Theorem efgval2 19938

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

Hypotheses

Ref	Expression
efgval.w	⊢ 𝑊 = ( I ‘Word (𝐼 × 2o))
efgval.r	⊢ ∼ = ( ~FG ‘𝐼)
efgval2.m	⊢ 𝑀 = (𝑦 ∈ 𝐼, 𝑧 ∈ 2o ↦ ⟨𝑦, (1o ∖ 𝑧)⟩)
efgval2.t	⊢ 𝑇 = (𝑣 ∈ 𝑊 ↦ (𝑛 ∈ (0...(♯‘𝑣)), 𝑤 ∈ (𝐼 × 2o) ↦ (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 × 2o))
2		efgval.r	. . 3 ⊢ ∼ = ( ~FG ‘𝐼)
3	1, 2	efgval 19931	. 2 ⊢ ∼ = ∩ {𝑟 ∣ (𝑟 Er 𝑊 ∧ ∀𝑥 ∈ 𝑊 ∀𝑚 ∈ (0...(♯‘𝑥))∀𝑎 ∈ 𝐼 ∀𝑏 ∈ 2o 𝑥𝑟(𝑥 splice ⟨𝑚, 𝑚, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o ∖ 𝑏)⟩”⟩⟩))}
4		efgval2.m	. . . . . . . . . . 11 ⊢ 𝑀 = (𝑦 ∈ 𝐼, 𝑧 ∈ 2o ↦ ⟨𝑦, (1o ∖ 𝑧)⟩)
5		efgval2.t	. . . . . . . . . . 11 ⊢ 𝑇 = (𝑣 ∈ 𝑊 ↦ (𝑛 ∈ (0...(♯‘𝑣)), 𝑤 ∈ (𝐼 × 2o) ↦ (𝑣 splice ⟨𝑛, 𝑛, ⟨“𝑤(𝑀‘𝑤)”⟩⟩)))
6	1, 2, 4, 5	efgtf 19936	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝑊 → ((𝑇‘𝑥) = (𝑚 ∈ (0...(♯‘𝑥)), 𝑢 ∈ (𝐼 × 2o) ↦ (𝑥 splice ⟨𝑚, 𝑚, ⟨“𝑢(𝑀‘𝑢)”⟩⟩)) ∧ (𝑇‘𝑥):((0...(♯‘𝑥)) × (𝐼 × 2o))⟶𝑊))
7	6	simpld 500	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝑊 → (𝑇‘𝑥) = (𝑚 ∈ (0...(♯‘𝑥)), 𝑢 ∈ (𝐼 × 2o) ↦ (𝑥 splice ⟨𝑚, 𝑚, ⟨“𝑢(𝑀‘𝑢)”⟩⟩)))
8	7	rneqd 5920	. . . . . . . 8 ⊢ (𝑥 ∈ 𝑊 → ran (𝑇‘𝑥) = ran (𝑚 ∈ (0...(♯‘𝑥)), 𝑢 ∈ (𝐼 × 2o) ↦ (𝑥 splice ⟨𝑚, 𝑚, ⟨“𝑢(𝑀‘𝑢)”⟩⟩)))
9	8	sseq1d 3962	. . . . . . 7 ⊢ (𝑥 ∈ 𝑊 → (ran (𝑇‘𝑥) ⊆ [𝑥]𝑟 ↔ ran (𝑚 ∈ (0...(♯‘𝑥)), 𝑢 ∈ (𝐼 × 2o) ↦ (𝑥 splice ⟨𝑚, 𝑚, ⟨“𝑢(𝑀‘𝑢)”⟩⟩)) ⊆ [𝑥]𝑟))
10		dfss3 3920	. . . . . . . 8 ⊢ (ran (𝑚 ∈ (0...(♯‘𝑥)), 𝑢 ∈ (𝐼 × 2o) ↦ (𝑥 splice ⟨𝑚, 𝑚, ⟨“𝑢(𝑀‘𝑢)”⟩⟩)) ⊆ [𝑥]𝑟 ↔ ∀𝑎 ∈ ran (𝑚 ∈ (0...(♯‘𝑥)), 𝑢 ∈ (𝐼 × 2o) ↦ (𝑥 splice ⟨𝑚, 𝑚, ⟨“𝑢(𝑀‘𝑢)”⟩⟩))𝑎 ∈ [𝑥]𝑟)
11		ovex 7453	. . . . . . . . . . 11 ⊢ (𝑥 splice ⟨𝑚, 𝑚, ⟨“𝑢(𝑀‘𝑢)”⟩⟩) ∈ V
12	11	rgen2w 3082	. . . . . . . . . 10 ⊢ ∀𝑚 ∈ (0...(♯‘𝑥))∀𝑢 ∈ (𝐼 × 2o)(𝑥 splice ⟨𝑚, 𝑚, ⟨“𝑢(𝑀‘𝑢)”⟩⟩) ∈ V
13		eqid 2761	. . . . . . . . . . 11 ⊢ (𝑚 ∈ (0...(♯‘𝑥)), 𝑢 ∈ (𝐼 × 2o) ↦ (𝑥 splice ⟨𝑚, 𝑚, ⟨“𝑢(𝑀‘𝑢)”⟩⟩)) = (𝑚 ∈ (0...(♯‘𝑥)), 𝑢 ∈ (𝐼 × 2o) ↦ (𝑥 splice ⟨𝑚, 𝑚, ⟨“𝑢(𝑀‘𝑢)”⟩⟩))
14		vex 3455	. . . . . . . . . . . . 13 ⊢ 𝑎 ∈ V
15		vex 3455	. . . . . . . . . . . . 13 ⊢ 𝑥 ∈ V
16	14, 15	elec 8764	. . . . . . . . . . . 12 ⊢ (𝑎 ∈ [𝑥]𝑟 ↔ 𝑥𝑟𝑎)
17		breq2 5107	. . . . . . . . . . . 12 ⊢ (𝑎 = (𝑥 splice ⟨𝑚, 𝑚, ⟨“𝑢(𝑀‘𝑢)”⟩⟩) → (𝑥𝑟𝑎 ↔ 𝑥𝑟(𝑥 splice ⟨𝑚, 𝑚, ⟨“𝑢(𝑀‘𝑢)”⟩⟩)))
18	16, 17	bitrid 286	. . . . . . . . . . 11 ⊢ (𝑎 = (𝑥 splice ⟨𝑚, 𝑚, ⟨“𝑢(𝑀‘𝑢)”⟩⟩) → (𝑎 ∈ [𝑥]𝑟 ↔ 𝑥𝑟(𝑥 splice ⟨𝑚, 𝑚, ⟨“𝑢(𝑀‘𝑢)”⟩⟩)))
19	13, 18	ralrnmpo 7559	. . . . . . . . . 10 ⊢ (∀𝑚 ∈ (0...(♯‘𝑥))∀𝑢 ∈ (𝐼 × 2o)(𝑥 splice ⟨𝑚, 𝑚, ⟨“𝑢(𝑀‘𝑢)”⟩⟩) ∈ V → (∀𝑎 ∈ ran (𝑚 ∈ (0...(♯‘𝑥)), 𝑢 ∈ (𝐼 × 2o) ↦ (𝑥 splice ⟨𝑚, 𝑚, ⟨“𝑢(𝑀‘𝑢)”⟩⟩))𝑎 ∈ [𝑥]𝑟 ↔ ∀𝑚 ∈ (0...(♯‘𝑥))∀𝑢 ∈ (𝐼 × 2o)𝑥𝑟(𝑥 splice ⟨𝑚, 𝑚, ⟨“𝑢(𝑀‘𝑢)”⟩⟩)))
20	12, 19	ax-mp 5	. . . . . . . . 9 ⊢ (∀𝑎 ∈ ran (𝑚 ∈ (0...(♯‘𝑥)), 𝑢 ∈ (𝐼 × 2o) ↦ (𝑥 splice ⟨𝑚, 𝑚, ⟨“𝑢(𝑀‘𝑢)”⟩⟩))𝑎 ∈ [𝑥]𝑟 ↔ ∀𝑚 ∈ (0...(♯‘𝑥))∀𝑢 ∈ (𝐼 × 2o)𝑥𝑟(𝑥 splice ⟨𝑚, 𝑚, ⟨“𝑢(𝑀‘𝑢)”⟩⟩))
21		id 23	. . . . . . . . . . . . . . . 16 ⊢ (𝑢 = ⟨𝑎, 𝑏⟩ → 𝑢 = ⟨𝑎, 𝑏⟩)
22		fveq2 6885	. . . . . . . . . . . . . . . . 17 ⊢ (𝑢 = ⟨𝑎, 𝑏⟩ → (𝑀‘𝑢) = (𝑀‘⟨𝑎, 𝑏⟩))
23		df-ov 7423	. . . . . . . . . . . . . . . . 17 ⊢ (𝑎𝑀𝑏) = (𝑀‘⟨𝑎, 𝑏⟩)
24	22, 23	eqtr4di 2814	. . . . . . . . . . . . . . . 16 ⊢ (𝑢 = ⟨𝑎, 𝑏⟩ → (𝑀‘𝑢) = (𝑎𝑀𝑏))
25	21, 24	s2eqd 15014	. . . . . . . . . . . . . . 15 ⊢ (𝑢 = ⟨𝑎, 𝑏⟩ → ⟨“𝑢(𝑀‘𝑢)”⟩ = ⟨“⟨𝑎, 𝑏⟩(𝑎𝑀𝑏)”⟩)
26	25	oteq3d 4847	. . . . . . . . . . . . . 14 ⊢ (𝑢 = ⟨𝑎, 𝑏⟩ → ⟨𝑚, 𝑚, ⟨“𝑢(𝑀‘𝑢)”⟩⟩ = ⟨𝑚, 𝑚, ⟨“⟨𝑎, 𝑏⟩(𝑎𝑀𝑏)”⟩⟩)
27	26	oveq2d 7436	. . . . . . . . . . . . 13 ⊢ (𝑢 = ⟨𝑎, 𝑏⟩ → (𝑥 splice ⟨𝑚, 𝑚, ⟨“𝑢(𝑀‘𝑢)”⟩⟩) = (𝑥 splice ⟨𝑚, 𝑚, ⟨“⟨𝑎, 𝑏⟩(𝑎𝑀𝑏)”⟩⟩))
28	27	breq2d 5115	. . . . . . . . . . . 12 ⊢ (𝑢 = ⟨𝑎, 𝑏⟩ → (𝑥𝑟(𝑥 splice ⟨𝑚, 𝑚, ⟨“𝑢(𝑀‘𝑢)”⟩⟩) ↔ 𝑥𝑟(𝑥 splice ⟨𝑚, 𝑚, ⟨“⟨𝑎, 𝑏⟩(𝑎𝑀𝑏)”⟩⟩)))
29	28	ralxp 5818	. . . . . . . . . . 11 ⊢ (∀𝑢 ∈ (𝐼 × 2o)𝑥𝑟(𝑥 splice ⟨𝑚, 𝑚, ⟨“𝑢(𝑀‘𝑢)”⟩⟩) ↔ ∀𝑎 ∈ 𝐼 ∀𝑏 ∈ 2o 𝑥𝑟(𝑥 splice ⟨𝑚, 𝑚, ⟨“⟨𝑎, 𝑏⟩(𝑎𝑀𝑏)”⟩⟩))
30		eqidd 2762	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑎 ∈ 𝐼 ∧ 𝑏 ∈ 2o) → ⟨𝑎, 𝑏⟩ = ⟨𝑎, 𝑏⟩)
31	4	efgmval 19926	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑎 ∈ 𝐼 ∧ 𝑏 ∈ 2o) → (𝑎𝑀𝑏) = ⟨𝑎, (1o ∖ 𝑏)⟩)
32	30, 31	s2eqd 15014	. . . . . . . . . . . . . . . 16 ⊢ ((𝑎 ∈ 𝐼 ∧ 𝑏 ∈ 2o) → ⟨“⟨𝑎, 𝑏⟩(𝑎𝑀𝑏)”⟩ = ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o ∖ 𝑏)⟩”⟩)
33	32	oteq3d 4847	. . . . . . . . . . . . . . 15 ⊢ ((𝑎 ∈ 𝐼 ∧ 𝑏 ∈ 2o) → ⟨𝑚, 𝑚, ⟨“⟨𝑎, 𝑏⟩(𝑎𝑀𝑏)”⟩⟩ = ⟨𝑚, 𝑚, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o ∖ 𝑏)⟩”⟩⟩)
34	33	oveq2d 7436	. . . . . . . . . . . . . 14 ⊢ ((𝑎 ∈ 𝐼 ∧ 𝑏 ∈ 2o) → (𝑥 splice ⟨𝑚, 𝑚, ⟨“⟨𝑎, 𝑏⟩(𝑎𝑀𝑏)”⟩⟩) = (𝑥 splice ⟨𝑚, 𝑚, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o ∖ 𝑏)⟩”⟩⟩))
35	34	breq2d 5115	. . . . . . . . . . . . 13 ⊢ ((𝑎 ∈ 𝐼 ∧ 𝑏 ∈ 2o) → (𝑥𝑟(𝑥 splice ⟨𝑚, 𝑚, ⟨“⟨𝑎, 𝑏⟩(𝑎𝑀𝑏)”⟩⟩) ↔ 𝑥𝑟(𝑥 splice ⟨𝑚, 𝑚, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o ∖ 𝑏)⟩”⟩⟩)))
36	35	ralbidva 3184	. . . . . . . . . . . 12 ⊢ (𝑎 ∈ 𝐼 → (∀𝑏 ∈ 2o 𝑥𝑟(𝑥 splice ⟨𝑚, 𝑚, ⟨“⟨𝑎, 𝑏⟩(𝑎𝑀𝑏)”⟩⟩) ↔ ∀𝑏 ∈ 2o 𝑥𝑟(𝑥 splice ⟨𝑚, 𝑚, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o ∖ 𝑏)⟩”⟩⟩)))
37	36	ralbiia 3107	. . . . . . . . . . 11 ⊢ (∀𝑎 ∈ 𝐼 ∀𝑏 ∈ 2o 𝑥𝑟(𝑥 splice ⟨𝑚, 𝑚, ⟨“⟨𝑎, 𝑏⟩(𝑎𝑀𝑏)”⟩⟩) ↔ ∀𝑎 ∈ 𝐼 ∀𝑏 ∈ 2o 𝑥𝑟(𝑥 splice ⟨𝑚, 𝑚, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o ∖ 𝑏)⟩”⟩⟩))
38	29, 37	bitri 278	. . . . . . . . . 10 ⊢ (∀𝑢 ∈ (𝐼 × 2o)𝑥𝑟(𝑥 splice ⟨𝑚, 𝑚, ⟨“𝑢(𝑀‘𝑢)”⟩⟩) ↔ ∀𝑎 ∈ 𝐼 ∀𝑏 ∈ 2o 𝑥𝑟(𝑥 splice ⟨𝑚, 𝑚, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o ∖ 𝑏)⟩”⟩⟩))
39	38	ralbii 3109	. . . . . . . . 9 ⊢ (∀𝑚 ∈ (0...(♯‘𝑥))∀𝑢 ∈ (𝐼 × 2o)𝑥𝑟(𝑥 splice ⟨𝑚, 𝑚, ⟨“𝑢(𝑀‘𝑢)”⟩⟩) ↔ ∀𝑚 ∈ (0...(♯‘𝑥))∀𝑎 ∈ 𝐼 ∀𝑏 ∈ 2o 𝑥𝑟(𝑥 splice ⟨𝑚, 𝑚, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o ∖ 𝑏)⟩”⟩⟩))
40	20, 39	bitri 278	. . . . . . . 8 ⊢ (∀𝑎 ∈ ran (𝑚 ∈ (0...(♯‘𝑥)), 𝑢 ∈ (𝐼 × 2o) ↦ (𝑥 splice ⟨𝑚, 𝑚, ⟨“𝑢(𝑀‘𝑢)”⟩⟩))𝑎 ∈ [𝑥]𝑟 ↔ ∀𝑚 ∈ (0...(♯‘𝑥))∀𝑎 ∈ 𝐼 ∀𝑏 ∈ 2o 𝑥𝑟(𝑥 splice ⟨𝑚, 𝑚, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o ∖ 𝑏)⟩”⟩⟩))
41	10, 40	bitri 278	. . . . . . 7 ⊢ (ran (𝑚 ∈ (0...(♯‘𝑥)), 𝑢 ∈ (𝐼 × 2o) ↦ (𝑥 splice ⟨𝑚, 𝑚, ⟨“𝑢(𝑀‘𝑢)”⟩⟩)) ⊆ [𝑥]𝑟 ↔ ∀𝑚 ∈ (0...(♯‘𝑥))∀𝑎 ∈ 𝐼 ∀𝑏 ∈ 2o 𝑥𝑟(𝑥 splice ⟨𝑚, 𝑚, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o ∖ 𝑏)⟩”⟩⟩))
42	9, 41	bitrdi 290	. . . . . 6 ⊢ (𝑥 ∈ 𝑊 → (ran (𝑇‘𝑥) ⊆ [𝑥]𝑟 ↔ ∀𝑚 ∈ (0...(♯‘𝑥))∀𝑎 ∈ 𝐼 ∀𝑏 ∈ 2o 𝑥𝑟(𝑥 splice ⟨𝑚, 𝑚, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o ∖ 𝑏)⟩”⟩⟩)))
43	42	ralbiia 3107	. . . . 5 ⊢ (∀𝑥 ∈ 𝑊 ran (𝑇‘𝑥) ⊆ [𝑥]𝑟 ↔ ∀𝑥 ∈ 𝑊 ∀𝑚 ∈ (0...(♯‘𝑥))∀𝑎 ∈ 𝐼 ∀𝑏 ∈ 2o 𝑥𝑟(𝑥 splice ⟨𝑚, 𝑚, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o ∖ 𝑏)⟩”⟩⟩))
44	43	anbi2i 635	. . . 4 ⊢ ((𝑟 Er 𝑊 ∧ ∀𝑥 ∈ 𝑊 ran (𝑇‘𝑥) ⊆ [𝑥]𝑟) ↔ (𝑟 Er 𝑊 ∧ ∀𝑥 ∈ 𝑊 ∀𝑚 ∈ (0...(♯‘𝑥))∀𝑎 ∈ 𝐼 ∀𝑏 ∈ 2o 𝑥𝑟(𝑥 splice ⟨𝑚, 𝑚, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o ∖ 𝑏)⟩”⟩⟩)))
45	44	abbii 2828	. . 3 ⊢ {𝑟 ∣ (𝑟 Er 𝑊 ∧ ∀𝑥 ∈ 𝑊 ran (𝑇‘𝑥) ⊆ [𝑥]𝑟)} = {𝑟 ∣ (𝑟 Er 𝑊 ∧ ∀𝑥 ∈ 𝑊 ∀𝑚 ∈ (0...(♯‘𝑥))∀𝑎 ∈ 𝐼 ∀𝑏 ∈ 2o 𝑥𝑟(𝑥 splice ⟨𝑚, 𝑚, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o ∖ 𝑏)⟩”⟩⟩))}
46	45	inteqi 4911	. 2 ⊢ ∩ {𝑟 ∣ (𝑟 Er 𝑊 ∧ ∀𝑥 ∈ 𝑊 ran (𝑇‘𝑥) ⊆ [𝑥]𝑟)} = ∩ {𝑟 ∣ (𝑟 Er 𝑊 ∧ ∀𝑥 ∈ 𝑊 ∀𝑚 ∈ (0...(♯‘𝑥))∀𝑎 ∈ 𝐼 ∀𝑏 ∈ 2o 𝑥𝑟(𝑥 splice ⟨𝑚, 𝑚, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o ∖ 𝑏)⟩”⟩⟩))}
47	3, 46	eqtr4i 2787	1 ⊢ ∼ = ∩ {𝑟 ∣ (𝑟 Er 𝑊 ∧ ∀𝑥 ∈ 𝑊 ran (𝑇‘𝑥) ⊆ [𝑥]𝑟)}

This proof depends on syntax axioms:   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2145  {cab 2739  ∀wral 3077  Vcvv 3451   ∖ cdif 3896   ⊆ wss 3899  ⟨cop 4590  ⟨cotp 4592  ∩ cint 4907   class class class wbr 5103   ↦ cmpt 5186   I cid 5545   × cxp 5649  ran crn 5652  ⟶wf 6534  ‘cfv 6538  (class class class)co 7420   ∈ cmpo 7422  1_oc1o 8469  2_oc2o 8470   Er wer 8714  [cec 8715  0cc0 11200  ...cfz 13639  ♯chash 14474  Word cword 14658   splice csplice 14898  ⟨“cs2 14992   ~_FG cefg 19920

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7751  ax-cnex 11256  ax-resscn 11257  ax-1cn 11258  ax-icn 11259  ax-addcl 11260  ax-addrcl 11261  ax-mulcl 11262  ax-mulrcl 11263  ax-mulcom 11264  ax-addass 11265  ax-mulass 11266  ax-distr 11267  ax-i2m1 11268  ax-1ne0 11269  ax-1rid 11270  ax-rnegex 11271  ax-rrecex 11272  ax-cnre 11273  ax-pre-lttri 11274  ax-pre-lttrn 11275  ax-pre-ltadd 11276  ax-pre-mulgt0 11277

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-nel 3063  df-ral 3078  df-rex 3088  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-ot 4593  df-uni 4868  df-int 4908  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-pred 6304  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-riota 7377  df-ov 7423  df-oprab 7424  df-mpo 7425  df-om 7878  df-1st 8001  df-2nd 8002  df-frecs 8299  df-wrecs 8330  df-recs 8379  df-rdg 8418  df-1o 8476  df-2o 8477  df-er 8717  df-ec 8719  df-map 8849  df-en 8974  df-dom 8975  df-sdom 8976  df-fin 8977  df-card 10020  df-pnf 11345  df-mnf 11346  df-xr 11347  df-ltxr 11348  df-le 11349  df-sub 11543  df-neg 11544  df-nn 12336  df-n0 12607  df-z 12694  df-uz 12966  df-fz 13640  df-fzo 13789  df-hash 14475  df-word 14659  df-concat 14716  df-s1 14743  df-substr 14789  df-pfx 14821  df-splice 14899  df-s2 14999  df-efg 19923

This theorem is used by:  efgi2  19939  efgrelexlemb  19964  efgcpbllemb  19969