Theorem clwlkclwwlkfo 28274

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 28273	. 2 ⊢ (𝐺 ∈ USPGraph → 𝐹:𝐶⟶(ClWWalks‘𝐺))
4		clwwlkgt0 28251	. . . . . 6 ⊢ (𝑤 ∈ (ClWWalks‘𝐺) → 0 < (♯‘𝑤))
5		eqid 2738	. . . . . . . 8 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
6	5	clwwlkbp 28250	. . . . . . 7 ⊢ (𝑤 ∈ (ClWWalks‘𝐺) → (𝐺 ∈ V ∧ 𝑤 ∈ Word (Vtx‘𝐺) ∧ 𝑤 ≠ ∅))
7		lencl 14164	. . . . . . . . . . . 12 ⊢ (𝑤 ∈ Word (Vtx‘𝐺) → (♯‘𝑤) ∈ ℕ0)
8	7	nn0zd 12353	. . . . . . . . . . 11 ⊢ (𝑤 ∈ Word (Vtx‘𝐺) → (♯‘𝑤) ∈ ℤ)
9		zgt0ge1 12304	. . . . . . . . . . 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 555	. . . . . . . 8 ⊢ (𝑤 ∈ Word (Vtx‘𝐺) → (0 < (♯‘𝑤) → (𝑤 ∈ Word (Vtx‘𝐺) ∧ 1 ≤ (♯‘𝑤))))
13	12	3ad2ant2 1132	. . . . . . 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 2738	. . . . . . . . 9 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺)
18	5, 17	clwlkclwwlk2 28268	. . . . . . . 8 ⊢ ((𝐺 ∈ USPGraph ∧ 𝑤 ∈ Word (Vtx‘𝐺) ∧ 1 ≤ (♯‘𝑤)) → (∃𝑓 𝑓(ClWalks‘𝐺)(𝑤 ++ ⟨“(𝑤‘0)”⟩) ↔ 𝑤 ∈ (ClWWalks‘𝐺)))
19		df-br 5071	. . . . . . . . . 10 ⊢ (𝑓(ClWalks‘𝐺)(𝑤 ++ ⟨“(𝑤‘0)”⟩) ↔ ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (ClWalks‘𝐺))
20		simpr2 1193	. . . . . . . . . . . . . 14 ⊢ ((⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (ClWalks‘𝐺) ∧ (𝐺 ∈ USPGraph ∧ 𝑤 ∈ Word (Vtx‘𝐺) ∧ 1 ≤ (♯‘𝑤))) → 𝑤 ∈ Word (Vtx‘𝐺))
21		simpr3 1194	. . . . . . . . . . . . . 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 28272	. . . . . . . . . . . . . 14 ⊢ ((𝑤 ∈ Word (Vtx‘𝐺) ∧ 1 ≤ (♯‘𝑤) ∧ ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (ClWalks‘𝐺)) → ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ 𝐶)
24	20, 21, 22, 23	syl3anc 1369	. . . . . . . . . . . . 13 ⊢ ((⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (ClWalks‘𝐺) ∧ (𝐺 ∈ USPGraph ∧ 𝑤 ∈ Word (Vtx‘𝐺) ∧ 1 ≤ (♯‘𝑤))) → ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ 𝐶)
25	23	3expa 1116	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑤 ∈ Word (Vtx‘𝐺) ∧ 1 ≤ (♯‘𝑤)) ∧ ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (ClWalks‘𝐺)) → ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ 𝐶)
26		ovex 7288	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑤 ++ ⟨“(𝑤‘0)”⟩) prefix ((♯‘(𝑤 ++ ⟨“(𝑤‘0)”⟩)) − 1)) ∈ V
27		fveq2 6756	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑐 = ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ → (2nd ‘𝑐) = (2nd ‘⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩))
28		2fveq3 6761	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑐 = ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ → (♯‘(2nd ‘𝑐)) = (♯‘(2nd ‘⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩)))
29	28	oveq1d 7270	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑐 = ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ → ((♯‘(2nd ‘𝑐)) − 1) = ((♯‘(2nd ‘⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩)) − 1))
30	27, 29	oveq12d 7273	. . . . . . . . . . . . . . . . . . . . 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 3426	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ 𝑓 ∈ V
32		ovex 7288	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑤 ++ ⟨“(𝑤‘0)”⟩) ∈ V
33	31, 32	op2nd 7813	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (2nd ‘⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩) = (𝑤 ++ ⟨“(𝑤‘0)”⟩)
34	33	fveq2i 6759	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (♯‘(2nd ‘⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩)) = (♯‘(𝑤 ++ ⟨“(𝑤‘0)”⟩))
35	34	oveq1i 7265	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((♯‘(2nd ‘⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩)) − 1) = ((♯‘(𝑤 ++ ⟨“(𝑤‘0)”⟩)) − 1)
36	33, 35	oveq12i 7267	. . . . . . . . . . . . . . . . . . . . 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 2795	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑐 = ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ → ((2nd ‘𝑐) prefix ((♯‘(2nd ‘𝑐)) − 1)) = ((𝑤 ++ ⟨“(𝑤‘0)”⟩) prefix ((♯‘(𝑤 ++ ⟨“(𝑤‘0)”⟩)) − 1)))
38	37, 2	fvmptg 6855	. . . . . . . . . . . . . . . . . . 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 585	. . . . . . . . . . . . . . . . . 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 14255	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑤 ∈ Word (Vtx‘𝐺) → ((♯‘(𝑤 ++ ⟨“(𝑤‘0)”⟩)) − 1) = (♯‘𝑤))
41	40	ad2antrr 722	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑤 ∈ Word (Vtx‘𝐺) ∧ 1 ≤ (♯‘𝑤)) ∧ ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (ClWalks‘𝐺)) → ((♯‘(𝑤 ++ ⟨“(𝑤‘0)”⟩)) − 1) = (♯‘𝑤))
42	41	oveq2d 7271	. . . . . . . . . . . . . . . . . 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 763	. . . . . . . . . . . . . . . . . . 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 14184	. . . . . . . . . . . . . . . . . . . . 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 14236	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑤 ∈ Word (Vtx‘𝐺) ∧ 1 ≤ (♯‘𝑤)) ∧ ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (ClWalks‘𝐺)) → ⟨“(𝑤‘0)”⟩ ∈ Word (Vtx‘𝐺))
48		eqidd 2739	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑤 ∈ Word (Vtx‘𝐺) ∧ 1 ≤ (♯‘𝑤)) ∧ ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (ClWalks‘𝐺)) → (♯‘𝑤) = (♯‘𝑤))
49		pfxccatid 14382	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑤 ∈ Word (Vtx‘𝐺) ∧ ⟨“(𝑤‘0)”⟩ ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑤) = (♯‘𝑤)) → ((𝑤 ++ ⟨“(𝑤‘0)”⟩) prefix (♯‘𝑤)) = 𝑤)
50	43, 47, 48, 49	syl3anc 1369	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑤 ∈ Word (Vtx‘𝐺) ∧ 1 ≤ (♯‘𝑤)) ∧ ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (ClWalks‘𝐺)) → ((𝑤 ++ ⟨“(𝑤‘0)”⟩) prefix (♯‘𝑤)) = 𝑤)
51	39, 42, 50	3eqtrrd 2783	. . . . . . . . . . . . . . . . 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 1128	. . . . . . . . . . . . . . 15 ⊢ ((𝐺 ∈ USPGraph ∧ 𝑤 ∈ Word (Vtx‘𝐺) ∧ 1 ≤ (♯‘𝑤)) → (⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (ClWalks‘𝐺) → 𝑤 = (𝐹‘⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩)))
54	53	ad2antlr 723	. . . . . . . . . . . . . 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 6756	. . . . . . . . . . . . . . . . 17 ⊢ (𝑐 = ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ → (𝐹‘𝑐) = (𝐹‘⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩))
56	55	eqeq2d 2749	. . . . . . . . . . . . . . . 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 256	. . . . . . . . . . . . 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 3542	. . . . . . . . . . . 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	syl5bi 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 259	. . . . . . 7 ⊢ ((𝐺 ∈ USPGraph ∧ 𝑤 ∈ Word (Vtx‘𝐺) ∧ 1 ≤ (♯‘𝑤)) → (𝑤 ∈ (ClWWalks‘𝐺) → ∃𝑐 ∈ 𝐶 𝑤 = (𝐹‘𝑐)))
66	65	3expib 1120	. . . . . 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 3107	. 2 ⊢ (𝐺 ∈ USPGraph → ∀𝑤 ∈ (ClWWalks‘𝐺)∃𝑐 ∈ 𝐶 𝑤 = (𝐹‘𝑐))
71		dffo3 6960	. 2 ⊢ (𝐹:𝐶–onto→(ClWWalks‘𝐺) ↔ (𝐹:𝐶⟶(ClWWalks‘𝐺) ∧ ∀𝑤 ∈ (ClWWalks‘𝐺)∃𝑐 ∈ 𝐶 𝑤 = (𝐹‘𝑐)))
72	3, 70, 71	sylanbrc 582	1 ⊢ (𝐺 ∈ USPGraph → 𝐹:𝐶–onto→(ClWWalks‘𝐺))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   ∧ w3a 1085   = wceq 1539  ∃wex 1783   ∈ wcel 2108   ≠ wne 2942  ∀wral 3063  ∃wrex 3064  {crab 3067  Vcvv 3422  ∅c0 4253  ⟨cop 4564   class class class wbr 5070   ↦ cmpt 5153  ⟶wf 6414  –onto→wfo 6416  ‘cfv 6418  (class class class)co 7255  1^st c1st 7802  2^nd c2nd 7803  0cc0 10802  1c1 10803   < clt 10940   ≤ cle 10941   − cmin 11135  ℤcz 12249  ♯chash 13972  Word cword 14145   ++ cconcat 14201  ⟨“cs1 14228   prefix cpfx 14311  Vtxcvtx 27269  iEdgciedg 27270  USPGraphcuspgr 27421  ClWalkscclwlks 28039  ClWWalkscclwwlk 28246

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566  ax-cnex 10858  ax-resscn 10859  ax-1cn 10860  ax-icn 10861  ax-addcl 10862  ax-addrcl 10863  ax-mulcl 10864  ax-mulrcl 10865  ax-mulcom 10866  ax-addass 10867  ax-mulass 10868  ax-distr 10869  ax-i2m1 10870  ax-1ne0 10871  ax-1rid 10872  ax-rnegex 10873  ax-rrecex 10874  ax-cnre 10875  ax-pre-lttri 10876  ax-pre-lttrn 10877  ax-pre-ltadd 10878  ax-pre-mulgt0 10879

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-ifp 1060  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-nel 3049  df-ral 3068  df-rex 3069  df-reu 3070  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-uni 4837  df-int 4877  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-tr 5188  df-id 5480  df-eprel 5486  df-po 5494  df-so 5495  df-fr 5535  df-we 5537  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-pred 6191  df-ord 6254  df-on 6255  df-lim 6256  df-suc 6257  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-riota 7212  df-ov 7258  df-oprab 7259  df-mpo 7260  df-om 7688  df-1st 7804  df-2nd 7805  df-frecs 8068  df-wrecs 8099  df-recs 8173  df-rdg 8212  df-1o 8267  df-2o 8268  df-oadd 8271  df-er 8456  df-map 8575  df-pm 8576  df-en 8692  df-dom 8693  df-sdom 8694  df-fin 8695  df-dju 9590  df-card 9628  df-pnf 10942  df-mnf 10943  df-xr 10944  df-ltxr 10945  df-le 10946  df-sub 11137  df-neg 11138  df-nn 11904  df-2 11966  df-n0 12164  df-xnn0 12236  df-z 12250  df-uz 12512  df-rp 12660  df-fz 13169  df-fzo 13312  df-hash 13973  df-word 14146  df-lsw 14194  df-concat 14202  df-s1 14229  df-substr 14282  df-pfx 14312  df-edg 27321  df-uhgr 27331  df-upgr 27355  df-uspgr 27423  df-wlks 27869  df-clwlks 28040  df-clwwlk 28247

This theorem is referenced by:  clwlkclwwlkf1o  28276