Theorem clwlkclwwlkfo 30067

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 ≤ (♯‘(1st ‘𝑤))}
clwlkclwwlkf.f	⊢ 𝐹 = (𝑐 ∈ 𝐶 ↦ ((2nd ‘𝑐) prefix ((♯‘(2nd ‘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 ≤ (♯‘(1st ‘𝑤))}
2		clwlkclwwlkf.f	. . 3 ⊢ 𝐹 = (𝑐 ∈ 𝐶 ↦ ((2nd ‘𝑐) prefix ((♯‘(2nd ‘𝑐)) − 1)))
3	1, 2	clwlkclwwlkf 30066	. 2 ⊢ (𝐺 ∈ USPGraph → 𝐹:𝐶⟶(ClWWalks‘𝐺))
4		clwwlkgt0 30044	. . . . . 6 ⊢ (𝑤 ∈ (ClWWalks‘𝐺) → 0 < (♯‘𝑤))
5		eqid 2737	. . . . . . . 8 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
6	5	clwwlkbp 30043	. . . . . . 7 ⊢ (𝑤 ∈ (ClWWalks‘𝐺) → (𝐺 ∈ V ∧ 𝑤 ∈ Word (Vtx‘𝐺) ∧ 𝑤 ≠ ∅))
7		lencl 14460	. . . . . . . . . . . 12 ⊢ (𝑤 ∈ Word (Vtx‘𝐺) → (♯‘𝑤) ∈ ℕ0)
8	7	nn0zd 12517	. . . . . . . . . . 11 ⊢ (𝑤 ∈ Word (Vtx‘𝐺) → (♯‘𝑤) ∈ ℤ)
9		zgt0ge1 12550	. . . . . . . . . . 11 ⊢ ((♯‘𝑤) ∈ ℤ → (0 < (♯‘𝑤) ↔ 1 ≤ (♯‘𝑤)))
10	8, 9	syl 17	. . . . . . . . . 10 ⊢ (𝑤 ∈ Word (Vtx‘𝐺) → (0 < (♯‘𝑤) ↔ 1 ≤ (♯‘𝑤)))
11	10	biimpd 229	. . . . . . . . 9 ⊢ (𝑤 ∈ Word (Vtx‘𝐺) → (0 < (♯‘𝑤) → 1 ≤ (♯‘𝑤)))
12	11	anc2li 555	. . . . . . . 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 481	. . . 4 ⊢ ((𝐺 ∈ USPGraph ∧ 𝑤 ∈ (ClWWalks‘𝐺)) → (𝑤 ∈ Word (Vtx‘𝐺) ∧ 1 ≤ (♯‘𝑤)))
17		eqid 2737	. . . . . . . . 9 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺)
18	5, 17	clwlkclwwlk2 30061	. . . . . . . 8 ⊢ ((𝐺 ∈ USPGraph ∧ 𝑤 ∈ Word (Vtx‘𝐺) ∧ 1 ≤ (♯‘𝑤)) → (∃𝑓 𝑓(ClWalks‘𝐺)(𝑤 ++ ⟨“(𝑤‘0)”⟩) ↔ 𝑤 ∈ (ClWWalks‘𝐺)))
19		df-br 5100	. . . . . . . . . 10 ⊢ (𝑓(ClWalks‘𝐺)(𝑤 ++ ⟨“(𝑤‘0)”⟩) ↔ ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (ClWalks‘𝐺))
20		simpr2 1197	. . . . . . . . . . . . . 14 ⊢ ((⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (ClWalks‘𝐺) ∧ (𝐺 ∈ USPGraph ∧ 𝑤 ∈ Word (Vtx‘𝐺) ∧ 1 ≤ (♯‘𝑤))) → 𝑤 ∈ Word (Vtx‘𝐺))
21		simpr3 1198	. . . . . . . . . . . . . 14 ⊢ ((⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (ClWalks‘𝐺) ∧ (𝐺 ∈ USPGraph ∧ 𝑤 ∈ Word (Vtx‘𝐺) ∧ 1 ≤ (♯‘𝑤))) → 1 ≤ (♯‘𝑤))
22		simpl 482	. . . . . . . . . . . . . 14 ⊢ ((⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (ClWalks‘𝐺) ∧ (𝐺 ∈ USPGraph ∧ 𝑤 ∈ Word (Vtx‘𝐺) ∧ 1 ≤ (♯‘𝑤))) → ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (ClWalks‘𝐺))
23	1	clwlkclwwlkfolem 30065	. . . . . . . . . . . . . 14 ⊢ ((𝑤 ∈ Word (Vtx‘𝐺) ∧ 1 ≤ (♯‘𝑤) ∧ ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (ClWalks‘𝐺)) → ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ 𝐶)
24	20, 21, 22, 23	syl3anc 1374	. . . . . . . . . . . . 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 7393	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑤 ++ ⟨“(𝑤‘0)”⟩) prefix ((♯‘(𝑤 ++ ⟨“(𝑤‘0)”⟩)) − 1)) ∈ V
27		fveq2 6835	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑐 = ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ → (2nd ‘𝑐) = (2nd ‘⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩))
28		2fveq3 6840	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑐 = ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ → (♯‘(2nd ‘𝑐)) = (♯‘(2nd ‘⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩)))
29	28	oveq1d 7375	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑐 = ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ → ((♯‘(2nd ‘𝑐)) − 1) = ((♯‘(2nd ‘⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩)) − 1))
30	27, 29	oveq12d 7378	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑐 = ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ → ((2nd ‘𝑐) prefix ((♯‘(2nd ‘𝑐)) − 1)) = ((2nd ‘⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩) prefix ((♯‘(2nd ‘⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩)) − 1)))
31		vex 3445	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ 𝑓 ∈ V
32		ovex 7393	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑤 ++ ⟨“(𝑤‘0)”⟩) ∈ V
33	31, 32	op2nd 7944	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (2nd ‘⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩) = (𝑤 ++ ⟨“(𝑤‘0)”⟩)
34	33	fveq2i 6838	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (♯‘(2nd ‘⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩)) = (♯‘(𝑤 ++ ⟨“(𝑤‘0)”⟩))
35	34	oveq1i 7370	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((♯‘(2nd ‘⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩)) − 1) = ((♯‘(𝑤 ++ ⟨“(𝑤‘0)”⟩)) − 1)
36	33, 35	oveq12i 7372	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((2nd ‘⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩) prefix ((♯‘(2nd ‘⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩)) − 1)) = ((𝑤 ++ ⟨“(𝑤‘0)”⟩) prefix ((♯‘(𝑤 ++ ⟨“(𝑤‘0)”⟩)) − 1))
37	30, 36	eqtrdi 2788	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑐 = ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ → ((2nd ‘𝑐) prefix ((♯‘(2nd ‘𝑐)) − 1)) = ((𝑤 ++ ⟨“(𝑤‘0)”⟩) prefix ((♯‘(𝑤 ++ ⟨“(𝑤‘0)”⟩)) − 1)))
38	37, 2	fvmptg 6940	. . . . . . . . . . . . . . . . . . 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 14550	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑤 ∈ Word (Vtx‘𝐺) → ((♯‘(𝑤 ++ ⟨“(𝑤‘0)”⟩)) − 1) = (♯‘𝑤))
41	40	ad2antrr 727	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑤 ∈ Word (Vtx‘𝐺) ∧ 1 ≤ (♯‘𝑤)) ∧ ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (ClWalks‘𝐺)) → ((♯‘(𝑤 ++ ⟨“(𝑤‘0)”⟩)) − 1) = (♯‘𝑤))
42	41	oveq2d 7376	. . . . . . . . . . . . . . . . . 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 767	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑤 ∈ Word (Vtx‘𝐺) ∧ 1 ≤ (♯‘𝑤)) ∧ ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (ClWalks‘𝐺)) → 𝑤 ∈ Word (Vtx‘𝐺))
44		simpl 482	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑤 ∈ Word (Vtx‘𝐺) ∧ 1 ≤ (♯‘𝑤)) ∧ ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (ClWalks‘𝐺)) → (𝑤 ∈ Word (Vtx‘𝐺) ∧ 1 ≤ (♯‘𝑤)))
45		wrdsymb1 14480	. . . . . . . . . . . . . . . . . . . . 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 14531	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑤 ∈ Word (Vtx‘𝐺) ∧ 1 ≤ (♯‘𝑤)) ∧ ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (ClWalks‘𝐺)) → ⟨“(𝑤‘0)”⟩ ∈ Word (Vtx‘𝐺))
48		eqidd 2738	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑤 ∈ Word (Vtx‘𝐺) ∧ 1 ≤ (♯‘𝑤)) ∧ ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (ClWalks‘𝐺)) → (♯‘𝑤) = (♯‘𝑤))
49		pfxccatid 14668	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑤 ∈ Word (Vtx‘𝐺) ∧ ⟨“(𝑤‘0)”⟩ ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑤) = (♯‘𝑤)) → ((𝑤 ++ ⟨“(𝑤‘0)”⟩) prefix (♯‘𝑤)) = 𝑤)
50	43, 47, 48, 49	syl3anc 1374	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑤 ∈ Word (Vtx‘𝐺) ∧ 1 ≤ (♯‘𝑤)) ∧ ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (ClWalks‘𝐺)) → ((𝑤 ++ ⟨“(𝑤‘0)”⟩) prefix (♯‘𝑤)) = 𝑤)
51	39, 42, 50	3eqtrrd 2777	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑤 ∈ Word (Vtx‘𝐺) ∧ 1 ≤ (♯‘𝑤)) ∧ ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (ClWalks‘𝐺)) → 𝑤 = (𝐹‘⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩))
52	51	ex 412	. . . . . . . . . . . . . . . 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 728	. . . . . . . . . . . . . 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 6835	. . . . . . . . . . . . . . . . 17 ⊢ (𝑐 = ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ → (𝐹‘𝑐) = (𝐹‘⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩))
56	55	eqeq2d 2748	. . . . . . . . . . . . . . . 16 ⊢ (𝑐 = ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ → (𝑤 = (𝐹‘𝑐) ↔ 𝑤 = (𝐹‘⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩)))
57	56	imbi2d 340	. . . . . . . . . . . . . . 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 481	. . . . . . . . . . . . . 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 3568	. . . . . . . . . . . 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 412	. . . . . . . . . . 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 242	. . . . . . . . 9 ⊢ ((𝐺 ∈ USPGraph ∧ 𝑤 ∈ Word (Vtx‘𝐺) ∧ 1 ≤ (♯‘𝑤)) → (𝑓(ClWalks‘𝐺)(𝑤 ++ ⟨“(𝑤‘0)”⟩) → ∃𝑐 ∈ 𝐶 𝑤 = (𝐹‘𝑐)))
64	63	exlimdv 1935	. . . . . . . 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 406	. . . 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 3129	. 2 ⊢ (𝐺 ∈ USPGraph → ∀𝑤 ∈ (ClWWalks‘𝐺)∃𝑐 ∈ 𝐶 𝑤 = (𝐹‘𝑐))
71		dffo3 7049	. 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))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1542  ∃wex 1781   ∈ wcel 2114   ≠ wne 2933  ∀wral 3052  ∃wrex 3061  {crab 3400  Vcvv 3441  ∅c0 4286  ⟨cop 4587   class class class wbr 5099   ↦ cmpt 5180  ⟶wf 6489  –onto→wfo 6491  ‘cfv 6493  (class class class)co 7360  1^st c1st 7933  2^nd c2nd 7934  0cc0 11030  1c1 11031   < clt 11170   ≤ cle 11171   − cmin 11368  ℤcz 12492  ♯chash 14257  Word cword 14440   ++ cconcat 14497  ⟨“cs1 14523   prefix cpfx 14598  Vtxcvtx 29052  iEdgciedg 29053  USPGraphcuspgr 29204  ClWalkscclwlks 29826  ClWWalkscclwwlk 30039

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5225  ax-sep 5242  ax-nul 5252  ax-pow 5311  ax-pr 5378  ax-un 7682  ax-cnex 11086  ax-resscn 11087  ax-1cn 11088  ax-icn 11089  ax-addcl 11090  ax-addrcl 11091  ax-mulcl 11092  ax-mulrcl 11093  ax-mulcom 11094  ax-addass 11095  ax-mulass 11096  ax-distr 11097  ax-i2m1 11098  ax-1ne0 11099  ax-1rid 11100  ax-rnegex 11101  ax-rrecex 11102  ax-cnre 11103  ax-pre-lttri 11104  ax-pre-lttrn 11105  ax-pre-ltadd 11106  ax-pre-mulgt0 11107

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-ifp 1064  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3062  df-reu 3352  df-rab 3401  df-v 3443  df-sbc 3742  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4287  df-if 4481  df-pw 4557  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-int 4904  df-iun 4949  df-br 5100  df-opab 5162  df-mpt 5181  df-tr 5207  df-id 5520  df-eprel 5525  df-po 5533  df-so 5534  df-fr 5578  df-we 5580  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-pred 6260  df-ord 6321  df-on 6322  df-lim 6323  df-suc 6324  df-iota 6449  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-riota 7317  df-ov 7363  df-oprab 7364  df-mpo 7365  df-om 7811  df-1st 7935  df-2nd 7936  df-frecs 8225  df-wrecs 8256  df-recs 8305  df-rdg 8343  df-1o 8399  df-2o 8400  df-oadd 8403  df-er 8637  df-map 8769  df-pm 8770  df-en 8888  df-dom 8889  df-sdom 8890  df-fin 8891  df-dju 9817  df-card 9855  df-pnf 11172  df-mnf 11173  df-xr 11174  df-ltxr 11175  df-le 11176  df-sub 11370  df-neg 11371  df-nn 12150  df-2 12212  df-n0 12406  df-xnn0 12479  df-z 12493  df-uz 12756  df-rp 12910  df-fz 13428  df-fzo 13575  df-hash 14258  df-word 14441  df-lsw 14490  df-concat 14498  df-s1 14524  df-substr 14569  df-pfx 14599  df-edg 29104  df-uhgr 29114  df-upgr 29138  df-uspgr 29206  df-wlks 29656  df-clwlks 29827  df-clwwlk 30040

This theorem is referenced by:  clwlkclwwlkf1o  30069