Theorem clwlkclwwlkfo 30371

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 30370	. 2 ⊢ (𝐺 ∈ USPGraph → 𝐹:𝐶⟶(ClWWalks‘𝐺))
4		clwwlkgt0 30348	. . . . . 6 ⊢ (𝑤 ∈ (ClWWalks‘𝐺) → 0 < (♯‘𝑤))
5		eqid 2762	. . . . . . . 8 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
6	5	clwwlkbp 30347	. . . . . . 7 ⊢ (𝑤 ∈ (ClWWalks‘𝐺) → (𝐺 ∈ V ∧ 𝑤 ∈ Word (Vtx‘𝐺) ∧ 𝑤 ≠ ∅))
7		lencl 14577	. . . . . . . . . . . 12 ⊢ (𝑤 ∈ Word (Vtx‘𝐺) → (♯‘𝑤) ∈ ℕ0)
8	7	nn0zd 12622	. . . . . . . . . . 11 ⊢ (𝑤 ∈ Word (Vtx‘𝐺) → (♯‘𝑤) ∈ ℤ)
9		zgt0ge1 12656	. . . . . . . . . . 11 ⊢ ((♯‘𝑤) ∈ ℤ → (0 < (♯‘𝑤) ↔ 1 ≤ (♯‘𝑤)))
10	8, 9	syl 18	. . . . . . . . . 10 ⊢ (𝑤 ∈ Word (Vtx‘𝐺) → (0 < (♯‘𝑤) ↔ 1 ≤ (♯‘𝑤)))
11	10	biimpd 232	. . . . . . . . 9 ⊢ (𝑤 ∈ Word (Vtx‘𝐺) → (0 < (♯‘𝑤) → 1 ≤ (♯‘𝑤)))
12	11	anc2li 564	. . . . . . . 8 ⊢ (𝑤 ∈ Word (Vtx‘𝐺) → (0 < (♯‘𝑤) → (𝑤 ∈ Word (Vtx‘𝐺) ∧ 1 ≤ (♯‘𝑤))))
13	12	3ad2ant2 1151	. . . . . . 7 ⊢ ((𝐺 ∈ V ∧ 𝑤 ∈ Word (Vtx‘𝐺) ∧ 𝑤 ≠ ∅) → (0 < (♯‘𝑤) → (𝑤 ∈ Word (Vtx‘𝐺) ∧ 1 ≤ (♯‘𝑤))))
14	6, 13	syl 18	. . . . . 6 ⊢ (𝑤 ∈ (ClWWalks‘𝐺) → (0 < (♯‘𝑤) → (𝑤 ∈ Word (Vtx‘𝐺) ∧ 1 ≤ (♯‘𝑤))))
15	4, 14	mpd 16	. . . . 5 ⊢ (𝑤 ∈ (ClWWalks‘𝐺) → (𝑤 ∈ Word (Vtx‘𝐺) ∧ 1 ≤ (♯‘𝑤)))
16	15	adantl 486	. . . 4 ⊢ ((𝐺 ∈ USPGraph ∧ 𝑤 ∈ (ClWWalks‘𝐺)) → (𝑤 ∈ Word (Vtx‘𝐺) ∧ 1 ≤ (♯‘𝑤)))
17		eqid 2762	. . . . . . . . 9 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺)
18	5, 17	clwlkclwwlk2 30365	. . . . . . . 8 ⊢ ((𝐺 ∈ USPGraph ∧ 𝑤 ∈ Word (Vtx‘𝐺) ∧ 1 ≤ (♯‘𝑤)) → (∃𝑓 𝑓(ClWalks‘𝐺)(𝑤 ++ ⟨“(𝑤‘0)”⟩) ↔ 𝑤 ∈ (ClWWalks‘𝐺)))
19		df-br 5109	. . . . . . . . . 10 ⊢ (𝑓(ClWalks‘𝐺)(𝑤 ++ ⟨“(𝑤‘0)”⟩) ↔ ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (ClWalks‘𝐺))
20		simpr2 1213	. . . . . . . . . . . . . 14 ⊢ ((⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (ClWalks‘𝐺) ∧ (𝐺 ∈ USPGraph ∧ 𝑤 ∈ Word (Vtx‘𝐺) ∧ 1 ≤ (♯‘𝑤))) → 𝑤 ∈ Word (Vtx‘𝐺))
21		simpr3 1214	. . . . . . . . . . . . . 14 ⊢ ((⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (ClWalks‘𝐺) ∧ (𝐺 ∈ USPGraph ∧ 𝑤 ∈ Word (Vtx‘𝐺) ∧ 1 ≤ (♯‘𝑤))) → 1 ≤ (♯‘𝑤))
22		simpl 487	. . . . . . . . . . . . . 14 ⊢ ((⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (ClWalks‘𝐺) ∧ (𝐺 ∈ USPGraph ∧ 𝑤 ∈ Word (Vtx‘𝐺) ∧ 1 ≤ (♯‘𝑤))) → ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (ClWalks‘𝐺))
23	1	clwlkclwwlkfolem 30369	. . . . . . . . . . . . . 14 ⊢ ((𝑤 ∈ Word (Vtx‘𝐺) ∧ 1 ≤ (♯‘𝑤) ∧ ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (ClWalks‘𝐺)) → ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ 𝐶)
24	20, 21, 22, 23	syl3anc 1397	. . . . . . . . . . . . 13 ⊢ ((⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (ClWalks‘𝐺) ∧ (𝐺 ∈ USPGraph ∧ 𝑤 ∈ Word (Vtx‘𝐺) ∧ 1 ≤ (♯‘𝑤))) → ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ 𝐶)
25	23	3expa 1135	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑤 ∈ Word (Vtx‘𝐺) ∧ 1 ≤ (♯‘𝑤)) ∧ ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (ClWalks‘𝐺)) → ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ 𝐶)
26		ovex 7445	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑤 ++ ⟨“(𝑤‘0)”⟩) prefix ((♯‘(𝑤 ++ ⟨“(𝑤‘0)”⟩)) − 1)) ∈ V
27		fveq2 6881	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑐 = ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ → (2nd ‘𝑐) = (2nd ‘⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩))
28		2fveq3 6886	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑐 = ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ → (♯‘(2nd ‘𝑐)) = (♯‘(2nd ‘⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩)))
29	28	oveq1d 7427	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑐 = ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ → ((♯‘(2nd ‘𝑐)) − 1) = ((♯‘(2nd ‘⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩)) − 1))
30	27, 29	oveq12d 7430	. . . . . . . . . . . . . . . . . . . . 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 3458	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ 𝑓 ∈ V
32		ovex 7445	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑤 ++ ⟨“(𝑤‘0)”⟩) ∈ V
33	31, 32	op2nd 7993	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (2nd ‘⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩) = (𝑤 ++ ⟨“(𝑤‘0)”⟩)
34	33	fveq2i 6884	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (♯‘(2nd ‘⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩)) = (♯‘(𝑤 ++ ⟨“(𝑤‘0)”⟩))
35	34	oveq1i 7422	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((♯‘(2nd ‘⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩)) − 1) = ((♯‘(𝑤 ++ ⟨“(𝑤‘0)”⟩)) − 1)
36	33, 35	oveq12i 7424	. . . . . . . . . . . . . . . . . . . . 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 2813	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑐 = ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ → ((2nd ‘𝑐) prefix ((♯‘(2nd ‘𝑐)) − 1)) = ((𝑤 ++ ⟨“(𝑤‘0)”⟩) prefix ((♯‘(𝑤 ++ ⟨“(𝑤‘0)”⟩)) − 1)))
38	37, 2	fvmptg 6987	. . . . . . . . . . . . . . . . . . 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 597	. . . . . . . . . . . . . . . . . 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 14667	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑤 ∈ Word (Vtx‘𝐺) → ((♯‘(𝑤 ++ ⟨“(𝑤‘0)”⟩)) − 1) = (♯‘𝑤))
41	40	ad2antrr 738	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑤 ∈ Word (Vtx‘𝐺) ∧ 1 ≤ (♯‘𝑤)) ∧ ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (ClWalks‘𝐺)) → ((♯‘(𝑤 ++ ⟨“(𝑤‘0)”⟩)) − 1) = (♯‘𝑤))
42	41	oveq2d 7428	. . . . . . . . . . . . . . . . . 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 778	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑤 ∈ Word (Vtx‘𝐺) ∧ 1 ≤ (♯‘𝑤)) ∧ ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (ClWalks‘𝐺)) → 𝑤 ∈ Word (Vtx‘𝐺))
44		simpl 487	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑤 ∈ Word (Vtx‘𝐺) ∧ 1 ≤ (♯‘𝑤)) ∧ ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (ClWalks‘𝐺)) → (𝑤 ∈ Word (Vtx‘𝐺) ∧ 1 ≤ (♯‘𝑤)))
45		wrdsymb1 14597	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑤 ∈ Word (Vtx‘𝐺) ∧ 1 ≤ (♯‘𝑤)) → (𝑤‘0) ∈ (Vtx‘𝐺))
46	44, 45	syl 18	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑤 ∈ Word (Vtx‘𝐺) ∧ 1 ≤ (♯‘𝑤)) ∧ ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (ClWalks‘𝐺)) → (𝑤‘0) ∈ (Vtx‘𝐺))
47	46	s1cld 14648	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑤 ∈ Word (Vtx‘𝐺) ∧ 1 ≤ (♯‘𝑤)) ∧ ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (ClWalks‘𝐺)) → ⟨“(𝑤‘0)”⟩ ∈ Word (Vtx‘𝐺))
48		eqidd 2763	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑤 ∈ Word (Vtx‘𝐺) ∧ 1 ≤ (♯‘𝑤)) ∧ ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (ClWalks‘𝐺)) → (♯‘𝑤) = (♯‘𝑤))
49		pfxccatid 14785	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑤 ∈ Word (Vtx‘𝐺) ∧ ⟨“(𝑤‘0)”⟩ ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑤) = (♯‘𝑤)) → ((𝑤 ++ ⟨“(𝑤‘0)”⟩) prefix (♯‘𝑤)) = 𝑤)
50	43, 47, 48, 49	syl3anc 1397	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑤 ∈ Word (Vtx‘𝐺) ∧ 1 ≤ (♯‘𝑤)) ∧ ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (ClWalks‘𝐺)) → ((𝑤 ++ ⟨“(𝑤‘0)”⟩) prefix (♯‘𝑤)) = 𝑤)
51	39, 42, 50	3eqtrrd 2802	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑤 ∈ Word (Vtx‘𝐺) ∧ 1 ≤ (♯‘𝑤)) ∧ ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (ClWalks‘𝐺)) → 𝑤 = (𝐹‘⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩))
52	51	ex 417	. . . . . . . . . . . . . . . 16 ⊢ ((𝑤 ∈ Word (Vtx‘𝐺) ∧ 1 ≤ (♯‘𝑤)) → (⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (ClWalks‘𝐺) → 𝑤 = (𝐹‘⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩)))
53	52	3adant1 1147	. . . . . . . . . . . . . . 15 ⊢ ((𝐺 ∈ USPGraph ∧ 𝑤 ∈ Word (Vtx‘𝐺) ∧ 1 ≤ (♯‘𝑤)) → (⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (ClWalks‘𝐺) → 𝑤 = (𝐹‘⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩)))
54	53	ad2antlr 739	. . . . . . . . . . . . . 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 6881	. . . . . . . . . . . . . . . . 17 ⊢ (𝑐 = ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ → (𝐹‘𝑐) = (𝐹‘⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩))
56	55	eqeq2d 2773	. . . . . . . . . . . . . . . 16 ⊢ (𝑐 = ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ → (𝑤 = (𝐹‘𝑐) ↔ 𝑤 = (𝐹‘⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩)))
57	56	imbi2d 343	. . . . . . . . . . . . . . 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 486	. . . . . . . . . . . . . 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 260	. . . . . . . . . . . . 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 3571	. . . . . . . . . . . 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 417	. . . . . . . . . . 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 56	. . . . . . . . . 10 ⊢ ((𝐺 ∈ USPGraph ∧ 𝑤 ∈ Word (Vtx‘𝐺) ∧ 1 ≤ (♯‘𝑤)) → (⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (ClWalks‘𝐺) → ∃𝑐 ∈ 𝐶 𝑤 = (𝐹‘𝑐)))
63	19, 62	biimtrid 245	. . . . . . . . 9 ⊢ ((𝐺 ∈ USPGraph ∧ 𝑤 ∈ Word (Vtx‘𝐺) ∧ 1 ≤ (♯‘𝑤)) → (𝑓(ClWalks‘𝐺)(𝑤 ++ ⟨“(𝑤‘0)”⟩) → ∃𝑐 ∈ 𝐶 𝑤 = (𝐹‘𝑐)))
64	63	exlimdv 1962	. . . . . . . 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 263	. . . . . . 7 ⊢ ((𝐺 ∈ USPGraph ∧ 𝑤 ∈ Word (Vtx‘𝐺) ∧ 1 ≤ (♯‘𝑤)) → (𝑤 ∈ (ClWWalks‘𝐺) → ∃𝑐 ∈ 𝐶 𝑤 = (𝐹‘𝑐)))
66	65	3expib 1139	. . . . . 6 ⊢ (𝐺 ∈ USPGraph → ((𝑤 ∈ Word (Vtx‘𝐺) ∧ 1 ≤ (♯‘𝑤)) → (𝑤 ∈ (ClWWalks‘𝐺) → ∃𝑐 ∈ 𝐶 𝑤 = (𝐹‘𝑐))))
67	66	com23 87	. . . . 5 ⊢ (𝐺 ∈ USPGraph → (𝑤 ∈ (ClWWalks‘𝐺) → ((𝑤 ∈ Word (Vtx‘𝐺) ∧ 1 ≤ (♯‘𝑤)) → ∃𝑐 ∈ 𝐶 𝑤 = (𝐹‘𝑐))))
68	67	imp 411	. . . 4 ⊢ ((𝐺 ∈ USPGraph ∧ 𝑤 ∈ (ClWWalks‘𝐺)) → ((𝑤 ∈ Word (Vtx‘𝐺) ∧ 1 ≤ (♯‘𝑤)) → ∃𝑐 ∈ 𝐶 𝑤 = (𝐹‘𝑐)))
69	16, 68	mpd 16	. . 3 ⊢ ((𝐺 ∈ USPGraph ∧ 𝑤 ∈ (ClWWalks‘𝐺)) → ∃𝑐 ∈ 𝐶 𝑤 = (𝐹‘𝑐))
70	69	ralrimiva 3156	. 2 ⊢ (𝐺 ∈ USPGraph → ∀𝑤 ∈ (ClWWalks‘𝐺)∃𝑐 ∈ 𝐶 𝑤 = (𝐹‘𝑐))
71		dffo3 7097	. 2 ⊢ (𝐹:𝐶–onto→(ClWWalks‘𝐺) ↔ (𝐹:𝐶⟶(ClWWalks‘𝐺) ∧ ∀𝑤 ∈ (ClWWalks‘𝐺)∃𝑐 ∈ 𝐶 𝑤 = (𝐹‘𝑐)))
72	3, 70, 71	sylanbrc 594	1 ⊢ (𝐺 ∈ USPGraph → 𝐹:𝐶–onto→(ClWWalks‘𝐺))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 400   ∧ w3a 1102   = wceq 1569  ∃wex 1808   ∈ wcel 2142   ≠ wne 2957  ∀wral 3078  ∃wrex 3088  {crab 3415  Vcvv 3454  ∅c0 4285  ⟨cop 4594   class class class wbr 5108   ↦ cmpt 5191  ⟶wf 6532  –onto→wfo 6534  ‘cfv 6536  (class class class)co 7412  1^st c1st 7982  2^nd c2nd 7983  0cc0 11106  1c1 11107   < clt 11249   ≤ cle 11250   − cmin 11447  ℤcz 12597  ♯chash 14373  Word cword 14557   ++ cconcat 14614  ⟨“cs1 14640   prefix cpfx 14715  Vtxcvtx 29357  iEdgciedg 29358  USPGraphcuspgr 29509  ClWalkscclwlks 30130  ClWWalkscclwwlk 30343

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-rep 5237  ax-sep 5256  ax-nul 5268  ax-pow 5335  ax-pr 5403  ax-un 7734  ax-cnex 11162  ax-resscn 11163  ax-1cn 11164  ax-icn 11165  ax-addcl 11166  ax-addrcl 11167  ax-mulcl 11168  ax-mulrcl 11169  ax-mulcom 11170  ax-addass 11171  ax-mulass 11172  ax-distr 11173  ax-i2m1 11174  ax-1ne0 11175  ax-1rid 11176  ax-rnegex 11177  ax-rrecex 11178  ax-cnre 11179  ax-pre-lttri 11180  ax-pre-lttrn 11181  ax-pre-ltadd 11182  ax-pre-mulgt0 11183

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-ifp 1078  df-3or 1103  df-3an 1104  df-tru 1572  df-fal 1582  df-ex 1809  df-nf 1813  df-sb 2096  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-nel 3064  df-ral 3079  df-rex 3089  df-reu 3369  df-rab 3416  df-v 3456  df-sbc 3744  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286  df-if 4487  df-pw 4563  df-sn 4589  df-pr 4591  df-op 4595  df-uni 4872  df-int 4912  df-iun 4957  df-br 5109  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5555  df-eprel 5560  df-po 5568  df-so 5569  df-fr 5613  df-we 5615  df-xp 5666  df-rel 5667  df-cnv 5668  df-co 5669  df-dm 5670  df-rn 5671  df-res 5672  df-ima 5673  df-pred 6302  df-ord 6363  df-on 6364  df-lim 6365  df-suc 6366  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-riota 7369  df-ov 7415  df-oprab 7416  df-mpo 7417  df-om 7861  df-1st 7984  df-2nd 7985  df-frecs 8276  df-wrecs 8307  df-recs 8356  df-rdg 8395  df-1o 8451  df-2o 8452  df-oadd 8455  df-er 8692  df-map 8824  df-pm 8825  df-en 8942  df-dom 8943  df-sdom 8944  df-fin 8945  df-dju 9894  df-card 9932  df-pnf 11251  df-mnf 11252  df-xr 11253  df-ltxr 11254  df-le 11255  df-sub 11449  df-neg 11450  df-nn 12240  df-2 12309  df-n0 12511  df-xnn0 12584  df-z 12598  df-uz 12869  df-rp 13023  df-fz 13542  df-fzo 13690  df-hash 14374  df-word 14558  df-lsw 14607  df-concat 14615  df-s1 14641  df-substr 14686  df-pfx 14716  df-edg 29409  df-uhgr 29419  df-upgr 29443  df-uspgr 29511  df-wlks 29960  df-clwlks 30131  df-clwwlk 30344

This theorem is used by:  clwlkclwwlkf1o  30373