Theorem efgtf 18135

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
efgtf	⊢ (𝑋 ∈ 𝑊 → ((𝑇‘𝑋) = (𝑎 ∈ (0...(#‘𝑋)), 𝑏 ∈ (𝐼 × 2𝑜) ↦ (𝑋 splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀‘𝑏)”⟩⟩)) ∧ (𝑇‘𝑋):((0...(#‘𝑋)) × (𝐼 × 2𝑜))⟶𝑊))

Distinct variable groups:   𝑎,𝑏,𝑦,𝑧   𝑣,𝑛,𝑤,𝑦,𝑧,𝑎   𝑀,𝑎   𝑛,𝑏,𝑣,𝑤,𝑀   𝑇,𝑎,𝑏   𝑋,𝑎,𝑏   𝑊,𝑎,𝑏,𝑛,𝑣,𝑤,𝑦,𝑧   ∼ ,𝑎,𝑏,𝑦,𝑧   𝐼,𝑎,𝑏,𝑛,𝑣,𝑤,𝑦,𝑧

Allowed substitution hints:   ∼ (𝑤,𝑣,𝑛)   𝑇(𝑦,𝑧,𝑤,𝑣,𝑛)   𝑀(𝑦,𝑧)   𝑋(𝑦,𝑧,𝑤,𝑣,𝑛)

Proof of Theorem efgtf

Dummy variable 𝑢 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		efgval.w	. . . . . . . . . 10 ⊢ 𝑊 = ( I ‘Word (𝐼 × 2𝑜))
2		fviss 6256	. . . . . . . . . 10 ⊢ ( I ‘Word (𝐼 × 2𝑜)) ⊆ Word (𝐼 × 2𝑜)
3	1, 2	eqsstri 3635	. . . . . . . . 9 ⊢ 𝑊 ⊆ Word (𝐼 × 2𝑜)
4		simpl 473	. . . . . . . . 9 ⊢ ((𝑋 ∈ 𝑊 ∧ (𝑎 ∈ (0...(#‘𝑋)) ∧ 𝑏 ∈ (𝐼 × 2𝑜))) → 𝑋 ∈ 𝑊)
5	3, 4	sseldi 3601	. . . . . . . 8 ⊢ ((𝑋 ∈ 𝑊 ∧ (𝑎 ∈ (0...(#‘𝑋)) ∧ 𝑏 ∈ (𝐼 × 2𝑜))) → 𝑋 ∈ Word (𝐼 × 2𝑜))
6		simprr 796	. . . . . . . . 9 ⊢ ((𝑋 ∈ 𝑊 ∧ (𝑎 ∈ (0...(#‘𝑋)) ∧ 𝑏 ∈ (𝐼 × 2𝑜))) → 𝑏 ∈ (𝐼 × 2𝑜))
7		efgval2.m	. . . . . . . . . . . 12 ⊢ 𝑀 = (𝑦 ∈ 𝐼, 𝑧 ∈ 2𝑜 ↦ ⟨𝑦, (1𝑜 ∖ 𝑧)⟩)
8	7	efgmf 18126	. . . . . . . . . . 11 ⊢ 𝑀:(𝐼 × 2𝑜)⟶(𝐼 × 2𝑜)
9	8	ffvelrni 6358	. . . . . . . . . 10 ⊢ (𝑏 ∈ (𝐼 × 2𝑜) → (𝑀‘𝑏) ∈ (𝐼 × 2𝑜))
10	9	ad2antll 765	. . . . . . . . 9 ⊢ ((𝑋 ∈ 𝑊 ∧ (𝑎 ∈ (0...(#‘𝑋)) ∧ 𝑏 ∈ (𝐼 × 2𝑜))) → (𝑀‘𝑏) ∈ (𝐼 × 2𝑜))
11	6, 10	s2cld 13616	. . . . . . . 8 ⊢ ((𝑋 ∈ 𝑊 ∧ (𝑎 ∈ (0...(#‘𝑋)) ∧ 𝑏 ∈ (𝐼 × 2𝑜))) → ⟨“𝑏(𝑀‘𝑏)”⟩ ∈ Word (𝐼 × 2𝑜))
12		splcl 13503	. . . . . . . 8 ⊢ ((𝑋 ∈ Word (𝐼 × 2𝑜) ∧ ⟨“𝑏(𝑀‘𝑏)”⟩ ∈ Word (𝐼 × 2𝑜)) → (𝑋 splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀‘𝑏)”⟩⟩) ∈ Word (𝐼 × 2𝑜))
13	5, 11, 12	syl2anc 693	. . . . . . 7 ⊢ ((𝑋 ∈ 𝑊 ∧ (𝑎 ∈ (0...(#‘𝑋)) ∧ 𝑏 ∈ (𝐼 × 2𝑜))) → (𝑋 splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀‘𝑏)”⟩⟩) ∈ Word (𝐼 × 2𝑜))
14	1	efgrcl 18128	. . . . . . . . 9 ⊢ (𝑋 ∈ 𝑊 → (𝐼 ∈ V ∧ 𝑊 = Word (𝐼 × 2𝑜)))
15	14	simprd 479	. . . . . . . 8 ⊢ (𝑋 ∈ 𝑊 → 𝑊 = Word (𝐼 × 2𝑜))
16	15	adantr 481	. . . . . . 7 ⊢ ((𝑋 ∈ 𝑊 ∧ (𝑎 ∈ (0...(#‘𝑋)) ∧ 𝑏 ∈ (𝐼 × 2𝑜))) → 𝑊 = Word (𝐼 × 2𝑜))
17	13, 16	eleqtrrd 2704	. . . . . 6 ⊢ ((𝑋 ∈ 𝑊 ∧ (𝑎 ∈ (0...(#‘𝑋)) ∧ 𝑏 ∈ (𝐼 × 2𝑜))) → (𝑋 splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀‘𝑏)”⟩⟩) ∈ 𝑊)
18	17	ralrimivva 2971	. . . . 5 ⊢ (𝑋 ∈ 𝑊 → ∀𝑎 ∈ (0...(#‘𝑋))∀𝑏 ∈ (𝐼 × 2𝑜)(𝑋 splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀‘𝑏)”⟩⟩) ∈ 𝑊)
19		eqid 2622	. . . . . 6 ⊢ (𝑎 ∈ (0...(#‘𝑋)), 𝑏 ∈ (𝐼 × 2𝑜) ↦ (𝑋 splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀‘𝑏)”⟩⟩)) = (𝑎 ∈ (0...(#‘𝑋)), 𝑏 ∈ (𝐼 × 2𝑜) ↦ (𝑋 splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀‘𝑏)”⟩⟩))
20	19	fmpt2 7237	. . . . 5 ⊢ (∀𝑎 ∈ (0...(#‘𝑋))∀𝑏 ∈ (𝐼 × 2𝑜)(𝑋 splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀‘𝑏)”⟩⟩) ∈ 𝑊 ↔ (𝑎 ∈ (0...(#‘𝑋)), 𝑏 ∈ (𝐼 × 2𝑜) ↦ (𝑋 splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀‘𝑏)”⟩⟩)):((0...(#‘𝑋)) × (𝐼 × 2𝑜))⟶𝑊)
21	18, 20	sylib 208	. . . 4 ⊢ (𝑋 ∈ 𝑊 → (𝑎 ∈ (0...(#‘𝑋)), 𝑏 ∈ (𝐼 × 2𝑜) ↦ (𝑋 splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀‘𝑏)”⟩⟩)):((0...(#‘𝑋)) × (𝐼 × 2𝑜))⟶𝑊)
22		ovex 6678	. . . . 5 ⊢ (0...(#‘𝑋)) ∈ V
23	14	simpld 475	. . . . . 6 ⊢ (𝑋 ∈ 𝑊 → 𝐼 ∈ V)
24		2on 7568	. . . . . 6 ⊢ 2𝑜 ∈ On
25		xpexg 6960	. . . . . 6 ⊢ ((𝐼 ∈ V ∧ 2𝑜 ∈ On) → (𝐼 × 2𝑜) ∈ V)
26	23, 24, 25	sylancl 694	. . . . 5 ⊢ (𝑋 ∈ 𝑊 → (𝐼 × 2𝑜) ∈ V)
27		xpexg 6960	. . . . 5 ⊢ (((0...(#‘𝑋)) ∈ V ∧ (𝐼 × 2𝑜) ∈ V) → ((0...(#‘𝑋)) × (𝐼 × 2𝑜)) ∈ V)
28	22, 26, 27	sylancr 695	. . . 4 ⊢ (𝑋 ∈ 𝑊 → ((0...(#‘𝑋)) × (𝐼 × 2𝑜)) ∈ V)
29		fvex 6201	. . . . . 6 ⊢ ( I ‘Word (𝐼 × 2𝑜)) ∈ V
30	1, 29	eqeltri 2697	. . . . 5 ⊢ 𝑊 ∈ V
31	30	a1i 11	. . . 4 ⊢ (𝑋 ∈ 𝑊 → 𝑊 ∈ V)
32		fex2 7121	. . . 4 ⊢ (((𝑎 ∈ (0...(#‘𝑋)), 𝑏 ∈ (𝐼 × 2𝑜) ↦ (𝑋 splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀‘𝑏)”⟩⟩)):((0...(#‘𝑋)) × (𝐼 × 2𝑜))⟶𝑊 ∧ ((0...(#‘𝑋)) × (𝐼 × 2𝑜)) ∈ V ∧ 𝑊 ∈ V) → (𝑎 ∈ (0...(#‘𝑋)), 𝑏 ∈ (𝐼 × 2𝑜) ↦ (𝑋 splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀‘𝑏)”⟩⟩)) ∈ V)
33	21, 28, 31, 32	syl3anc 1326	. . 3 ⊢ (𝑋 ∈ 𝑊 → (𝑎 ∈ (0...(#‘𝑋)), 𝑏 ∈ (𝐼 × 2𝑜) ↦ (𝑋 splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀‘𝑏)”⟩⟩)) ∈ V)
34		fveq2 6191	. . . . . 6 ⊢ (𝑢 = 𝑋 → (#‘𝑢) = (#‘𝑋))
35	34	oveq2d 6666	. . . . 5 ⊢ (𝑢 = 𝑋 → (0...(#‘𝑢)) = (0...(#‘𝑋)))
36		eqidd 2623	. . . . 5 ⊢ (𝑢 = 𝑋 → (𝐼 × 2𝑜) = (𝐼 × 2𝑜))
37		oveq1 6657	. . . . 5 ⊢ (𝑢 = 𝑋 → (𝑢 splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀‘𝑏)”⟩⟩) = (𝑋 splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀‘𝑏)”⟩⟩))
38	35, 36, 37	mpt2eq123dv 6717	. . . 4 ⊢ (𝑢 = 𝑋 → (𝑎 ∈ (0...(#‘𝑢)), 𝑏 ∈ (𝐼 × 2𝑜) ↦ (𝑢 splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀‘𝑏)”⟩⟩)) = (𝑎 ∈ (0...(#‘𝑋)), 𝑏 ∈ (𝐼 × 2𝑜) ↦ (𝑋 splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀‘𝑏)”⟩⟩)))
39		efgval2.t	. . . . 5 ⊢ 𝑇 = (𝑣 ∈ 𝑊 ↦ (𝑛 ∈ (0...(#‘𝑣)), 𝑤 ∈ (𝐼 × 2𝑜) ↦ (𝑣 splice ⟨𝑛, 𝑛, ⟨“𝑤(𝑀‘𝑤)”⟩⟩)))
40		oteq1 4411	. . . . . . . . . 10 ⊢ (𝑛 = 𝑎 → ⟨𝑛, 𝑛, ⟨“𝑤(𝑀‘𝑤)”⟩⟩ = ⟨𝑎, 𝑛, ⟨“𝑤(𝑀‘𝑤)”⟩⟩)
41		oteq2 4412	. . . . . . . . . 10 ⊢ (𝑛 = 𝑎 → ⟨𝑎, 𝑛, ⟨“𝑤(𝑀‘𝑤)”⟩⟩ = ⟨𝑎, 𝑎, ⟨“𝑤(𝑀‘𝑤)”⟩⟩)
42	40, 41	eqtrd 2656	. . . . . . . . 9 ⊢ (𝑛 = 𝑎 → ⟨𝑛, 𝑛, ⟨“𝑤(𝑀‘𝑤)”⟩⟩ = ⟨𝑎, 𝑎, ⟨“𝑤(𝑀‘𝑤)”⟩⟩)
43	42	oveq2d 6666	. . . . . . . 8 ⊢ (𝑛 = 𝑎 → (𝑣 splice ⟨𝑛, 𝑛, ⟨“𝑤(𝑀‘𝑤)”⟩⟩) = (𝑣 splice ⟨𝑎, 𝑎, ⟨“𝑤(𝑀‘𝑤)”⟩⟩))
44		id 22	. . . . . . . . . . 11 ⊢ (𝑤 = 𝑏 → 𝑤 = 𝑏)
45		fveq2 6191	. . . . . . . . . . 11 ⊢ (𝑤 = 𝑏 → (𝑀‘𝑤) = (𝑀‘𝑏))
46	44, 45	s2eqd 13608	. . . . . . . . . 10 ⊢ (𝑤 = 𝑏 → ⟨“𝑤(𝑀‘𝑤)”⟩ = ⟨“𝑏(𝑀‘𝑏)”⟩)
47	46	oteq3d 4416	. . . . . . . . 9 ⊢ (𝑤 = 𝑏 → ⟨𝑎, 𝑎, ⟨“𝑤(𝑀‘𝑤)”⟩⟩ = ⟨𝑎, 𝑎, ⟨“𝑏(𝑀‘𝑏)”⟩⟩)
48	47	oveq2d 6666	. . . . . . . 8 ⊢ (𝑤 = 𝑏 → (𝑣 splice ⟨𝑎, 𝑎, ⟨“𝑤(𝑀‘𝑤)”⟩⟩) = (𝑣 splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀‘𝑏)”⟩⟩))
49	43, 48	cbvmpt2v 6735	. . . . . . 7 ⊢ (𝑛 ∈ (0...(#‘𝑣)), 𝑤 ∈ (𝐼 × 2𝑜) ↦ (𝑣 splice ⟨𝑛, 𝑛, ⟨“𝑤(𝑀‘𝑤)”⟩⟩)) = (𝑎 ∈ (0...(#‘𝑣)), 𝑏 ∈ (𝐼 × 2𝑜) ↦ (𝑣 splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀‘𝑏)”⟩⟩))
50		fveq2 6191	. . . . . . . . 9 ⊢ (𝑣 = 𝑢 → (#‘𝑣) = (#‘𝑢))
51	50	oveq2d 6666	. . . . . . . 8 ⊢ (𝑣 = 𝑢 → (0...(#‘𝑣)) = (0...(#‘𝑢)))
52		eqidd 2623	. . . . . . . 8 ⊢ (𝑣 = 𝑢 → (𝐼 × 2𝑜) = (𝐼 × 2𝑜))
53		oveq1 6657	. . . . . . . 8 ⊢ (𝑣 = 𝑢 → (𝑣 splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀‘𝑏)”⟩⟩) = (𝑢 splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀‘𝑏)”⟩⟩))
54	51, 52, 53	mpt2eq123dv 6717	. . . . . . 7 ⊢ (𝑣 = 𝑢 → (𝑎 ∈ (0...(#‘𝑣)), 𝑏 ∈ (𝐼 × 2𝑜) ↦ (𝑣 splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀‘𝑏)”⟩⟩)) = (𝑎 ∈ (0...(#‘𝑢)), 𝑏 ∈ (𝐼 × 2𝑜) ↦ (𝑢 splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀‘𝑏)”⟩⟩)))
55	49, 54	syl5eq 2668	. . . . . 6 ⊢ (𝑣 = 𝑢 → (𝑛 ∈ (0...(#‘𝑣)), 𝑤 ∈ (𝐼 × 2𝑜) ↦ (𝑣 splice ⟨𝑛, 𝑛, ⟨“𝑤(𝑀‘𝑤)”⟩⟩)) = (𝑎 ∈ (0...(#‘𝑢)), 𝑏 ∈ (𝐼 × 2𝑜) ↦ (𝑢 splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀‘𝑏)”⟩⟩)))
56	55	cbvmptv 4750	. . . . 5 ⊢ (𝑣 ∈ 𝑊 ↦ (𝑛 ∈ (0...(#‘𝑣)), 𝑤 ∈ (𝐼 × 2𝑜) ↦ (𝑣 splice ⟨𝑛, 𝑛, ⟨“𝑤(𝑀‘𝑤)”⟩⟩))) = (𝑢 ∈ 𝑊 ↦ (𝑎 ∈ (0...(#‘𝑢)), 𝑏 ∈ (𝐼 × 2𝑜) ↦ (𝑢 splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀‘𝑏)”⟩⟩)))
57	39, 56	eqtri 2644	. . . 4 ⊢ 𝑇 = (𝑢 ∈ 𝑊 ↦ (𝑎 ∈ (0...(#‘𝑢)), 𝑏 ∈ (𝐼 × 2𝑜) ↦ (𝑢 splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀‘𝑏)”⟩⟩)))
58	38, 57	fvmptg 6280	. . 3 ⊢ ((𝑋 ∈ 𝑊 ∧ (𝑎 ∈ (0...(#‘𝑋)), 𝑏 ∈ (𝐼 × 2𝑜) ↦ (𝑋 splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀‘𝑏)”⟩⟩)) ∈ V) → (𝑇‘𝑋) = (𝑎 ∈ (0...(#‘𝑋)), 𝑏 ∈ (𝐼 × 2𝑜) ↦ (𝑋 splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀‘𝑏)”⟩⟩)))
59	33, 58	mpdan 702	. 2 ⊢ (𝑋 ∈ 𝑊 → (𝑇‘𝑋) = (𝑎 ∈ (0...(#‘𝑋)), 𝑏 ∈ (𝐼 × 2𝑜) ↦ (𝑋 splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀‘𝑏)”⟩⟩)))
60	59	feq1d 6030	. . 3 ⊢ (𝑋 ∈ 𝑊 → ((𝑇‘𝑋):((0...(#‘𝑋)) × (𝐼 × 2𝑜))⟶𝑊 ↔ (𝑎 ∈ (0...(#‘𝑋)), 𝑏 ∈ (𝐼 × 2𝑜) ↦ (𝑋 splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀‘𝑏)”⟩⟩)):((0...(#‘𝑋)) × (𝐼 × 2𝑜))⟶𝑊))
61	21, 60	mpbird 247	. 2 ⊢ (𝑋 ∈ 𝑊 → (𝑇‘𝑋):((0...(#‘𝑋)) × (𝐼 × 2𝑜))⟶𝑊)
62	59, 61	jca 554	1 ⊢ (𝑋 ∈ 𝑊 → ((𝑇‘𝑋) = (𝑎 ∈ (0...(#‘𝑋)), 𝑏 ∈ (𝐼 × 2𝑜) ↦ (𝑋 splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀‘𝑏)”⟩⟩)) ∧ (𝑇‘𝑋):((0...(#‘𝑋)) × (𝐼 × 2𝑜))⟶𝑊))

Syntax hints:   → wi 4   ∧ wa 384   = wceq 1483   ∈ wcel 1990  ∀wral 2912  Vcvv 3200   ∖ cdif 3571  ⟨cop 4183  ⟨cotp 4185   ↦ cmpt 4729   I cid 5023   × cxp 5112  Oncon0 5723  ⟶wf 5884  ‘cfv 5888  (class class class)co 6650   ↦ cmpt2 6652  1_𝑜c1o 7553  2_𝑜c2o 7554  0cc0 9936  ...cfz 12326  #chash 13117  Word cword 13291   splice csplice 13296  ⟨“cs2 13586   ~_FG cefg 18119

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-ot 4186  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-er 7742  df-map 7859  df-pm 7860  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-card 8765  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-n0 11293  df-z 11378  df-uz 11688  df-fz 12327  df-fzo 12466  df-hash 13118  df-word 13299  df-concat 13301  df-s1 13302  df-substr 13303  df-splice 13304  df-s2 13593

This theorem is referenced by:  efgtval  18136  efgval2  18137  efgtlen  18139  efginvrel2  18140  efgsp1  18150  efgredleme  18156  efgredlem  18160  efgrelexlemb  18163  efgcpbllemb  18168  frgpnabllem1  18276