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Theorem clwlkclwwlkfo 29262

Description: 𝐹 is a function from the nonempty closed walks onto the closed walks as words in a simple pseudograph. (Contributed by Alexander van der Vekens, 30-Jun-2018.) (Revised by AV, 2-May-2021.) (Revised by AV, 29-Oct-2022.)

**Hypotheses**
Ref	Expression
clwlkclwwlkf.c	⊢ 𝐶 = {𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1^st ‘𝑤))}
clwlkclwwlkf.f	⊢ 𝐹 = (𝑐 ∈ 𝐶 ↦ ((2^nd ‘𝑐) prefix ((♯‘(2^nd ‘𝑐)) − 1)))

**Assertion**
Ref	Expression
clwlkclwwlkfo	⊢ (𝐺 ∈ USPGraph → 𝐹:𝐶–onto→(ClWWalks‘𝐺))

Distinct variable groups: 𝑤,𝐺,𝑐 𝐶,𝑐,𝑤 𝐹,𝑐,𝑤

Proof of Theorem clwlkclwwlkfo Dummy variable 𝑓 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		clwlkclwwlkf.c	. . 3 ⊢ 𝐶 = {𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1^st ‘𝑤))}
2		clwlkclwwlkf.f	. . 3 ⊢ 𝐹 = (𝑐 ∈ 𝐶 ↦ ((2^nd ‘𝑐) prefix ((♯‘(2^nd ‘𝑐)) − 1)))
3	1, 2	clwlkclwwlkf 29261	. 2 ⊢ (𝐺 ∈ USPGraph → 𝐹:𝐶⟶(ClWWalks‘𝐺))
4		clwwlkgt0 29239	. . . . . 6 ⊢ (𝑤 ∈ (ClWWalks‘𝐺) → 0 < (♯‘𝑤))
5		eqid 2733	. . . . . . . 8 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
6	5	clwwlkbp 29238	. . . . . . 7 ⊢ (𝑤 ∈ (ClWWalks‘𝐺) → (𝐺 ∈ V ∧ 𝑤 ∈ Word (Vtx‘𝐺) ∧ 𝑤 ≠ ∅))
7		lencl 14483	. . . . . . . . . . . 12 ⊢ (𝑤 ∈ Word (Vtx‘𝐺) → (♯‘𝑤) ∈ ℕ₀)
8	7	nn0zd 12584	. . . . . . . . . . 11 ⊢ (𝑤 ∈ Word (Vtx‘𝐺) → (♯‘𝑤) ∈ ℤ)
9		zgt0ge1 12616	. . . . . . . . . . 11 ⊢ ((♯‘𝑤) ∈ ℤ → (0 < (♯‘𝑤) ↔ 1 ≤ (♯‘𝑤)))
10	8, 9	syl 17	. . . . . . . . . 10 ⊢ (𝑤 ∈ Word (Vtx‘𝐺) → (0 < (♯‘𝑤) ↔ 1 ≤ (♯‘𝑤)))
11	10	biimpd 228	. . . . . . . . 9 ⊢ (𝑤 ∈ Word (Vtx‘𝐺) → (0 < (♯‘𝑤) → 1 ≤ (♯‘𝑤)))
12	11	anc2li 557	. . . . . . . 8 ⊢ (𝑤 ∈ Word (Vtx‘𝐺) → (0 < (♯‘𝑤) → (𝑤 ∈ Word (Vtx‘𝐺) ∧ 1 ≤ (♯‘𝑤))))
13	12	3ad2ant2 1135	. . . . . . 7 ⊢ ((𝐺 ∈ V ∧ 𝑤 ∈ Word (Vtx‘𝐺) ∧ 𝑤 ≠ ∅) → (0 < (♯‘𝑤) → (𝑤 ∈ Word (Vtx‘𝐺) ∧ 1 ≤ (♯‘𝑤))))
14	6, 13	syl 17	. . . . . 6 ⊢ (𝑤 ∈ (ClWWalks‘𝐺) → (0 < (♯‘𝑤) → (𝑤 ∈ Word (Vtx‘𝐺) ∧ 1 ≤ (♯‘𝑤))))
15	4, 14	mpd 15	. . . . 5 ⊢ (𝑤 ∈ (ClWWalks‘𝐺) → (𝑤 ∈ Word (Vtx‘𝐺) ∧ 1 ≤ (♯‘𝑤)))
16	15	adantl 483	. . . 4 ⊢ ((𝐺 ∈ USPGraph ∧ 𝑤 ∈ (ClWWalks‘𝐺)) → (𝑤 ∈ Word (Vtx‘𝐺) ∧ 1 ≤ (♯‘𝑤)))
17		eqid 2733	. . . . . . . . 9 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺)
18	5, 17	clwlkclwwlk2 29256	. . . . . . . 8 ⊢ ((𝐺 ∈ USPGraph ∧ 𝑤 ∈ Word (Vtx‘𝐺) ∧ 1 ≤ (♯‘𝑤)) → (∃𝑓 𝑓(ClWalks‘𝐺)(𝑤 ++ ⟨“(𝑤‘0)”⟩) ↔ 𝑤 ∈ (ClWWalks‘𝐺)))
19		df-br 5150	. . . . . . . . . 10 ⊢ (𝑓(ClWalks‘𝐺)(𝑤 ++ ⟨“(𝑤‘0)”⟩) ↔ ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (ClWalks‘𝐺))
20		simpr2 1196	. . . . . . . . . . . . . 14 ⊢ ((⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (ClWalks‘𝐺) ∧ (𝐺 ∈ USPGraph ∧ 𝑤 ∈ Word (Vtx‘𝐺) ∧ 1 ≤ (♯‘𝑤))) → 𝑤 ∈ Word (Vtx‘𝐺))
21		simpr3 1197	. . . . . . . . . . . . . 14 ⊢ ((⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (ClWalks‘𝐺) ∧ (𝐺 ∈ USPGraph ∧ 𝑤 ∈ Word (Vtx‘𝐺) ∧ 1 ≤ (♯‘𝑤))) → 1 ≤ (♯‘𝑤))
22		simpl 484	. . . . . . . . . . . . . 14 ⊢ ((⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (ClWalks‘𝐺) ∧ (𝐺 ∈ USPGraph ∧ 𝑤 ∈ Word (Vtx‘𝐺) ∧ 1 ≤ (♯‘𝑤))) → ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (ClWalks‘𝐺))
23	1	clwlkclwwlkfolem 29260	. . . . . . . . . . . . . 14 ⊢ ((𝑤 ∈ Word (Vtx‘𝐺) ∧ 1 ≤ (♯‘𝑤) ∧ ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (ClWalks‘𝐺)) → ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ 𝐶)
24	20, 21, 22, 23	syl3anc 1372	. . . . . . . . . . . . 13 ⊢ ((⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (ClWalks‘𝐺) ∧ (𝐺 ∈ USPGraph ∧ 𝑤 ∈ Word (Vtx‘𝐺) ∧ 1 ≤ (♯‘𝑤))) → ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ 𝐶)
25	23	3expa 1119	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑤 ∈ Word (Vtx‘𝐺) ∧ 1 ≤ (♯‘𝑤)) ∧ ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (ClWalks‘𝐺)) → ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ 𝐶)
26		ovex 7442	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑤 ++ ⟨“(𝑤‘0)”⟩) prefix ((♯‘(𝑤 ++ ⟨“(𝑤‘0)”⟩)) − 1)) ∈ V
27		fveq2 6892	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑐 = ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ → (2^nd ‘𝑐) = (2^nd ‘⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩))
28		2fveq3 6897	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑐 = ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ → (♯‘(2^nd ‘𝑐)) = (♯‘(2^nd ‘⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩)))
29	28	oveq1d 7424	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑐 = ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ → ((♯‘(2^nd ‘𝑐)) − 1) = ((♯‘(2^nd ‘⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩)) − 1))
30	27, 29	oveq12d 7427	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑐 = ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ → ((2^nd ‘𝑐) prefix ((♯‘(2^nd ‘𝑐)) − 1)) = ((2^nd ‘⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩) prefix ((♯‘(2^nd ‘⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩)) − 1)))
31		vex 3479	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ 𝑓 ∈ V
32		ovex 7442	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑤 ++ ⟨“(𝑤‘0)”⟩) ∈ V
33	31, 32	op2nd 7984	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (2^nd ‘⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩) = (𝑤 ++ ⟨“(𝑤‘0)”⟩)
34	33	fveq2i 6895	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (♯‘(2^nd ‘⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩)) = (♯‘(𝑤 ++ ⟨“(𝑤‘0)”⟩))
35	34	oveq1i 7419	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((♯‘(2^nd ‘⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩)) − 1) = ((♯‘(𝑤 ++ ⟨“(𝑤‘0)”⟩)) − 1)
36	33, 35	oveq12i 7421	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((2^nd ‘⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩) prefix ((♯‘(2^nd ‘⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩)) − 1)) = ((𝑤 ++ ⟨“(𝑤‘0)”⟩) prefix ((♯‘(𝑤 ++ ⟨“(𝑤‘0)”⟩)) − 1))
37	30, 36	eqtrdi 2789	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑐 = ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ → ((2^nd ‘𝑐) prefix ((♯‘(2^nd ‘𝑐)) − 1)) = ((𝑤 ++ ⟨“(𝑤‘0)”⟩) prefix ((♯‘(𝑤 ++ ⟨“(𝑤‘0)”⟩)) − 1)))
38	37, 2	fvmptg 6997	. . . . . . . . . . . . . . . . . . 19 ⊢ ((⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ 𝐶 ∧ ((𝑤 ++ ⟨“(𝑤‘0)”⟩) prefix ((♯‘(𝑤 ++ ⟨“(𝑤‘0)”⟩)) − 1)) ∈ V) → (𝐹‘⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩) = ((𝑤 ++ ⟨“(𝑤‘0)”⟩) prefix ((♯‘(𝑤 ++ ⟨“(𝑤‘0)”⟩)) − 1)))
39	25, 26, 38	sylancl 587	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑤 ∈ Word (Vtx‘𝐺) ∧ 1 ≤ (♯‘𝑤)) ∧ ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (ClWalks‘𝐺)) → (𝐹‘⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩) = ((𝑤 ++ ⟨“(𝑤‘0)”⟩) prefix ((♯‘(𝑤 ++ ⟨“(𝑤‘0)”⟩)) − 1)))
40		wrdlenccats1lenm1 14572	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑤 ∈ Word (Vtx‘𝐺) → ((♯‘(𝑤 ++ ⟨“(𝑤‘0)”⟩)) − 1) = (♯‘𝑤))
41	40	ad2antrr 725	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑤 ∈ Word (Vtx‘𝐺) ∧ 1 ≤ (♯‘𝑤)) ∧ ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (ClWalks‘𝐺)) → ((♯‘(𝑤 ++ ⟨“(𝑤‘0)”⟩)) − 1) = (♯‘𝑤))
42	41	oveq2d 7425	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑤 ∈ Word (Vtx‘𝐺) ∧ 1 ≤ (♯‘𝑤)) ∧ ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (ClWalks‘𝐺)) → ((𝑤 ++ ⟨“(𝑤‘0)”⟩) prefix ((♯‘(𝑤 ++ ⟨“(𝑤‘0)”⟩)) − 1)) = ((𝑤 ++ ⟨“(𝑤‘0)”⟩) prefix (♯‘𝑤)))
43		simpll 766	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑤 ∈ Word (Vtx‘𝐺) ∧ 1 ≤ (♯‘𝑤)) ∧ ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (ClWalks‘𝐺)) → 𝑤 ∈ Word (Vtx‘𝐺))
44		simpl 484	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑤 ∈ Word (Vtx‘𝐺) ∧ 1 ≤ (♯‘𝑤)) ∧ ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (ClWalks‘𝐺)) → (𝑤 ∈ Word (Vtx‘𝐺) ∧ 1 ≤ (♯‘𝑤)))
45		wrdsymb1 14503	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑤 ∈ Word (Vtx‘𝐺) ∧ 1 ≤ (♯‘𝑤)) → (𝑤‘0) ∈ (Vtx‘𝐺))
46	44, 45	syl 17	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑤 ∈ Word (Vtx‘𝐺) ∧ 1 ≤ (♯‘𝑤)) ∧ ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (ClWalks‘𝐺)) → (𝑤‘0) ∈ (Vtx‘𝐺))
47	46	s1cld 14553	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑤 ∈ Word (Vtx‘𝐺) ∧ 1 ≤ (♯‘𝑤)) ∧ ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (ClWalks‘𝐺)) → ⟨“(𝑤‘0)”⟩ ∈ Word (Vtx‘𝐺))
48		eqidd 2734	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑤 ∈ Word (Vtx‘𝐺) ∧ 1 ≤ (♯‘𝑤)) ∧ ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (ClWalks‘𝐺)) → (♯‘𝑤) = (♯‘𝑤))
49		pfxccatid 14691	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑤 ∈ Word (Vtx‘𝐺) ∧ ⟨“(𝑤‘0)”⟩ ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑤) = (♯‘𝑤)) → ((𝑤 ++ ⟨“(𝑤‘0)”⟩) prefix (♯‘𝑤)) = 𝑤)
50	43, 47, 48, 49	syl3anc 1372	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑤 ∈ Word (Vtx‘𝐺) ∧ 1 ≤ (♯‘𝑤)) ∧ ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (ClWalks‘𝐺)) → ((𝑤 ++ ⟨“(𝑤‘0)”⟩) prefix (♯‘𝑤)) = 𝑤)
51	39, 42, 50	3eqtrrd 2778	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑤 ∈ Word (Vtx‘𝐺) ∧ 1 ≤ (♯‘𝑤)) ∧ ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (ClWalks‘𝐺)) → 𝑤 = (𝐹‘⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩))
52	51	ex 414	. . . . . . . . . . . . . . . 16 ⊢ ((𝑤 ∈ Word (Vtx‘𝐺) ∧ 1 ≤ (♯‘𝑤)) → (⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (ClWalks‘𝐺) → 𝑤 = (𝐹‘⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩)))
53	52	3adant1 1131	. . . . . . . . . . . . . . 15 ⊢ ((𝐺 ∈ USPGraph ∧ 𝑤 ∈ Word (Vtx‘𝐺) ∧ 1 ≤ (♯‘𝑤)) → (⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (ClWalks‘𝐺) → 𝑤 = (𝐹‘⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩)))
54	53	ad2antlr 726	. . . . . . . . . . . . . 14 ⊢ (((⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (ClWalks‘𝐺) ∧ (𝐺 ∈ USPGraph ∧ 𝑤 ∈ Word (Vtx‘𝐺) ∧ 1 ≤ (♯‘𝑤))) ∧ 𝑐 = ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩) → (⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (ClWalks‘𝐺) → 𝑤 = (𝐹‘⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩)))
55		fveq2 6892	. . . . . . . . . . . . . . . . 17 ⊢ (𝑐 = ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ → (𝐹‘𝑐) = (𝐹‘⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩))
56	55	eqeq2d 2744	. . . . . . . . . . . . . . . 16 ⊢ (𝑐 = ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ → (𝑤 = (𝐹‘𝑐) ↔ 𝑤 = (𝐹‘⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩)))
57	56	imbi2d 341	. . . . . . . . . . . . . . 15 ⊢ (𝑐 = ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ → ((⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (ClWalks‘𝐺) → 𝑤 = (𝐹‘𝑐)) ↔ (⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (ClWalks‘𝐺) → 𝑤 = (𝐹‘⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩))))
58	57	adantl 483	. . . . . . . . . . . . . 14 ⊢ (((⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (ClWalks‘𝐺) ∧ (𝐺 ∈ USPGraph ∧ 𝑤 ∈ Word (Vtx‘𝐺) ∧ 1 ≤ (♯‘𝑤))) ∧ 𝑐 = ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩) → ((⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (ClWalks‘𝐺) → 𝑤 = (𝐹‘𝑐)) ↔ (⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (ClWalks‘𝐺) → 𝑤 = (𝐹‘⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩))))
59	54, 58	mpbird 257	. . . . . . . . . . . . 13 ⊢ (((⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (ClWalks‘𝐺) ∧ (𝐺 ∈ USPGraph ∧ 𝑤 ∈ Word (Vtx‘𝐺) ∧ 1 ≤ (♯‘𝑤))) ∧ 𝑐 = ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩) → (⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (ClWalks‘𝐺) → 𝑤 = (𝐹‘𝑐)))
60	24, 59	rspcimedv 3604	. . . . . . . . . . . 12 ⊢ ((⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (ClWalks‘𝐺) ∧ (𝐺 ∈ USPGraph ∧ 𝑤 ∈ Word (Vtx‘𝐺) ∧ 1 ≤ (♯‘𝑤))) → (⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (ClWalks‘𝐺) → ∃𝑐 ∈ 𝐶 𝑤 = (𝐹‘𝑐)))
61	60	ex 414	. . . . . . . . . . 11 ⊢ (⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (ClWalks‘𝐺) → ((𝐺 ∈ USPGraph ∧ 𝑤 ∈ Word (Vtx‘𝐺) ∧ 1 ≤ (♯‘𝑤)) → (⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (ClWalks‘𝐺) → ∃𝑐 ∈ 𝐶 𝑤 = (𝐹‘𝑐))))
62	61	pm2.43b 55	. . . . . . . . . 10 ⊢ ((𝐺 ∈ USPGraph ∧ 𝑤 ∈ Word (Vtx‘𝐺) ∧ 1 ≤ (♯‘𝑤)) → (⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (ClWalks‘𝐺) → ∃𝑐 ∈ 𝐶 𝑤 = (𝐹‘𝑐)))
63	19, 62	biimtrid 241	. . . . . . . . 9 ⊢ ((𝐺 ∈ USPGraph ∧ 𝑤 ∈ Word (Vtx‘𝐺) ∧ 1 ≤ (♯‘𝑤)) → (𝑓(ClWalks‘𝐺)(𝑤 ++ ⟨“(𝑤‘0)”⟩) → ∃𝑐 ∈ 𝐶 𝑤 = (𝐹‘𝑐)))
64	63	exlimdv 1937	. . . . . . . 8 ⊢ ((𝐺 ∈ USPGraph ∧ 𝑤 ∈ Word (Vtx‘𝐺) ∧ 1 ≤ (♯‘𝑤)) → (∃𝑓 𝑓(ClWalks‘𝐺)(𝑤 ++ ⟨“(𝑤‘0)”⟩) → ∃𝑐 ∈ 𝐶 𝑤 = (𝐹‘𝑐)))
65	18, 64	sylbird 260	. . . . . . 7 ⊢ ((𝐺 ∈ USPGraph ∧ 𝑤 ∈ Word (Vtx‘𝐺) ∧ 1 ≤ (♯‘𝑤)) → (𝑤 ∈ (ClWWalks‘𝐺) → ∃𝑐 ∈ 𝐶 𝑤 = (𝐹‘𝑐)))
66	65	3expib 1123	. . . . . 6 ⊢ (𝐺 ∈ USPGraph → ((𝑤 ∈ Word (Vtx‘𝐺) ∧ 1 ≤ (♯‘𝑤)) → (𝑤 ∈ (ClWWalks‘𝐺) → ∃𝑐 ∈ 𝐶 𝑤 = (𝐹‘𝑐))))
67	66	com23 86	. . . . 5 ⊢ (𝐺 ∈ USPGraph → (𝑤 ∈ (ClWWalks‘𝐺) → ((𝑤 ∈ Word (Vtx‘𝐺) ∧ 1 ≤ (♯‘𝑤)) → ∃𝑐 ∈ 𝐶 𝑤 = (𝐹‘𝑐))))
68	67	imp 408	. . . 4 ⊢ ((𝐺 ∈ USPGraph ∧ 𝑤 ∈ (ClWWalks‘𝐺)) → ((𝑤 ∈ Word (Vtx‘𝐺) ∧ 1 ≤ (♯‘𝑤)) → ∃𝑐 ∈ 𝐶 𝑤 = (𝐹‘𝑐)))
69	16, 68	mpd 15	. . 3 ⊢ ((𝐺 ∈ USPGraph ∧ 𝑤 ∈ (ClWWalks‘𝐺)) → ∃𝑐 ∈ 𝐶 𝑤 = (𝐹‘𝑐))
70	69	ralrimiva 3147	. 2 ⊢ (𝐺 ∈ USPGraph → ∀𝑤 ∈ (ClWWalks‘𝐺)∃𝑐 ∈ 𝐶 𝑤 = (𝐹‘𝑐))
71		dffo3 7104	. 2 ⊢ (𝐹:𝐶–onto→(ClWWalks‘𝐺) ↔ (𝐹:𝐶⟶(ClWWalks‘𝐺) ∧ ∀𝑤 ∈ (ClWWalks‘𝐺)∃𝑐 ∈ 𝐶 𝑤 = (𝐹‘𝑐)))
72	3, 70, 71	sylanbrc 584	1 ⊢ (𝐺 ∈ USPGraph → 𝐹:𝐶–onto→(ClWWalks‘𝐺))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 ∧ w3a 1088 = wceq 1542 ∃wex 1782 ∈ wcel 2107 ≠ wne 2941 ∀wral 3062 ∃wrex 3071 {crab 3433 Vcvv 3475 ∅c0 4323 ⟨cop 4635 class class class wbr 5149 ↦ cmpt 5232 ⟶wf 6540 –onto→wfo 6542 ‘cfv 6544 (class class class)co 7409 1^st c1st 7973 2^nd c2nd 7974 0cc0 11110 1c1 11111 < clt 11248 ≤ cle 11249 − cmin 11444 ℤcz 12558 ♯chash 14290 Word cword 14464 ++ cconcat 14520 ⟨“cs1 14545 prefix cpfx 14620 Vtxcvtx 28256 iEdgciedg 28257 USPGraphcuspgr 28408 ClWalkscclwlks 29027 ClWWalkscclwwlk 29234

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 ax-cnex 11166 ax-resscn 11167 ax-1cn 11168 ax-icn 11169 ax-addcl 11170 ax-addrcl 11171 ax-mulcl 11172 ax-mulrcl 11173 ax-mulcom 11174 ax-addass 11175 ax-mulass 11176 ax-distr 11177 ax-i2m1 11178 ax-1ne0 11179 ax-1rid 11180 ax-rnegex 11181 ax-rrecex 11182 ax-cnre 11183 ax-pre-lttri 11184 ax-pre-lttrn 11185 ax-pre-ltadd 11186 ax-pre-mulgt0 11187

This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-ifp 1063 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-int 4952 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-riota 7365 df-ov 7412 df-oprab 7413 df-mpo 7414 df-om 7856 df-1st 7975 df-2nd 7976 df-frecs 8266 df-wrecs 8297 df-recs 8371 df-rdg 8410 df-1o 8466 df-2o 8467 df-oadd 8470 df-er 8703 df-map 8822 df-pm 8823 df-en 8940 df-dom 8941 df-sdom 8942 df-fin 8943 df-dju 9896 df-card 9934 df-pnf 11250 df-mnf 11251 df-xr 11252 df-ltxr 11253 df-le 11254 df-sub 11446 df-neg 11447 df-nn 12213 df-2 12275 df-n0 12473 df-xnn0 12545 df-z 12559 df-uz 12823 df-rp 12975 df-fz 13485 df-fzo 13628 df-hash 14291 df-word 14465 df-lsw 14513 df-concat 14521 df-s1 14546 df-substr 14591 df-pfx 14621 df-edg 28308 df-uhgr 28318 df-upgr 28342 df-uspgr 28410 df-wlks 28856 df-clwlks 29028 df-clwwlk 29235

This theorem is referenced by: clwlkclwwlkf1o 29264

W3C validator