfc 2883 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-reu 3375 df-rab 3431 df-v 3474 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-ot 4636 df-uni 4908 df-int 4950 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7367 df-ov 7414 df-oprab 7415 df-mpo 7416 df-om 7858 df-1st 7977 df-2nd 7978 df-frecs 8268 df-wrecs 8299 df-recs 8373 df-rdg 8412 df-1o 8468 df-2o 8469 df-er 8705 df-ec 8707 df-map 8824 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-card 9936 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-nn 12217 df-n0 12477 df-z 12563 df-uz 12827 df-fz 13489 df-fzo 13632 df-hash 14295 df-word 14469 df-concat 14525 df-s1 14550 df-substr 14595 df-pfx 14625 df-splice 14704 df-s2 14803 df-efg 19618

This theorem is referenced by: efgi2 19634 efgrelexlemb 19659 efgcpbllemb 19664

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