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Theorem numclwlk2lem2f1o 27850

Description: 𝑅 is a 1-1 onto function. (Contributed by Alexander van der Vekens, 6-Oct-2018.) (Revised by AV, 21-Jan-2022.) (Proof shortened by AV, 17-Mar-2022.) (Revised by AV, 1-Nov-2022.)

**Hypotheses**
Ref	Expression
numclwwlk.v	⊢ 𝑉 = (Vtx‘𝐺)
numclwwlk.q	⊢ 𝑄 = (𝑣 ∈ 𝑉, 𝑛 ∈ ℕ ↦ {𝑤 ∈ (𝑛 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝑣 ∧ (lastS‘𝑤) ≠ 𝑣)})
numclwwlk.h	⊢ 𝐻 = (𝑣 ∈ 𝑉, 𝑛 ∈ (ℤ_≥‘2) ↦ {𝑤 ∈ (𝑣(ClWWalksNOn‘𝐺)𝑛) ∣ (𝑤‘(𝑛 − 2)) ≠ 𝑣})
numclwwlk.r	⊢ 𝑅 = (𝑥 ∈ (𝑋𝐻(𝑁 + 2)) ↦ (𝑥 prefix (𝑁 + 1)))

**Assertion**
Ref	Expression
numclwlk2lem2f1o	⊢ ((𝐺 ∈ FriendGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → 𝑅:(𝑋𝐻(𝑁 + 2))–1-1-onto→(𝑋𝑄𝑁))

Distinct variable groups: 𝑛,𝐺,𝑣,𝑤 𝑛,𝑁,𝑣,𝑤 𝑛,𝑉,𝑣 𝑛,𝑋,𝑣,𝑤 𝑤,𝑉 𝑥,𝐺,𝑤 𝑥,𝐻 𝑥,𝑁 𝑥,𝑄 𝑥,𝑉 𝑥,𝑋,𝑣 Allowed substitution hints: 𝑄(𝑤,𝑣,𝑛) 𝑅(𝑥,𝑤,𝑣,𝑛) 𝐻(𝑤,𝑣,𝑛)

Proof of Theorem numclwlk2lem2f1o Dummy variables 𝑦 𝑢 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eleq1w 2865	. . . . . . . . 9 ⊢ (𝑦 = 𝑥 → (𝑦 ∈ (𝑋𝐻(𝑁 + 2)) ↔ 𝑥 ∈ (𝑋𝐻(𝑁 + 2))))
2		fveq2 6538	. . . . . . . . . 10 ⊢ (𝑦 = 𝑥 → (𝑅‘𝑦) = (𝑅‘𝑥))
3		oveq1 7023	. . . . . . . . . 10 ⊢ (𝑦 = 𝑥 → (𝑦 prefix (𝑁 + 1)) = (𝑥 prefix (𝑁 + 1)))
4	2, 3	eqeq12d 2810	. . . . . . . . 9 ⊢ (𝑦 = 𝑥 → ((𝑅‘𝑦) = (𝑦 prefix (𝑁 + 1)) ↔ (𝑅‘𝑥) = (𝑥 prefix (𝑁 + 1))))
5	1, 4	imbi12d 346	. . . . . . . 8 ⊢ (𝑦 = 𝑥 → ((𝑦 ∈ (𝑋𝐻(𝑁 + 2)) → (𝑅‘𝑦) = (𝑦 prefix (𝑁 + 1))) ↔ (𝑥 ∈ (𝑋𝐻(𝑁 + 2)) → (𝑅‘𝑥) = (𝑥 prefix (𝑁 + 1)))))
6	5	imbi2d 342	. . . . . . 7 ⊢ (𝑦 = 𝑥 → (((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → (𝑦 ∈ (𝑋𝐻(𝑁 + 2)) → (𝑅‘𝑦) = (𝑦 prefix (𝑁 + 1)))) ↔ ((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → (𝑥 ∈ (𝑋𝐻(𝑁 + 2)) → (𝑅‘𝑥) = (𝑥 prefix (𝑁 + 1))))))
7		numclwwlk.v	. . . . . . . 8 ⊢ 𝑉 = (Vtx‘𝐺)
8		numclwwlk.q	. . . . . . . 8 ⊢ 𝑄 = (𝑣 ∈ 𝑉, 𝑛 ∈ ℕ ↦ {𝑤 ∈ (𝑛 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝑣 ∧ (lastS‘𝑤) ≠ 𝑣)})
9		numclwwlk.h	. . . . . . . 8 ⊢ 𝐻 = (𝑣 ∈ 𝑉, 𝑛 ∈ (ℤ_≥‘2) ↦ {𝑤 ∈ (𝑣(ClWWalksNOn‘𝐺)𝑛) ∣ (𝑤‘(𝑛 − 2)) ≠ 𝑣})
10		numclwwlk.r	. . . . . . . 8 ⊢ 𝑅 = (𝑥 ∈ (𝑋𝐻(𝑁 + 2)) ↦ (𝑥 prefix (𝑁 + 1)))
11	7, 8, 9, 10	numclwlk2lem2fv 27849	. . . . . . 7 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → (𝑦 ∈ (𝑋𝐻(𝑁 + 2)) → (𝑅‘𝑦) = (𝑦 prefix (𝑁 + 1))))
12	6, 11	chvarv 2370	. . . . . 6 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → (𝑥 ∈ (𝑋𝐻(𝑁 + 2)) → (𝑅‘𝑥) = (𝑥 prefix (𝑁 + 1))))
13	12	3adant1 1123	. . . . 5 ⊢ ((𝐺 ∈ FriendGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → (𝑥 ∈ (𝑋𝐻(𝑁 + 2)) → (𝑅‘𝑥) = (𝑥 prefix (𝑁 + 1))))
14	13	imp 407	. . . 4 ⊢ (((𝐺 ∈ FriendGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) ∧ 𝑥 ∈ (𝑋𝐻(𝑁 + 2))) → (𝑅‘𝑥) = (𝑥 prefix (𝑁 + 1)))
15	7, 8, 9, 10	numclwlk2lem2f 27848	. . . . 5 ⊢ ((𝐺 ∈ FriendGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → 𝑅:(𝑋𝐻(𝑁 + 2))⟶(𝑋𝑄𝑁))
16	15	ffvelrnda 6716	. . . 4 ⊢ (((𝐺 ∈ FriendGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) ∧ 𝑥 ∈ (𝑋𝐻(𝑁 + 2))) → (𝑅‘𝑥) ∈ (𝑋𝑄𝑁))
17	14, 16	eqeltrrd 2884	. . 3 ⊢ (((𝐺 ∈ FriendGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) ∧ 𝑥 ∈ (𝑋𝐻(𝑁 + 2))) → (𝑥 prefix (𝑁 + 1)) ∈ (𝑋𝑄𝑁))
18	17	ralrimiva 3149	. 2 ⊢ ((𝐺 ∈ FriendGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → ∀𝑥 ∈ (𝑋𝐻(𝑁 + 2))(𝑥 prefix (𝑁 + 1)) ∈ (𝑋𝑄𝑁))
19	7, 8, 9	numclwwlk2lem1 27847	. . . . 5 ⊢ ((𝐺 ∈ FriendGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → (𝑢 ∈ (𝑋𝑄𝑁) → ∃!𝑣 ∈ 𝑉 (𝑢 ++ ⟨“𝑣”⟩) ∈ (𝑋𝐻(𝑁 + 2))))
20	19	imp 407	. . . 4 ⊢ (((𝐺 ∈ FriendGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) ∧ 𝑢 ∈ (𝑋𝑄𝑁)) → ∃!𝑣 ∈ 𝑉 (𝑢 ++ ⟨“𝑣”⟩) ∈ (𝑋𝐻(𝑁 + 2)))
21	7, 8	numclwwlkovq 27845	. . . . . . . . 9 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → (𝑋𝑄𝑁) = {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝑋 ∧ (lastS‘𝑤) ≠ 𝑋)})
22	21	eleq2d 2868	. . . . . . . 8 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → (𝑢 ∈ (𝑋𝑄𝑁) ↔ 𝑢 ∈ {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝑋 ∧ (lastS‘𝑤) ≠ 𝑋)}))
23	22	3adant1 1123	. . . . . . 7 ⊢ ((𝐺 ∈ FriendGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → (𝑢 ∈ (𝑋𝑄𝑁) ↔ 𝑢 ∈ {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝑋 ∧ (lastS‘𝑤) ≠ 𝑋)}))
24		fveq1 6537	. . . . . . . . . 10 ⊢ (𝑤 = 𝑢 → (𝑤‘0) = (𝑢‘0))
25	24	eqeq1d 2797	. . . . . . . . 9 ⊢ (𝑤 = 𝑢 → ((𝑤‘0) = 𝑋 ↔ (𝑢‘0) = 𝑋))
26		fveq2 6538	. . . . . . . . . 10 ⊢ (𝑤 = 𝑢 → (lastS‘𝑤) = (lastS‘𝑢))
27	26	neeq1d 3043	. . . . . . . . 9 ⊢ (𝑤 = 𝑢 → ((lastS‘𝑤) ≠ 𝑋 ↔ (lastS‘𝑢) ≠ 𝑋))
28	25, 27	anbi12d 630	. . . . . . . 8 ⊢ (𝑤 = 𝑢 → (((𝑤‘0) = 𝑋 ∧ (lastS‘𝑤) ≠ 𝑋) ↔ ((𝑢‘0) = 𝑋 ∧ (lastS‘𝑢) ≠ 𝑋)))
29	28	elrab 3618	. . . . . . 7 ⊢ (𝑢 ∈ {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝑋 ∧ (lastS‘𝑤) ≠ 𝑋)} ↔ (𝑢 ∈ (𝑁 WWalksN 𝐺) ∧ ((𝑢‘0) = 𝑋 ∧ (lastS‘𝑢) ≠ 𝑋)))
30	23, 29	syl6bb 288	. . . . . 6 ⊢ ((𝐺 ∈ FriendGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → (𝑢 ∈ (𝑋𝑄𝑁) ↔ (𝑢 ∈ (𝑁 WWalksN 𝐺) ∧ ((𝑢‘0) = 𝑋 ∧ (lastS‘𝑢) ≠ 𝑋))))
31		wwlknbp1 27309	. . . . . . . . . . . . . . . 16 ⊢ (𝑢 ∈ (𝑁 WWalksN 𝐺) → (𝑁 ∈ ℕ₀ ∧ 𝑢 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑢) = (𝑁 + 1)))
32		3simpc 1143	. . . . . . . . . . . . . . . 16 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝑢 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑢) = (𝑁 + 1)) → (𝑢 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑢) = (𝑁 + 1)))
33	31, 32	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝑢 ∈ (𝑁 WWalksN 𝐺) → (𝑢 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑢) = (𝑁 + 1)))
34	7	wrdeqi 13733	. . . . . . . . . . . . . . . . 17 ⊢ Word 𝑉 = Word (Vtx‘𝐺)
35	34	eleq2i 2874	. . . . . . . . . . . . . . . 16 ⊢ (𝑢 ∈ Word 𝑉 ↔ 𝑢 ∈ Word (Vtx‘𝐺))
36	35	anbi1i 623	. . . . . . . . . . . . . . 15 ⊢ ((𝑢 ∈ Word 𝑉 ∧ (♯‘𝑢) = (𝑁 + 1)) ↔ (𝑢 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑢) = (𝑁 + 1)))
37	33, 36	sylibr 235	. . . . . . . . . . . . . 14 ⊢ (𝑢 ∈ (𝑁 WWalksN 𝐺) → (𝑢 ∈ Word 𝑉 ∧ (♯‘𝑢) = (𝑁 + 1)))
38		simpll 763	. . . . . . . . . . . . . . . 16 ⊢ (((𝑢 ∈ Word 𝑉 ∧ (♯‘𝑢) = (𝑁 + 1)) ∧ (𝐺 ∈ FriendGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ)) → 𝑢 ∈ Word 𝑉)
39		nnnn0 11752	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℕ₀)
40		2nn 11558	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ 2 ∈ ℕ
41	40	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑁 ∈ ℕ → 2 ∈ ℕ)
42	41	nnzd 11935	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑁 ∈ ℕ → 2 ∈ ℤ)
43		nn0pzuz 12154	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑁 ∈ ℕ₀ ∧ 2 ∈ ℤ) → (𝑁 + 2) ∈ (ℤ_≥‘2))
44	39, 42, 43	syl2anc 584	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑁 ∈ ℕ → (𝑁 + 2) ∈ (ℤ_≥‘2))
45	9	numclwwlkovh 27844	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑋 ∈ 𝑉 ∧ (𝑁 + 2) ∈ (ℤ_≥‘2)) → (𝑋𝐻(𝑁 + 2)) = {𝑤 ∈ ((𝑁 + 2) ClWWalksN 𝐺) ∣ ((𝑤‘0) = 𝑋 ∧ (𝑤‘((𝑁 + 2) − 2)) ≠ (𝑤‘0))})
46	44, 45	sylan2 592	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → (𝑋𝐻(𝑁 + 2)) = {𝑤 ∈ ((𝑁 + 2) ClWWalksN 𝐺) ∣ ((𝑤‘0) = 𝑋 ∧ (𝑤‘((𝑁 + 2) − 2)) ≠ (𝑤‘0))})
47	46	eleq2d 2868	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → (𝑥 ∈ (𝑋𝐻(𝑁 + 2)) ↔ 𝑥 ∈ {𝑤 ∈ ((𝑁 + 2) ClWWalksN 𝐺) ∣ ((𝑤‘0) = 𝑋 ∧ (𝑤‘((𝑁 + 2) − 2)) ≠ (𝑤‘0))}))
48		fveq1 6537	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑤 = 𝑥 → (𝑤‘0) = (𝑥‘0))
49	48	eqeq1d 2797	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑤 = 𝑥 → ((𝑤‘0) = 𝑋 ↔ (𝑥‘0) = 𝑋))
50		fveq1 6537	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑤 = 𝑥 → (𝑤‘((𝑁 + 2) − 2)) = (𝑥‘((𝑁 + 2) − 2)))
51	50, 48	neeq12d 3045	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑤 = 𝑥 → ((𝑤‘((𝑁 + 2) − 2)) ≠ (𝑤‘0) ↔ (𝑥‘((𝑁 + 2) − 2)) ≠ (𝑥‘0)))
52	49, 51	anbi12d 630	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑤 = 𝑥 → (((𝑤‘0) = 𝑋 ∧ (𝑤‘((𝑁 + 2) − 2)) ≠ (𝑤‘0)) ↔ ((𝑥‘0) = 𝑋 ∧ (𝑥‘((𝑁 + 2) − 2)) ≠ (𝑥‘0))))
53	52	elrab 3618	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑥 ∈ {𝑤 ∈ ((𝑁 + 2) ClWWalksN 𝐺) ∣ ((𝑤‘0) = 𝑋 ∧ (𝑤‘((𝑁 + 2) − 2)) ≠ (𝑤‘0))} ↔ (𝑥 ∈ ((𝑁 + 2) ClWWalksN 𝐺) ∧ ((𝑥‘0) = 𝑋 ∧ (𝑥‘((𝑁 + 2) − 2)) ≠ (𝑥‘0))))
54	47, 53	syl6bb 288	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → (𝑥 ∈ (𝑋𝐻(𝑁 + 2)) ↔ (𝑥 ∈ ((𝑁 + 2) ClWWalksN 𝐺) ∧ ((𝑥‘0) = 𝑋 ∧ (𝑥‘((𝑁 + 2) − 2)) ≠ (𝑥‘0)))))
55	54	3adant1 1123	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐺 ∈ FriendGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → (𝑥 ∈ (𝑋𝐻(𝑁 + 2)) ↔ (𝑥 ∈ ((𝑁 + 2) ClWWalksN 𝐺) ∧ ((𝑥‘0) = 𝑋 ∧ (𝑥‘((𝑁 + 2) − 2)) ≠ (𝑥‘0)))))
56	55	adantl 482	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑢 ∈ Word 𝑉 ∧ (♯‘𝑢) = (𝑁 + 1)) ∧ (𝐺 ∈ FriendGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ)) → (𝑥 ∈ (𝑋𝐻(𝑁 + 2)) ↔ (𝑥 ∈ ((𝑁 + 2) ClWWalksN 𝐺) ∧ ((𝑥‘0) = 𝑋 ∧ (𝑥‘((𝑁 + 2) − 2)) ≠ (𝑥‘0)))))
57	7	clwwlknbp 27500	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑥 ∈ ((𝑁 + 2) ClWWalksN 𝐺) → (𝑥 ∈ Word 𝑉 ∧ (♯‘𝑥) = (𝑁 + 2)))
58		lencl 13729	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑢 ∈ Word 𝑉 → (♯‘𝑢) ∈ ℕ₀)
59		simprr 769	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((((♯‘𝑢) ∈ ℕ₀ ∧ (♯‘𝑢) = (𝑁 + 1)) ∧ 𝑁 ∈ ℕ) ∧ ((♯‘𝑥) = (𝑁 + 2) ∧ 𝑥 ∈ Word 𝑉)) → 𝑥 ∈ Word 𝑉)
60		df-2 11548	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ⊢ 2 = (1 + 1)
61	60	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ (𝑁 ∈ ℕ → 2 = (1 + 1))
62	61	oveq2d 7032	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ (𝑁 ∈ ℕ → (𝑁 + 2) = (𝑁 + (1 + 1)))
63		nncn 11494	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℂ)
64		1cnd 10482	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ (𝑁 ∈ ℕ → 1 ∈ ℂ)
65	63, 64, 64	addassd 10509	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ (𝑁 ∈ ℕ → ((𝑁 + 1) + 1) = (𝑁 + (1 + 1)))
66	62, 65	eqtr4d 2834	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (𝑁 ∈ ℕ → (𝑁 + 2) = ((𝑁 + 1) + 1))
67	66	adantl 482	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ ((((♯‘𝑢) ∈ ℕ₀ ∧ (♯‘𝑢) = (𝑁 + 1)) ∧ 𝑁 ∈ ℕ) → (𝑁 + 2) = ((𝑁 + 1) + 1))
68	67	eqeq2d 2805	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((((♯‘𝑢) ∈ ℕ₀ ∧ (♯‘𝑢) = (𝑁 + 1)) ∧ 𝑁 ∈ ℕ) → ((♯‘𝑥) = (𝑁 + 2) ↔ (♯‘𝑥) = ((𝑁 + 1) + 1)))
69	68	biimpcd 250	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((♯‘𝑥) = (𝑁 + 2) → ((((♯‘𝑢) ∈ ℕ₀ ∧ (♯‘𝑢) = (𝑁 + 1)) ∧ 𝑁 ∈ ℕ) → (♯‘𝑥) = ((𝑁 + 1) + 1)))
70	69	adantr 481	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (((♯‘𝑥) = (𝑁 + 2) ∧ 𝑥 ∈ Word 𝑉) → ((((♯‘𝑢) ∈ ℕ₀ ∧ (♯‘𝑢) = (𝑁 + 1)) ∧ 𝑁 ∈ ℕ) → (♯‘𝑥) = ((𝑁 + 1) + 1)))
71	70	impcom 408	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (((((♯‘𝑢) ∈ ℕ₀ ∧ (♯‘𝑢) = (𝑁 + 1)) ∧ 𝑁 ∈ ℕ) ∧ ((♯‘𝑥) = (𝑁 + 2) ∧ 𝑥 ∈ Word 𝑉)) → (♯‘𝑥) = ((𝑁 + 1) + 1))
72		oveq1 7023	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((♯‘𝑢) = (𝑁 + 1) → ((♯‘𝑢) + 1) = ((𝑁 + 1) + 1))
73	72	ad3antlr 727	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (((((♯‘𝑢) ∈ ℕ₀ ∧ (♯‘𝑢) = (𝑁 + 1)) ∧ 𝑁 ∈ ℕ) ∧ ((♯‘𝑥) = (𝑁 + 2) ∧ 𝑥 ∈ Word 𝑉)) → ((♯‘𝑢) + 1) = ((𝑁 + 1) + 1))
74	71, 73	eqtr4d 2834	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((((♯‘𝑢) ∈ ℕ₀ ∧ (♯‘𝑢) = (𝑁 + 1)) ∧ 𝑁 ∈ ℕ) ∧ ((♯‘𝑥) = (𝑁 + 2) ∧ 𝑥 ∈ Word 𝑉)) → (♯‘𝑥) = ((♯‘𝑢) + 1))
75	59, 74	jca 512	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((((♯‘𝑢) ∈ ℕ₀ ∧ (♯‘𝑢) = (𝑁 + 1)) ∧ 𝑁 ∈ ℕ) ∧ ((♯‘𝑥) = (𝑁 + 2) ∧ 𝑥 ∈ Word 𝑉)) → (𝑥 ∈ Word 𝑉 ∧ (♯‘𝑥) = ((♯‘𝑢) + 1)))
76	75	exp31 420	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((♯‘𝑢) ∈ ℕ₀ ∧ (♯‘𝑢) = (𝑁 + 1)) → (𝑁 ∈ ℕ → (((♯‘𝑥) = (𝑁 + 2) ∧ 𝑥 ∈ Word 𝑉) → (𝑥 ∈ Word 𝑉 ∧ (♯‘𝑥) = ((♯‘𝑢) + 1)))))
77	58, 76	sylan 580	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑢 ∈ Word 𝑉 ∧ (♯‘𝑢) = (𝑁 + 1)) → (𝑁 ∈ ℕ → (((♯‘𝑥) = (𝑁 + 2) ∧ 𝑥 ∈ Word 𝑉) → (𝑥 ∈ Word 𝑉 ∧ (♯‘𝑥) = ((♯‘𝑢) + 1)))))
78	77	com12 32	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑁 ∈ ℕ → ((𝑢 ∈ Word 𝑉 ∧ (♯‘𝑢) = (𝑁 + 1)) → (((♯‘𝑥) = (𝑁 + 2) ∧ 𝑥 ∈ Word 𝑉) → (𝑥 ∈ Word 𝑉 ∧ (♯‘𝑥) = ((♯‘𝑢) + 1)))))
79	78	3ad2ant3 1128	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝐺 ∈ FriendGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → ((𝑢 ∈ Word 𝑉 ∧ (♯‘𝑢) = (𝑁 + 1)) → (((♯‘𝑥) = (𝑁 + 2) ∧ 𝑥 ∈ Word 𝑉) → (𝑥 ∈ Word 𝑉 ∧ (♯‘𝑥) = ((♯‘𝑢) + 1)))))
80	79	impcom 408	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑢 ∈ Word 𝑉 ∧ (♯‘𝑢) = (𝑁 + 1)) ∧ (𝐺 ∈ FriendGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ)) → (((♯‘𝑥) = (𝑁 + 2) ∧ 𝑥 ∈ Word 𝑉) → (𝑥 ∈ Word 𝑉 ∧ (♯‘𝑥) = ((♯‘𝑢) + 1))))
81	80	com12 32	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((♯‘𝑥) = (𝑁 + 2) ∧ 𝑥 ∈ Word 𝑉) → (((𝑢 ∈ Word 𝑉 ∧ (♯‘𝑢) = (𝑁 + 1)) ∧ (𝐺 ∈ FriendGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ)) → (𝑥 ∈ Word 𝑉 ∧ (♯‘𝑥) = ((♯‘𝑢) + 1))))
82	81	ancoms 459	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑥 ∈ Word 𝑉 ∧ (♯‘𝑥) = (𝑁 + 2)) → (((𝑢 ∈ Word 𝑉 ∧ (♯‘𝑢) = (𝑁 + 1)) ∧ (𝐺 ∈ FriendGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ)) → (𝑥 ∈ Word 𝑉 ∧ (♯‘𝑥) = ((♯‘𝑢) + 1))))
83	57, 82	syl 17	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑥 ∈ ((𝑁 + 2) ClWWalksN 𝐺) → (((𝑢 ∈ Word 𝑉 ∧ (♯‘𝑢) = (𝑁 + 1)) ∧ (𝐺 ∈ FriendGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ)) → (𝑥 ∈ Word 𝑉 ∧ (♯‘𝑥) = ((♯‘𝑢) + 1))))
84	83	adantr 481	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑥 ∈ ((𝑁 + 2) ClWWalksN 𝐺) ∧ ((𝑥‘0) = 𝑋 ∧ (𝑥‘((𝑁 + 2) − 2)) ≠ (𝑥‘0))) → (((𝑢 ∈ Word 𝑉 ∧ (♯‘𝑢) = (𝑁 + 1)) ∧ (𝐺 ∈ FriendGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ)) → (𝑥 ∈ Word 𝑉 ∧ (♯‘𝑥) = ((♯‘𝑢) + 1))))
85	84	com12 32	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑢 ∈ Word 𝑉 ∧ (♯‘𝑢) = (𝑁 + 1)) ∧ (𝐺 ∈ FriendGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ)) → ((𝑥 ∈ ((𝑁 + 2) ClWWalksN 𝐺) ∧ ((𝑥‘0) = 𝑋 ∧ (𝑥‘((𝑁 + 2) − 2)) ≠ (𝑥‘0))) → (𝑥 ∈ Word 𝑉 ∧ (♯‘𝑥) = ((♯‘𝑢) + 1))))
86	56, 85	sylbid 241	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑢 ∈ Word 𝑉 ∧ (♯‘𝑢) = (𝑁 + 1)) ∧ (𝐺 ∈ FriendGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ)) → (𝑥 ∈ (𝑋𝐻(𝑁 + 2)) → (𝑥 ∈ Word 𝑉 ∧ (♯‘𝑥) = ((♯‘𝑢) + 1))))
87	86	ralrimiv 3148	. . . . . . . . . . . . . . . 16 ⊢ (((𝑢 ∈ Word 𝑉 ∧ (♯‘𝑢) = (𝑁 + 1)) ∧ (𝐺 ∈ FriendGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ)) → ∀𝑥 ∈ (𝑋𝐻(𝑁 + 2))(𝑥 ∈ Word 𝑉 ∧ (♯‘𝑥) = ((♯‘𝑢) + 1)))
88	38, 87	jca 512	. . . . . . . . . . . . . . 15 ⊢ (((𝑢 ∈ Word 𝑉 ∧ (♯‘𝑢) = (𝑁 + 1)) ∧ (𝐺 ∈ FriendGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ)) → (𝑢 ∈ Word 𝑉 ∧ ∀𝑥 ∈ (𝑋𝐻(𝑁 + 2))(𝑥 ∈ Word 𝑉 ∧ (♯‘𝑥) = ((♯‘𝑢) + 1))))
89	88	ex 413	. . . . . . . . . . . . . 14 ⊢ ((𝑢 ∈ Word 𝑉 ∧ (♯‘𝑢) = (𝑁 + 1)) → ((𝐺 ∈ FriendGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → (𝑢 ∈ Word 𝑉 ∧ ∀𝑥 ∈ (𝑋𝐻(𝑁 + 2))(𝑥 ∈ Word 𝑉 ∧ (♯‘𝑥) = ((♯‘𝑢) + 1)))))
90	37, 89	syl 17	. . . . . . . . . . . . 13 ⊢ (𝑢 ∈ (𝑁 WWalksN 𝐺) → ((𝐺 ∈ FriendGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → (𝑢 ∈ Word 𝑉 ∧ ∀𝑥 ∈ (𝑋𝐻(𝑁 + 2))(𝑥 ∈ Word 𝑉 ∧ (♯‘𝑥) = ((♯‘𝑢) + 1)))))
91	90	adantr 481	. . . . . . . . . . . 12 ⊢ ((𝑢 ∈ (𝑁 WWalksN 𝐺) ∧ ((𝑢‘0) = 𝑋 ∧ (lastS‘𝑢) ≠ 𝑋)) → ((𝐺 ∈ FriendGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → (𝑢 ∈ Word 𝑉 ∧ ∀𝑥 ∈ (𝑋𝐻(𝑁 + 2))(𝑥 ∈ Word 𝑉 ∧ (♯‘𝑥) = ((♯‘𝑢) + 1)))))
92	91	imp 407	. . . . . . . . . . 11 ⊢ (((𝑢 ∈ (𝑁 WWalksN 𝐺) ∧ ((𝑢‘0) = 𝑋 ∧ (lastS‘𝑢) ≠ 𝑋)) ∧ (𝐺 ∈ FriendGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ)) → (𝑢 ∈ Word 𝑉 ∧ ∀𝑥 ∈ (𝑋𝐻(𝑁 + 2))(𝑥 ∈ Word 𝑉 ∧ (♯‘𝑥) = ((♯‘𝑢) + 1))))
93		nfcv 2949	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑣𝑋
94		nfmpo1 7092	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑣(𝑣 ∈ 𝑉, 𝑛 ∈ (ℤ_≥‘2) ↦ {𝑤 ∈ (𝑣(ClWWalksNOn‘𝐺)𝑛) ∣ (𝑤‘(𝑛 − 2)) ≠ 𝑣})
95	9, 94	nfcxfr 2947	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑣𝐻
96		nfcv 2949	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑣(𝑁 + 2)
97	93, 95, 96	nfov 7046	. . . . . . . . . . . 12 ⊢ Ⅎ𝑣(𝑋𝐻(𝑁 + 2))
98	97	reuccatpfxs1 13945	. . . . . . . . . . 11 ⊢ ((𝑢 ∈ Word 𝑉 ∧ ∀𝑥 ∈ (𝑋𝐻(𝑁 + 2))(𝑥 ∈ Word 𝑉 ∧ (♯‘𝑥) = ((♯‘𝑢) + 1))) → (∃!𝑣 ∈ 𝑉 (𝑢 ++ ⟨“𝑣”⟩) ∈ (𝑋𝐻(𝑁 + 2)) → ∃!𝑥 ∈ (𝑋𝐻(𝑁 + 2))𝑢 = (𝑥 prefix (♯‘𝑢))))
99	92, 98	syl 17	. . . . . . . . . 10 ⊢ (((𝑢 ∈ (𝑁 WWalksN 𝐺) ∧ ((𝑢‘0) = 𝑋 ∧ (lastS‘𝑢) ≠ 𝑋)) ∧ (𝐺 ∈ FriendGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ)) → (∃!𝑣 ∈ 𝑉 (𝑢 ++ ⟨“𝑣”⟩) ∈ (𝑋𝐻(𝑁 + 2)) → ∃!𝑥 ∈ (𝑋𝐻(𝑁 + 2))𝑢 = (𝑥 prefix (♯‘𝑢))))
100	99	imp 407	. . . . . . . . 9 ⊢ ((((𝑢 ∈ (𝑁 WWalksN 𝐺) ∧ ((𝑢‘0) = 𝑋 ∧ (lastS‘𝑢) ≠ 𝑋)) ∧ (𝐺 ∈ FriendGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ)) ∧ ∃!𝑣 ∈ 𝑉 (𝑢 ++ ⟨“𝑣”⟩) ∈ (𝑋𝐻(𝑁 + 2))) → ∃!𝑥 ∈ (𝑋𝐻(𝑁 + 2))𝑢 = (𝑥 prefix (♯‘𝑢)))
101	31	simp3d 1137	. . . . . . . . . . . . . 14 ⊢ (𝑢 ∈ (𝑁 WWalksN 𝐺) → (♯‘𝑢) = (𝑁 + 1))
102	101	eqcomd 2801	. . . . . . . . . . . . 13 ⊢ (𝑢 ∈ (𝑁 WWalksN 𝐺) → (𝑁 + 1) = (♯‘𝑢))
103	102	ad4antr 728	. . . . . . . . . . . 12 ⊢ (((((𝑢 ∈ (𝑁 WWalksN 𝐺) ∧ ((𝑢‘0) = 𝑋 ∧ (lastS‘𝑢) ≠ 𝑋)) ∧ (𝐺 ∈ FriendGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ)) ∧ ∃!𝑣 ∈ 𝑉 (𝑢 ++ ⟨“𝑣”⟩) ∈ (𝑋𝐻(𝑁 + 2))) ∧ 𝑥 ∈ (𝑋𝐻(𝑁 + 2))) → (𝑁 + 1) = (♯‘𝑢))
104	103	oveq2d 7032	. . . . . . . . . . 11 ⊢ (((((𝑢 ∈ (𝑁 WWalksN 𝐺) ∧ ((𝑢‘0) = 𝑋 ∧ (lastS‘𝑢) ≠ 𝑋)) ∧ (𝐺 ∈ FriendGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ)) ∧ ∃!𝑣 ∈ 𝑉 (𝑢 ++ ⟨“𝑣”⟩) ∈ (𝑋𝐻(𝑁 + 2))) ∧ 𝑥 ∈ (𝑋𝐻(𝑁 + 2))) → (𝑥 prefix (𝑁 + 1)) = (𝑥 prefix (♯‘𝑢)))
105	104	eqeq2d 2805	. . . . . . . . . 10 ⊢ (((((𝑢 ∈ (𝑁 WWalksN 𝐺) ∧ ((𝑢‘0) = 𝑋 ∧ (lastS‘𝑢) ≠ 𝑋)) ∧ (𝐺 ∈ FriendGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ)) ∧ ∃!𝑣 ∈ 𝑉 (𝑢 ++ ⟨“𝑣”⟩) ∈ (𝑋𝐻(𝑁 + 2))) ∧ 𝑥 ∈ (𝑋𝐻(𝑁 + 2))) → (𝑢 = (𝑥 prefix (𝑁 + 1)) ↔ 𝑢 = (𝑥 prefix (♯‘𝑢))))
106	105	reubidva 3347	. . . . . . . . 9 ⊢ ((((𝑢 ∈ (𝑁 WWalksN 𝐺) ∧ ((𝑢‘0) = 𝑋 ∧ (lastS‘𝑢) ≠ 𝑋)) ∧ (𝐺 ∈ FriendGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ)) ∧ ∃!𝑣 ∈ 𝑉 (𝑢 ++ ⟨“𝑣”⟩) ∈ (𝑋𝐻(𝑁 + 2))) → (∃!𝑥 ∈ (𝑋𝐻(𝑁 + 2))𝑢 = (𝑥 prefix (𝑁 + 1)) ↔ ∃!𝑥 ∈ (𝑋𝐻(𝑁 + 2))𝑢 = (𝑥 prefix (♯‘𝑢))))
107	100, 106	mpbird 258	. . . . . . . 8 ⊢ ((((𝑢 ∈ (𝑁 WWalksN 𝐺) ∧ ((𝑢‘0) = 𝑋 ∧ (lastS‘𝑢) ≠ 𝑋)) ∧ (𝐺 ∈ FriendGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ)) ∧ ∃!𝑣 ∈ 𝑉 (𝑢 ++ ⟨“𝑣”⟩) ∈ (𝑋𝐻(𝑁 + 2))) → ∃!𝑥 ∈ (𝑋𝐻(𝑁 + 2))𝑢 = (𝑥 prefix (𝑁 + 1)))
108	107	exp31 420	. . . . . . 7 ⊢ ((𝑢 ∈ (𝑁 WWalksN 𝐺) ∧ ((𝑢‘0) = 𝑋 ∧ (lastS‘𝑢) ≠ 𝑋)) → ((𝐺 ∈ FriendGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → (∃!𝑣 ∈ 𝑉 (𝑢 ++ ⟨“𝑣”⟩) ∈ (𝑋𝐻(𝑁 + 2)) → ∃!𝑥 ∈ (𝑋𝐻(𝑁 + 2))𝑢 = (𝑥 prefix (𝑁 + 1)))))
109	108	com12 32	. . . . . 6 ⊢ ((𝐺 ∈ FriendGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → ((𝑢 ∈ (𝑁 WWalksN 𝐺) ∧ ((𝑢‘0) = 𝑋 ∧ (lastS‘𝑢) ≠ 𝑋)) → (∃!𝑣 ∈ 𝑉 (𝑢 ++ ⟨“𝑣”⟩) ∈ (𝑋𝐻(𝑁 + 2)) → ∃!𝑥 ∈ (𝑋𝐻(𝑁 + 2))𝑢 = (𝑥 prefix (𝑁 + 1)))))
110	30, 109	sylbid 241	. . . . 5 ⊢ ((𝐺 ∈ FriendGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → (𝑢 ∈ (𝑋𝑄𝑁) → (∃!𝑣 ∈ 𝑉 (𝑢 ++ ⟨“𝑣”⟩) ∈ (𝑋𝐻(𝑁 + 2)) → ∃!𝑥 ∈ (𝑋𝐻(𝑁 + 2))𝑢 = (𝑥 prefix (𝑁 + 1)))))
111	110	imp 407	. . . 4 ⊢ (((𝐺 ∈ FriendGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) ∧ 𝑢 ∈ (𝑋𝑄𝑁)) → (∃!𝑣 ∈ 𝑉 (𝑢 ++ ⟨“𝑣”⟩) ∈ (𝑋𝐻(𝑁 + 2)) → ∃!𝑥 ∈ (𝑋𝐻(𝑁 + 2))𝑢 = (𝑥 prefix (𝑁 + 1))))
112	20, 111	mpd 15	. . 3 ⊢ (((𝐺 ∈ FriendGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) ∧ 𝑢 ∈ (𝑋𝑄𝑁)) → ∃!𝑥 ∈ (𝑋𝐻(𝑁 + 2))𝑢 = (𝑥 prefix (𝑁 + 1)))
113	112	ralrimiva 3149	. 2 ⊢ ((𝐺 ∈ FriendGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → ∀𝑢 ∈ (𝑋𝑄𝑁)∃!𝑥 ∈ (𝑋𝐻(𝑁 + 2))𝑢 = (𝑥 prefix (𝑁 + 1)))
114	10	f1ompt 6738	. 2 ⊢ (𝑅:(𝑋𝐻(𝑁 + 2))–1-1-onto→(𝑋𝑄𝑁) ↔ (∀𝑥 ∈ (𝑋𝐻(𝑁 + 2))(𝑥 prefix (𝑁 + 1)) ∈ (𝑋𝑄𝑁) ∧ ∀𝑢 ∈ (𝑋𝑄𝑁)∃!𝑥 ∈ (𝑋𝐻(𝑁 + 2))𝑢 = (𝑥 prefix (𝑁 + 1))))
115	18, 113, 114	sylanbrc 583	1 ⊢ ((𝐺 ∈ FriendGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → 𝑅:(𝑋𝐻(𝑁 + 2))–1-1-onto→(𝑋𝑄𝑁))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 ∧ w3a 1080 = wceq 1522 ∈ wcel 2081 ≠ wne 2984 ∀wral 3105 ∃!wreu 3107 {crab 3109 ↦ cmpt 5041 –1-1-onto→wf1o 6224 ‘cfv 6225 (class class class)co 7016 ∈ cmpo 7018 0cc0 10383 1c1 10384 + caddc 10386 − cmin 10717 ℕcn 11486 2c2 11540 ℕ₀cn0 11745 ℤcz 11829 ℤ_≥cuz 12093 ♯chash 13540 Word cword 13707 lastSclsw 13760 ++ cconcat 13768 ⟨“cs1 13793 prefix cpfx 13868 Vtxcvtx 26464 WWalksN cwwlksn 27291 ClWWalksN cclwwlkn 27489 ClWWalksNOncclwwlknon 27553 FriendGraph cfrgr 27727

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1777 ax-4 1791 ax-5 1888 ax-6 1947 ax-7 1992 ax-8 2083 ax-9 2091 ax-10 2112 ax-11 2126 ax-12 2141 ax-13 2344 ax-ext 2769 ax-rep 5081 ax-sep 5094 ax-nul 5101 ax-pow 5157 ax-pr 5221 ax-un 7319 ax-cnex 10439 ax-resscn 10440 ax-1cn 10441 ax-icn 10442 ax-addcl 10443 ax-addrcl 10444 ax-mulcl 10445 ax-mulrcl 10446 ax-mulcom 10447 ax-addass 10448 ax-mulass 10449 ax-distr 10450 ax-i2m1 10451 ax-1ne0 10452 ax-1rid 10453 ax-rnegex 10454 ax-rrecex 10455 ax-cnre 10456 ax-pre-lttri 10457 ax-pre-lttrn 10458 ax-pre-ltadd 10459 ax-pre-mulgt0 10460

This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3or 1081 df-3an 1082 df-tru 1525 df-ex 1762 df-nf 1766 df-sb 2043 df-mo 2576 df-eu 2612 df-clab 2776 df-cleq 2788 df-clel 2863 df-nfc 2935 df-ne 2985 df-nel 3091 df-ral 3110 df-rex 3111 df-reu 3112 df-rab 3114 df-v 3439 df-sbc 3707 df-csb 3812 df-dif 3862 df-un 3864 df-in 3866 df-ss 3874 df-pss 3876 df-nul 4212 df-if 4382 df-pw 4455 df-sn 4473 df-pr 4475 df-tp 4477 df-op 4479 df-uni 4746 df-int 4783 df-iun 4827 df-br 4963 df-opab 5025 df-mpt 5042 df-tr 5064 df-id 5348 df-eprel 5353 df-po 5362 df-so 5363 df-fr 5402 df-we 5404 df-xp 5449 df-rel 5450 df-cnv 5451 df-co 5452 df-dm 5453 df-rn 5454 df-res 5455 df-ima 5456 df-pred 6023 df-ord 6069 df-on 6070 df-lim 6071 df-suc 6072 df-iota 6189 df-fun 6227 df-fn 6228 df-f 6229 df-f1 6230 df-fo 6231 df-f1o 6232 df-fv 6233 df-riota 6977 df-ov 7019 df-oprab 7020 df-mpo 7021 df-om 7437 df-1st 7545 df-2nd 7546 df-wrecs 7798 df-recs 7860 df-rdg 7898 df-1o 7953 df-oadd 7957 df-er 8139 df-map 8258 df-en 8358 df-dom 8359 df-sdom 8360 df-fin 8361 df-card 9214 df-pnf 10523 df-mnf 10524 df-xr 10525 df-ltxr 10526 df-le 10527 df-sub 10719 df-neg 10720 df-nn 11487 df-2 11548 df-n0 11746 df-xnn0 11816 df-z 11830 df-uz 12094 df-rp 12240 df-fz 12743 df-fzo 12884 df-hash 13541 df-word 13708 df-lsw 13761 df-concat 13769 df-s1 13794 df-substr 13839 df-pfx 13869 df-wwlks 27295 df-wwlksn 27296 df-clwwlk 27447 df-clwwlkn 27490 df-clwwlknon 27554 df-frgr 27728

This theorem is referenced by: numclwwlk2lem3 27851

W3C validator