Theorem efgval2 19506

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 19499	. 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 19504	. . . . . . . . . 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 495	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝑊 → (𝑇‘𝑥) = (𝑚 ∈ (0...(♯‘𝑥)), 𝑢 ∈ (𝐼 × 2o) ↦ (𝑥 splice ⟨𝑚, 𝑚, ⟨“𝑢(𝑀‘𝑢)”⟩⟩)))
8	7	rneqd 5893	. . . . . . . 8 ⊢ (𝑥 ∈ 𝑊 → ran (𝑇‘𝑥) = ran (𝑚 ∈ (0...(♯‘𝑥)), 𝑢 ∈ (𝐼 × 2o) ↦ (𝑥 splice ⟨𝑚, 𝑚, ⟨“𝑢(𝑀‘𝑢)”⟩⟩)))
9	8	sseq1d 3975	. . . . . . 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 3932	. . . . . . . 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 7390	. . . . . . . . . . 11 ⊢ (𝑥 splice ⟨𝑚, 𝑚, ⟨“𝑢(𝑀‘𝑢)”⟩⟩) ∈ V
12	11	rgen2w 3069	. . . . . . . . . 10 ⊢ ∀𝑚 ∈ (0...(♯‘𝑥))∀𝑢 ∈ (𝐼 × 2o)(𝑥 splice ⟨𝑚, 𝑚, ⟨“𝑢(𝑀‘𝑢)”⟩⟩) ∈ V
13		eqid 2736	. . . . . . . . . . 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 3449	. . . . . . . . . . . . 13 ⊢ 𝑎 ∈ V
15		vex 3449	. . . . . . . . . . . . 13 ⊢ 𝑥 ∈ V
16	14, 15	elec 8692	. . . . . . . . . . . 12 ⊢ (𝑎 ∈ [𝑥]𝑟 ↔ 𝑥𝑟𝑎)
17		breq2 5109	. . . . . . . . . . . 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 7494	. . . . . . . . . 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 22	. . . . . . . . . . . . . . . 16 ⊢ (𝑢 = ⟨𝑎, 𝑏⟩ → 𝑢 = ⟨𝑎, 𝑏⟩)
22		fveq2 6842	. . . . . . . . . . . . . . . . 17 ⊢ (𝑢 = ⟨𝑎, 𝑏⟩ → (𝑀‘𝑢) = (𝑀‘⟨𝑎, 𝑏⟩))
23		df-ov 7360	. . . . . . . . . . . . . . . . 17 ⊢ (𝑎𝑀𝑏) = (𝑀‘⟨𝑎, 𝑏⟩)
24	22, 23	eqtr4di 2794	. . . . . . . . . . . . . . . 16 ⊢ (𝑢 = ⟨𝑎, 𝑏⟩ → (𝑀‘𝑢) = (𝑎𝑀𝑏))
25	21, 24	s2eqd 14752	. . . . . . . . . . . . . . 15 ⊢ (𝑢 = ⟨𝑎, 𝑏⟩ → ⟨“𝑢(𝑀‘𝑢)”⟩ = ⟨“⟨𝑎, 𝑏⟩(𝑎𝑀𝑏)”⟩)
26	25	oteq3d 4844	. . . . . . . . . . . . . 14 ⊢ (𝑢 = ⟨𝑎, 𝑏⟩ → ⟨𝑚, 𝑚, ⟨“𝑢(𝑀‘𝑢)”⟩⟩ = ⟨𝑚, 𝑚, ⟨“⟨𝑎, 𝑏⟩(𝑎𝑀𝑏)”⟩⟩)
27	26	oveq2d 7373	. . . . . . . . . . . . 13 ⊢ (𝑢 = ⟨𝑎, 𝑏⟩ → (𝑥 splice ⟨𝑚, 𝑚, ⟨“𝑢(𝑀‘𝑢)”⟩⟩) = (𝑥 splice ⟨𝑚, 𝑚, ⟨“⟨𝑎, 𝑏⟩(𝑎𝑀𝑏)”⟩⟩))
28	27	breq2d 5117	. . . . . . . . . . . 12 ⊢ (𝑢 = ⟨𝑎, 𝑏⟩ → (𝑥𝑟(𝑥 splice ⟨𝑚, 𝑚, ⟨“𝑢(𝑀‘𝑢)”⟩⟩) ↔ 𝑥𝑟(𝑥 splice ⟨𝑚, 𝑚, ⟨“⟨𝑎, 𝑏⟩(𝑎𝑀𝑏)”⟩⟩)))
29	28	ralxp 5797	. . . . . . . . . . 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 2737	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑎 ∈ 𝐼 ∧ 𝑏 ∈ 2o) → ⟨𝑎, 𝑏⟩ = ⟨𝑎, 𝑏⟩)
31	4	efgmval 19494	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑎 ∈ 𝐼 ∧ 𝑏 ∈ 2o) → (𝑎𝑀𝑏) = ⟨𝑎, (1o ∖ 𝑏)⟩)
32	30, 31	s2eqd 14752	. . . . . . . . . . . . . . . 16 ⊢ ((𝑎 ∈ 𝐼 ∧ 𝑏 ∈ 2o) → ⟨“⟨𝑎, 𝑏⟩(𝑎𝑀𝑏)”⟩ = ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o ∖ 𝑏)⟩”⟩)
33	32	oteq3d 4844	. . . . . . . . . . . . . . 15 ⊢ ((𝑎 ∈ 𝐼 ∧ 𝑏 ∈ 2o) → ⟨𝑚, 𝑚, ⟨“⟨𝑎, 𝑏⟩(𝑎𝑀𝑏)”⟩⟩ = ⟨𝑚, 𝑚, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o ∖ 𝑏)⟩”⟩⟩)
34	33	oveq2d 7373	. . . . . . . . . . . . . 14 ⊢ ((𝑎 ∈ 𝐼 ∧ 𝑏 ∈ 2o) → (𝑥 splice ⟨𝑚, 𝑚, ⟨“⟨𝑎, 𝑏⟩(𝑎𝑀𝑏)”⟩⟩) = (𝑥 splice ⟨𝑚, 𝑚, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o ∖ 𝑏)⟩”⟩⟩))
35	34	breq2d 5117	. . . . . . . . . . . . 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 3172	. . . . . . . . . . . 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 3094	. . . . . . . . . . 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 274	. . . . . . . . . 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 3096	. . . . . . . . 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 274	. . . . . . . 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 274	. . . . . . 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 286	. . . . . 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 3094	. . . . 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 623	. . . 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 2806	. . 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 2767	1 ⊢ ∼ = ∩ {𝑟 ∣ (𝑟 Er 𝑊 ∧ ∀𝑥 ∈ 𝑊 ran (𝑇‘𝑥) ⊆ [𝑥]𝑟)}

Syntax hints:   ↔ wb 205   ∧ wa 396   = wceq 1541   ∈ wcel 2106  {cab 2713  ∀wral 3064  Vcvv 3445   ∖ cdif 3907   ⊆ wss 3910  ⟨cop 4592  ⟨cotp 4594  ∩ cint 4907   class class class wbr 5105   ↦ cmpt 5188   I cid 5530   × cxp 5631  ran crn 5634  ⟶wf 6492  ‘cfv 6496  (class class class)co 7357   ∈ cmpo 7359  1_oc1o 8405  2_oc2o 8406   Er wer 8645  [cec 8646  0cc0 11051  ...cfz 13424  ♯chash 14230  Word cword 14402   splice csplice 14637  ⟨“cs2 14730   ~_FG cefg 19488

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-ot 4595  df-uni 4866  df-int 4908  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-1st 7921  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-2o 8413  df-er 8648  df-ec 8650  df-map 8767  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-card 9875  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-nn 12154  df-n0 12414  df-z 12500  df-uz 12764  df-fz 13425  df-fzo 13568  df-hash 14231  df-word 14403  df-concat 14459  df-s1 14484  df-substr 14529  df-pfx 14559  df-splice 14638  df-s2 14737  df-efg 19491

This theorem is referenced by:  efgi2  19507  efgrelexlemb  19532  efgcpbllemb  19537