Theorem wlkiswwlks2 30205

Description: A walk as word corresponds to the sequence of vertices in a walk in a simple pseudograph. (Contributed by Alexander van der Vekens, 21-Jul-2018.) (Revised by AV, 10-Apr-2021.)

Assertion

Ref	Expression
wlkiswwlks2	⊢ (𝐺 ∈ USPGraph → (𝑃 ∈ (WWalks‘𝐺) → ∃𝑓 𝑓(Walks‘𝐺)𝑃))

Distinct variable groups:   𝑓,𝐺   𝑃,𝑓

Proof of Theorem wlkiswwlks2

Dummy variables 𝑖 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2763	. . . 4 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
2	1	wwlkbp 30171	. . 3 ⊢ (𝑃 ∈ (WWalks‘𝐺) → (𝐺 ∈ V ∧ 𝑃 ∈ Word (Vtx‘𝐺)))
3		eqid 2763	. . . . 5 ⊢ (Edg‘𝐺) = (Edg‘𝐺)
4	1, 3	iswwlks 30166	. . . 4 ⊢ (𝑃 ∈ (WWalks‘𝐺) ↔ (𝑃 ≠ ∅ ∧ 𝑃 ∈ Word (Vtx‘𝐺) ∧ ∀𝑖 ∈ (0..^((♯‘𝑃) − 1)){(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ∈ (Edg‘𝐺)))
5		ovex 7445	. . . . . . . . . . . . . . 15 ⊢ (0..^((♯‘𝑃) − 1)) ∈ V
6		mptexg 7221	. . . . . . . . . . . . . . 15 ⊢ ((0..^((♯‘𝑃) − 1)) ∈ V → (𝑥 ∈ (0..^((♯‘𝑃) − 1)) ↦ (◡(iEdg‘𝐺)‘{(𝑃‘𝑥), (𝑃‘(𝑥 + 1))})) ∈ V)
7	5, 6	mp1i 14	. . . . . . . . . . . . . 14 ⊢ (((𝑃 ≠ ∅ ∧ 𝑃 ∈ Word (Vtx‘𝐺)) ∧ ((𝐺 ∈ V ∧ 𝑃 ∈ Word (Vtx‘𝐺)) ∧ 𝐺 ∈ USPGraph)) → (𝑥 ∈ (0..^((♯‘𝑃) − 1)) ↦ (◡(iEdg‘𝐺)‘{(𝑃‘𝑥), (𝑃‘(𝑥 + 1))})) ∈ V)
8		simprr 784	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑃 ≠ ∅ ∧ 𝑃 ∈ Word (Vtx‘𝐺)) ∧ ((𝐺 ∈ V ∧ 𝑃 ∈ Word (Vtx‘𝐺)) ∧ 𝐺 ∈ USPGraph)) → 𝐺 ∈ USPGraph)
9		simplr 780	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑃 ≠ ∅ ∧ 𝑃 ∈ Word (Vtx‘𝐺)) ∧ ((𝐺 ∈ V ∧ 𝑃 ∈ Word (Vtx‘𝐺)) ∧ 𝐺 ∈ USPGraph)) → 𝑃 ∈ Word (Vtx‘𝐺))
10		hashge1 14427	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑃 ∈ Word (Vtx‘𝐺) ∧ 𝑃 ≠ ∅) → 1 ≤ (♯‘𝑃))
11	10	ancoms 463	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑃 ≠ ∅ ∧ 𝑃 ∈ Word (Vtx‘𝐺)) → 1 ≤ (♯‘𝑃))
12	11	adantr 485	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑃 ≠ ∅ ∧ 𝑃 ∈ Word (Vtx‘𝐺)) ∧ ((𝐺 ∈ V ∧ 𝑃 ∈ Word (Vtx‘𝐺)) ∧ 𝐺 ∈ USPGraph)) → 1 ≤ (♯‘𝑃))
13	8, 9, 12	3jca 1146	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑃 ≠ ∅ ∧ 𝑃 ∈ Word (Vtx‘𝐺)) ∧ ((𝐺 ∈ V ∧ 𝑃 ∈ Word (Vtx‘𝐺)) ∧ 𝐺 ∈ USPGraph)) → (𝐺 ∈ USPGraph ∧ 𝑃 ∈ Word (Vtx‘𝐺) ∧ 1 ≤ (♯‘𝑃)))
14	13	adantr 485	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑃 ≠ ∅ ∧ 𝑃 ∈ Word (Vtx‘𝐺)) ∧ ((𝐺 ∈ V ∧ 𝑃 ∈ Word (Vtx‘𝐺)) ∧ 𝐺 ∈ USPGraph)) ∧ 𝑓 = (𝑥 ∈ (0..^((♯‘𝑃) − 1)) ↦ (◡(iEdg‘𝐺)‘{(𝑃‘𝑥), (𝑃‘(𝑥 + 1))}))) → (𝐺 ∈ USPGraph ∧ 𝑃 ∈ Word (Vtx‘𝐺) ∧ 1 ≤ (♯‘𝑃)))
15		edgval 29380	. . . . . . . . . . . . . . . . . . . 20 ⊢ (Edg‘𝐺) = ran (iEdg‘𝐺)
16	15	a1i 11	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝑃 ≠ ∅ ∧ 𝑃 ∈ Word (Vtx‘𝐺)) ∧ ((𝐺 ∈ V ∧ 𝑃 ∈ Word (Vtx‘𝐺)) ∧ 𝐺 ∈ USPGraph)) ∧ 𝑓 = (𝑥 ∈ (0..^((♯‘𝑃) − 1)) ↦ (◡(iEdg‘𝐺)‘{(𝑃‘𝑥), (𝑃‘(𝑥 + 1))}))) → (Edg‘𝐺) = ran (iEdg‘𝐺))
17	16	eleq2d 2849	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝑃 ≠ ∅ ∧ 𝑃 ∈ Word (Vtx‘𝐺)) ∧ ((𝐺 ∈ V ∧ 𝑃 ∈ Word (Vtx‘𝐺)) ∧ 𝐺 ∈ USPGraph)) ∧ 𝑓 = (𝑥 ∈ (0..^((♯‘𝑃) − 1)) ↦ (◡(iEdg‘𝐺)‘{(𝑃‘𝑥), (𝑃‘(𝑥 + 1))}))) → ({(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ∈ (Edg‘𝐺) ↔ {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ∈ ran (iEdg‘𝐺)))
18	17	ralbidv 3188	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑃 ≠ ∅ ∧ 𝑃 ∈ Word (Vtx‘𝐺)) ∧ ((𝐺 ∈ V ∧ 𝑃 ∈ Word (Vtx‘𝐺)) ∧ 𝐺 ∈ USPGraph)) ∧ 𝑓 = (𝑥 ∈ (0..^((♯‘𝑃) − 1)) ↦ (◡(iEdg‘𝐺)‘{(𝑃‘𝑥), (𝑃‘(𝑥 + 1))}))) → (∀𝑖 ∈ (0..^((♯‘𝑃) − 1)){(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ∈ (Edg‘𝐺) ↔ ∀𝑖 ∈ (0..^((♯‘𝑃) − 1)){(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ∈ ran (iEdg‘𝐺)))
19	18	biimpd 232	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑃 ≠ ∅ ∧ 𝑃 ∈ Word (Vtx‘𝐺)) ∧ ((𝐺 ∈ V ∧ 𝑃 ∈ Word (Vtx‘𝐺)) ∧ 𝐺 ∈ USPGraph)) ∧ 𝑓 = (𝑥 ∈ (0..^((♯‘𝑃) − 1)) ↦ (◡(iEdg‘𝐺)‘{(𝑃‘𝑥), (𝑃‘(𝑥 + 1))}))) → (∀𝑖 ∈ (0..^((♯‘𝑃) − 1)){(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ∈ (Edg‘𝐺) → ∀𝑖 ∈ (0..^((♯‘𝑃) − 1)){(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ∈ ran (iEdg‘𝐺)))
20		eqid 2763	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ (0..^((♯‘𝑃) − 1)) ↦ (◡(iEdg‘𝐺)‘{(𝑃‘𝑥), (𝑃‘(𝑥 + 1))})) = (𝑥 ∈ (0..^((♯‘𝑃) − 1)) ↦ (◡(iEdg‘𝐺)‘{(𝑃‘𝑥), (𝑃‘(𝑥 + 1))}))
21		eqid 2763	. . . . . . . . . . . . . . . . 17 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺)
22	20, 21	wlkiswwlks2lem6 30204	. . . . . . . . . . . . . . . 16 ⊢ ((𝐺 ∈ USPGraph ∧ 𝑃 ∈ Word (Vtx‘𝐺) ∧ 1 ≤ (♯‘𝑃)) → (∀𝑖 ∈ (0..^((♯‘𝑃) − 1)){(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ∈ ran (iEdg‘𝐺) → ((𝑥 ∈ (0..^((♯‘𝑃) − 1)) ↦ (◡(iEdg‘𝐺)‘{(𝑃‘𝑥), (𝑃‘(𝑥 + 1))})) ∈ Word dom (iEdg‘𝐺) ∧ 𝑃:(0...(♯‘(𝑥 ∈ (0..^((♯‘𝑃) − 1)) ↦ (◡(iEdg‘𝐺)‘{(𝑃‘𝑥), (𝑃‘(𝑥 + 1))}))))⟶(Vtx‘𝐺) ∧ ∀𝑖 ∈ (0..^(♯‘(𝑥 ∈ (0..^((♯‘𝑃) − 1)) ↦ (◡(iEdg‘𝐺)‘{(𝑃‘𝑥), (𝑃‘(𝑥 + 1))}))))((iEdg‘𝐺)‘((𝑥 ∈ (0..^((♯‘𝑃) − 1)) ↦ (◡(iEdg‘𝐺)‘{(𝑃‘𝑥), (𝑃‘(𝑥 + 1))}))‘𝑖)) = {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))})))
23	14, 19, 22	sylsyld 62	. . . . . . . . . . . . . . 15 ⊢ ((((𝑃 ≠ ∅ ∧ 𝑃 ∈ Word (Vtx‘𝐺)) ∧ ((𝐺 ∈ V ∧ 𝑃 ∈ Word (Vtx‘𝐺)) ∧ 𝐺 ∈ USPGraph)) ∧ 𝑓 = (𝑥 ∈ (0..^((♯‘𝑃) − 1)) ↦ (◡(iEdg‘𝐺)‘{(𝑃‘𝑥), (𝑃‘(𝑥 + 1))}))) → (∀𝑖 ∈ (0..^((♯‘𝑃) − 1)){(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ∈ (Edg‘𝐺) → ((𝑥 ∈ (0..^((♯‘𝑃) − 1)) ↦ (◡(iEdg‘𝐺)‘{(𝑃‘𝑥), (𝑃‘(𝑥 + 1))})) ∈ Word dom (iEdg‘𝐺) ∧ 𝑃:(0...(♯‘(𝑥 ∈ (0..^((♯‘𝑃) − 1)) ↦ (◡(iEdg‘𝐺)‘{(𝑃‘𝑥), (𝑃‘(𝑥 + 1))}))))⟶(Vtx‘𝐺) ∧ ∀𝑖 ∈ (0..^(♯‘(𝑥 ∈ (0..^((♯‘𝑃) − 1)) ↦ (◡(iEdg‘𝐺)‘{(𝑃‘𝑥), (𝑃‘(𝑥 + 1))}))))((iEdg‘𝐺)‘((𝑥 ∈ (0..^((♯‘𝑃) − 1)) ↦ (◡(iEdg‘𝐺)‘{(𝑃‘𝑥), (𝑃‘(𝑥 + 1))}))‘𝑖)) = {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))})))
24		eleq1 2851	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑓 = (𝑥 ∈ (0..^((♯‘𝑃) − 1)) ↦ (◡(iEdg‘𝐺)‘{(𝑃‘𝑥), (𝑃‘(𝑥 + 1))})) → (𝑓 ∈ Word dom (iEdg‘𝐺) ↔ (𝑥 ∈ (0..^((♯‘𝑃) − 1)) ↦ (◡(iEdg‘𝐺)‘{(𝑃‘𝑥), (𝑃‘(𝑥 + 1))})) ∈ Word dom (iEdg‘𝐺)))
25		fveq2 6883	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑓 = (𝑥 ∈ (0..^((♯‘𝑃) − 1)) ↦ (◡(iEdg‘𝐺)‘{(𝑃‘𝑥), (𝑃‘(𝑥 + 1))})) → (♯‘𝑓) = (♯‘(𝑥 ∈ (0..^((♯‘𝑃) − 1)) ↦ (◡(iEdg‘𝐺)‘{(𝑃‘𝑥), (𝑃‘(𝑥 + 1))}))))
26	25	oveq2d 7428	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑓 = (𝑥 ∈ (0..^((♯‘𝑃) − 1)) ↦ (◡(iEdg‘𝐺)‘{(𝑃‘𝑥), (𝑃‘(𝑥 + 1))})) → (0...(♯‘𝑓)) = (0...(♯‘(𝑥 ∈ (0..^((♯‘𝑃) − 1)) ↦ (◡(iEdg‘𝐺)‘{(𝑃‘𝑥), (𝑃‘(𝑥 + 1))})))))
27	26	feq2d 6691	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑓 = (𝑥 ∈ (0..^((♯‘𝑃) − 1)) ↦ (◡(iEdg‘𝐺)‘{(𝑃‘𝑥), (𝑃‘(𝑥 + 1))})) → (𝑃:(0...(♯‘𝑓))⟶(Vtx‘𝐺) ↔ 𝑃:(0...(♯‘(𝑥 ∈ (0..^((♯‘𝑃) − 1)) ↦ (◡(iEdg‘𝐺)‘{(𝑃‘𝑥), (𝑃‘(𝑥 + 1))}))))⟶(Vtx‘𝐺)))
28	25	oveq2d 7428	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑓 = (𝑥 ∈ (0..^((♯‘𝑃) − 1)) ↦ (◡(iEdg‘𝐺)‘{(𝑃‘𝑥), (𝑃‘(𝑥 + 1))})) → (0..^(♯‘𝑓)) = (0..^(♯‘(𝑥 ∈ (0..^((♯‘𝑃) − 1)) ↦ (◡(iEdg‘𝐺)‘{(𝑃‘𝑥), (𝑃‘(𝑥 + 1))})))))
29		fveq1 6882	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑓 = (𝑥 ∈ (0..^((♯‘𝑃) − 1)) ↦ (◡(iEdg‘𝐺)‘{(𝑃‘𝑥), (𝑃‘(𝑥 + 1))})) → (𝑓‘𝑖) = ((𝑥 ∈ (0..^((♯‘𝑃) − 1)) ↦ (◡(iEdg‘𝐺)‘{(𝑃‘𝑥), (𝑃‘(𝑥 + 1))}))‘𝑖))
30	29	fveqeq2d 6891	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑓 = (𝑥 ∈ (0..^((♯‘𝑃) − 1)) ↦ (◡(iEdg‘𝐺)‘{(𝑃‘𝑥), (𝑃‘(𝑥 + 1))})) → (((iEdg‘𝐺)‘(𝑓‘𝑖)) = {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ↔ ((iEdg‘𝐺)‘((𝑥 ∈ (0..^((♯‘𝑃) − 1)) ↦ (◡(iEdg‘𝐺)‘{(𝑃‘𝑥), (𝑃‘(𝑥 + 1))}))‘𝑖)) = {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))}))
31	28, 30	raleqbidv 3338	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑓 = (𝑥 ∈ (0..^((♯‘𝑃) − 1)) ↦ (◡(iEdg‘𝐺)‘{(𝑃‘𝑥), (𝑃‘(𝑥 + 1))})) → (∀𝑖 ∈ (0..^(♯‘𝑓))((iEdg‘𝐺)‘(𝑓‘𝑖)) = {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ↔ ∀𝑖 ∈ (0..^(♯‘(𝑥 ∈ (0..^((♯‘𝑃) − 1)) ↦ (◡(iEdg‘𝐺)‘{(𝑃‘𝑥), (𝑃‘(𝑥 + 1))}))))((iEdg‘𝐺)‘((𝑥 ∈ (0..^((♯‘𝑃) − 1)) ↦ (◡(iEdg‘𝐺)‘{(𝑃‘𝑥), (𝑃‘(𝑥 + 1))}))‘𝑖)) = {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))}))
32	24, 27, 31	3anbi123d 1464	. . . . . . . . . . . . . . . . 17 ⊢ (𝑓 = (𝑥 ∈ (0..^((♯‘𝑃) − 1)) ↦ (◡(iEdg‘𝐺)‘{(𝑃‘𝑥), (𝑃‘(𝑥 + 1))})) → ((𝑓 ∈ Word dom (iEdg‘𝐺) ∧ 𝑃:(0...(♯‘𝑓))⟶(Vtx‘𝐺) ∧ ∀𝑖 ∈ (0..^(♯‘𝑓))((iEdg‘𝐺)‘(𝑓‘𝑖)) = {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))}) ↔ ((𝑥 ∈ (0..^((♯‘𝑃) − 1)) ↦ (◡(iEdg‘𝐺)‘{(𝑃‘𝑥), (𝑃‘(𝑥 + 1))})) ∈ Word dom (iEdg‘𝐺) ∧ 𝑃:(0...(♯‘(𝑥 ∈ (0..^((♯‘𝑃) − 1)) ↦ (◡(iEdg‘𝐺)‘{(𝑃‘𝑥), (𝑃‘(𝑥 + 1))}))))⟶(Vtx‘𝐺) ∧ ∀𝑖 ∈ (0..^(♯‘(𝑥 ∈ (0..^((♯‘𝑃) − 1)) ↦ (◡(iEdg‘𝐺)‘{(𝑃‘𝑥), (𝑃‘(𝑥 + 1))}))))((iEdg‘𝐺)‘((𝑥 ∈ (0..^((♯‘𝑃) − 1)) ↦ (◡(iEdg‘𝐺)‘{(𝑃‘𝑥), (𝑃‘(𝑥 + 1))}))‘𝑖)) = {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))})))
33	32	imbi2d 343	. . . . . . . . . . . . . . . 16 ⊢ (𝑓 = (𝑥 ∈ (0..^((♯‘𝑃) − 1)) ↦ (◡(iEdg‘𝐺)‘{(𝑃‘𝑥), (𝑃‘(𝑥 + 1))})) → ((∀𝑖 ∈ (0..^((♯‘𝑃) − 1)){(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ∈ (Edg‘𝐺) → (𝑓 ∈ Word dom (iEdg‘𝐺) ∧ 𝑃:(0...(♯‘𝑓))⟶(Vtx‘𝐺) ∧ ∀𝑖 ∈ (0..^(♯‘𝑓))((iEdg‘𝐺)‘(𝑓‘𝑖)) = {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))})) ↔ (∀𝑖 ∈ (0..^((♯‘𝑃) − 1)){(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ∈ (Edg‘𝐺) → ((𝑥 ∈ (0..^((♯‘𝑃) − 1)) ↦ (◡(iEdg‘𝐺)‘{(𝑃‘𝑥), (𝑃‘(𝑥 + 1))})) ∈ Word dom (iEdg‘𝐺) ∧ 𝑃:(0...(♯‘(𝑥 ∈ (0..^((♯‘𝑃) − 1)) ↦ (◡(iEdg‘𝐺)‘{(𝑃‘𝑥), (𝑃‘(𝑥 + 1))}))))⟶(Vtx‘𝐺) ∧ ∀𝑖 ∈ (0..^(♯‘(𝑥 ∈ (0..^((♯‘𝑃) − 1)) ↦ (◡(iEdg‘𝐺)‘{(𝑃‘𝑥), (𝑃‘(𝑥 + 1))}))))((iEdg‘𝐺)‘((𝑥 ∈ (0..^((♯‘𝑃) − 1)) ↦ (◡(iEdg‘𝐺)‘{(𝑃‘𝑥), (𝑃‘(𝑥 + 1))}))‘𝑖)) = {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))}))))
34	33	adantl 486	. . . . . . . . . . . . . . 15 ⊢ ((((𝑃 ≠ ∅ ∧ 𝑃 ∈ Word (Vtx‘𝐺)) ∧ ((𝐺 ∈ V ∧ 𝑃 ∈ Word (Vtx‘𝐺)) ∧ 𝐺 ∈ USPGraph)) ∧ 𝑓 = (𝑥 ∈ (0..^((♯‘𝑃) − 1)) ↦ (◡(iEdg‘𝐺)‘{(𝑃‘𝑥), (𝑃‘(𝑥 + 1))}))) → ((∀𝑖 ∈ (0..^((♯‘𝑃) − 1)){(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ∈ (Edg‘𝐺) → (𝑓 ∈ Word dom (iEdg‘𝐺) ∧ 𝑃:(0...(♯‘𝑓))⟶(Vtx‘𝐺) ∧ ∀𝑖 ∈ (0..^(♯‘𝑓))((iEdg‘𝐺)‘(𝑓‘𝑖)) = {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))})) ↔ (∀𝑖 ∈ (0..^((♯‘𝑃) − 1)){(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ∈ (Edg‘𝐺) → ((𝑥 ∈ (0..^((♯‘𝑃) − 1)) ↦ (◡(iEdg‘𝐺)‘{(𝑃‘𝑥), (𝑃‘(𝑥 + 1))})) ∈ Word dom (iEdg‘𝐺) ∧ 𝑃:(0...(♯‘(𝑥 ∈ (0..^((♯‘𝑃) − 1)) ↦ (◡(iEdg‘𝐺)‘{(𝑃‘𝑥), (𝑃‘(𝑥 + 1))}))))⟶(Vtx‘𝐺) ∧ ∀𝑖 ∈ (0..^(♯‘(𝑥 ∈ (0..^((♯‘𝑃) − 1)) ↦ (◡(iEdg‘𝐺)‘{(𝑃‘𝑥), (𝑃‘(𝑥 + 1))}))))((iEdg‘𝐺)‘((𝑥 ∈ (0..^((♯‘𝑃) − 1)) ↦ (◡(iEdg‘𝐺)‘{(𝑃‘𝑥), (𝑃‘(𝑥 + 1))}))‘𝑖)) = {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))}))))
35	23, 34	mpbird 260	. . . . . . . . . . . . . 14 ⊢ ((((𝑃 ≠ ∅ ∧ 𝑃 ∈ Word (Vtx‘𝐺)) ∧ ((𝐺 ∈ V ∧ 𝑃 ∈ Word (Vtx‘𝐺)) ∧ 𝐺 ∈ USPGraph)) ∧ 𝑓 = (𝑥 ∈ (0..^((♯‘𝑃) − 1)) ↦ (◡(iEdg‘𝐺)‘{(𝑃‘𝑥), (𝑃‘(𝑥 + 1))}))) → (∀𝑖 ∈ (0..^((♯‘𝑃) − 1)){(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ∈ (Edg‘𝐺) → (𝑓 ∈ Word dom (iEdg‘𝐺) ∧ 𝑃:(0...(♯‘𝑓))⟶(Vtx‘𝐺) ∧ ∀𝑖 ∈ (0..^(♯‘𝑓))((iEdg‘𝐺)‘(𝑓‘𝑖)) = {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))})))
36	7, 35	spcimedv 3555	. . . . . . . . . . . . 13 ⊢ (((𝑃 ≠ ∅ ∧ 𝑃 ∈ Word (Vtx‘𝐺)) ∧ ((𝐺 ∈ V ∧ 𝑃 ∈ Word (Vtx‘𝐺)) ∧ 𝐺 ∈ USPGraph)) → (∀𝑖 ∈ (0..^((♯‘𝑃) − 1)){(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ∈ (Edg‘𝐺) → ∃𝑓(𝑓 ∈ Word dom (iEdg‘𝐺) ∧ 𝑃:(0...(♯‘𝑓))⟶(Vtx‘𝐺) ∧ ∀𝑖 ∈ (0..^(♯‘𝑓))((iEdg‘𝐺)‘(𝑓‘𝑖)) = {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))})))
37	36	ex 417	. . . . . . . . . . . 12 ⊢ ((𝑃 ≠ ∅ ∧ 𝑃 ∈ Word (Vtx‘𝐺)) → (((𝐺 ∈ V ∧ 𝑃 ∈ Word (Vtx‘𝐺)) ∧ 𝐺 ∈ USPGraph) → (∀𝑖 ∈ (0..^((♯‘𝑃) − 1)){(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ∈ (Edg‘𝐺) → ∃𝑓(𝑓 ∈ Word dom (iEdg‘𝐺) ∧ 𝑃:(0...(♯‘𝑓))⟶(Vtx‘𝐺) ∧ ∀𝑖 ∈ (0..^(♯‘𝑓))((iEdg‘𝐺)‘(𝑓‘𝑖)) = {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))}))))
38	37	com23 87	. . . . . . . . . . 11 ⊢ ((𝑃 ≠ ∅ ∧ 𝑃 ∈ Word (Vtx‘𝐺)) → (∀𝑖 ∈ (0..^((♯‘𝑃) − 1)){(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ∈ (Edg‘𝐺) → (((𝐺 ∈ V ∧ 𝑃 ∈ Word (Vtx‘𝐺)) ∧ 𝐺 ∈ USPGraph) → ∃𝑓(𝑓 ∈ Word dom (iEdg‘𝐺) ∧ 𝑃:(0...(♯‘𝑓))⟶(Vtx‘𝐺) ∧ ∀𝑖 ∈ (0..^(♯‘𝑓))((iEdg‘𝐺)‘(𝑓‘𝑖)) = {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))}))))
39	38	3impia 1135	. . . . . . . . . 10 ⊢ ((𝑃 ≠ ∅ ∧ 𝑃 ∈ Word (Vtx‘𝐺) ∧ ∀𝑖 ∈ (0..^((♯‘𝑃) − 1)){(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ∈ (Edg‘𝐺)) → (((𝐺 ∈ V ∧ 𝑃 ∈ Word (Vtx‘𝐺)) ∧ 𝐺 ∈ USPGraph) → ∃𝑓(𝑓 ∈ Word dom (iEdg‘𝐺) ∧ 𝑃:(0...(♯‘𝑓))⟶(Vtx‘𝐺) ∧ ∀𝑖 ∈ (0..^(♯‘𝑓))((iEdg‘𝐺)‘(𝑓‘𝑖)) = {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))})))
40	39	expd 420	. . . . . . . . 9 ⊢ ((𝑃 ≠ ∅ ∧ 𝑃 ∈ Word (Vtx‘𝐺) ∧ ∀𝑖 ∈ (0..^((♯‘𝑃) − 1)){(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ∈ (Edg‘𝐺)) → ((𝐺 ∈ V ∧ 𝑃 ∈ Word (Vtx‘𝐺)) → (𝐺 ∈ USPGraph → ∃𝑓(𝑓 ∈ Word dom (iEdg‘𝐺) ∧ 𝑃:(0...(♯‘𝑓))⟶(Vtx‘𝐺) ∧ ∀𝑖 ∈ (0..^(♯‘𝑓))((iEdg‘𝐺)‘(𝑓‘𝑖)) = {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))}))))
41	40	impcom 412	. . . . . . . 8 ⊢ (((𝐺 ∈ V ∧ 𝑃 ∈ Word (Vtx‘𝐺)) ∧ (𝑃 ≠ ∅ ∧ 𝑃 ∈ Word (Vtx‘𝐺) ∧ ∀𝑖 ∈ (0..^((♯‘𝑃) − 1)){(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ∈ (Edg‘𝐺))) → (𝐺 ∈ USPGraph → ∃𝑓(𝑓 ∈ Word dom (iEdg‘𝐺) ∧ 𝑃:(0...(♯‘𝑓))⟶(Vtx‘𝐺) ∧ ∀𝑖 ∈ (0..^(♯‘𝑓))((iEdg‘𝐺)‘(𝑓‘𝑖)) = {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))})))
42	41	imp 411	. . . . . . 7 ⊢ ((((𝐺 ∈ V ∧ 𝑃 ∈ Word (Vtx‘𝐺)) ∧ (𝑃 ≠ ∅ ∧ 𝑃 ∈ Word (Vtx‘𝐺) ∧ ∀𝑖 ∈ (0..^((♯‘𝑃) − 1)){(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ∈ (Edg‘𝐺))) ∧ 𝐺 ∈ USPGraph) → ∃𝑓(𝑓 ∈ Word dom (iEdg‘𝐺) ∧ 𝑃:(0...(♯‘𝑓))⟶(Vtx‘𝐺) ∧ ∀𝑖 ∈ (0..^(♯‘𝑓))((iEdg‘𝐺)‘(𝑓‘𝑖)) = {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))}))
43		uspgrupgr 29509	. . . . . . . . . 10 ⊢ (𝐺 ∈ USPGraph → 𝐺 ∈ UPGraph)
44	1, 21	upgriswlk 29971	. . . . . . . . . 10 ⊢ (𝐺 ∈ UPGraph → (𝑓(Walks‘𝐺)𝑃 ↔ (𝑓 ∈ Word dom (iEdg‘𝐺) ∧ 𝑃:(0...(♯‘𝑓))⟶(Vtx‘𝐺) ∧ ∀𝑖 ∈ (0..^(♯‘𝑓))((iEdg‘𝐺)‘(𝑓‘𝑖)) = {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))})))
45	43, 44	syl 18	. . . . . . . . 9 ⊢ (𝐺 ∈ USPGraph → (𝑓(Walks‘𝐺)𝑃 ↔ (𝑓 ∈ Word dom (iEdg‘𝐺) ∧ 𝑃:(0...(♯‘𝑓))⟶(Vtx‘𝐺) ∧ ∀𝑖 ∈ (0..^(♯‘𝑓))((iEdg‘𝐺)‘(𝑓‘𝑖)) = {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))})))
46	45	adantl 486	. . . . . . . 8 ⊢ ((((𝐺 ∈ V ∧ 𝑃 ∈ Word (Vtx‘𝐺)) ∧ (𝑃 ≠ ∅ ∧ 𝑃 ∈ Word (Vtx‘𝐺) ∧ ∀𝑖 ∈ (0..^((♯‘𝑃) − 1)){(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ∈ (Edg‘𝐺))) ∧ 𝐺 ∈ USPGraph) → (𝑓(Walks‘𝐺)𝑃 ↔ (𝑓 ∈ Word dom (iEdg‘𝐺) ∧ 𝑃:(0...(♯‘𝑓))⟶(Vtx‘𝐺) ∧ ∀𝑖 ∈ (0..^(♯‘𝑓))((iEdg‘𝐺)‘(𝑓‘𝑖)) = {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))})))
47	46	exbidv 1951	. . . . . . 7 ⊢ ((((𝐺 ∈ V ∧ 𝑃 ∈ Word (Vtx‘𝐺)) ∧ (𝑃 ≠ ∅ ∧ 𝑃 ∈ Word (Vtx‘𝐺) ∧ ∀𝑖 ∈ (0..^((♯‘𝑃) − 1)){(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ∈ (Edg‘𝐺))) ∧ 𝐺 ∈ USPGraph) → (∃𝑓 𝑓(Walks‘𝐺)𝑃 ↔ ∃𝑓(𝑓 ∈ Word dom (iEdg‘𝐺) ∧ 𝑃:(0...(♯‘𝑓))⟶(Vtx‘𝐺) ∧ ∀𝑖 ∈ (0..^(♯‘𝑓))((iEdg‘𝐺)‘(𝑓‘𝑖)) = {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))})))
48	42, 47	mpbird 260	. . . . . 6 ⊢ ((((𝐺 ∈ V ∧ 𝑃 ∈ Word (Vtx‘𝐺)) ∧ (𝑃 ≠ ∅ ∧ 𝑃 ∈ Word (Vtx‘𝐺) ∧ ∀𝑖 ∈ (0..^((♯‘𝑃) − 1)){(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ∈ (Edg‘𝐺))) ∧ 𝐺 ∈ USPGraph) → ∃𝑓 𝑓(Walks‘𝐺)𝑃)
49	48	ex 417	. . . . 5 ⊢ (((𝐺 ∈ V ∧ 𝑃 ∈ Word (Vtx‘𝐺)) ∧ (𝑃 ≠ ∅ ∧ 𝑃 ∈ Word (Vtx‘𝐺) ∧ ∀𝑖 ∈ (0..^((♯‘𝑃) − 1)){(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ∈ (Edg‘𝐺))) → (𝐺 ∈ USPGraph → ∃𝑓 𝑓(Walks‘𝐺)𝑃))
50	49	ex 417	. . . 4 ⊢ ((𝐺 ∈ V ∧ 𝑃 ∈ Word (Vtx‘𝐺)) → ((𝑃 ≠ ∅ ∧ 𝑃 ∈ Word (Vtx‘𝐺) ∧ ∀𝑖 ∈ (0..^((♯‘𝑃) − 1)){(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ∈ (Edg‘𝐺)) → (𝐺 ∈ USPGraph → ∃𝑓 𝑓(Walks‘𝐺)𝑃)))
51	4, 50	biimtrid 245	. . 3 ⊢ ((𝐺 ∈ V ∧ 𝑃 ∈ Word (Vtx‘𝐺)) → (𝑃 ∈ (WWalks‘𝐺) → (𝐺 ∈ USPGraph → ∃𝑓 𝑓(Walks‘𝐺)𝑃)))
52	2, 51	mpcom 39	. 2 ⊢ (𝑃 ∈ (WWalks‘𝐺) → (𝐺 ∈ USPGraph → ∃𝑓 𝑓(Walks‘𝐺)𝑃))
53	52	com12 33	1 ⊢ (𝐺 ∈ USPGraph → (𝑃 ∈ (WWalks‘𝐺) → ∃𝑓 𝑓(Walks‘𝐺)𝑃))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   ∧ w3a 1103   = wceq 1570  ∃wex 1809   ∈ wcel 2143   ≠ wne 2958  ∀wral 3079  Vcvv 3455  ∅c0 4287  {cpr 4592   class class class wbr 5110   ↦ cmpt 5193  ^◡ccnv 5662  dom cdm 5663  ran crn 5664  ⟶wf 6534  ‘cfv 6538  (class class class)co 7412  0cc0 11101  1c1 11102   + caddc 11104   ≤ cle 11245   − cmin 11442  ...cfz 13536  ..^cfzo 13684  ♯chash 14368  Word cword 14552  Vtxcvtx 29327  iEdgciedg 29328  Edgcedg 29378  UPGraphcupgr 29411  USPGraphcuspgr 29479  Walkscwlks 29927  WWalkscwwlks 30155

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5239  ax-sep 5258  ax-nul 5270  ax-pow 5338  ax-pr 5406  ax-un 7734  ax-cnex 11157  ax-resscn 11158  ax-1cn 11159  ax-icn 11160  ax-addcl 11161  ax-addrcl 11162  ax-mulcl 11163  ax-mulrcl 11164  ax-mulcom 11165  ax-addass 11166  ax-mulass 11167  ax-distr 11168  ax-i2m1 11169  ax-1ne0 11170  ax-1rid 11171  ax-rnegex 11172  ax-rrecex 11173  ax-cnre 11174  ax-pre-lttri 11175  ax-pre-lttrn 11176  ax-pre-ltadd 11177  ax-pre-mulgt0 11178

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-ifp 1079  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-nel 3065  df-ral 3080  df-rex 3090  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3746  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4288  df-if 4489  df-pw 4565  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4874  df-int 4914  df-iun 4959  df-br 5111  df-opab 5175  df-mpt 5194  df-tr 5220  df-id 5558  df-eprel 5563  df-po 5571  df-so 5572  df-fr 5616  df-we 5618  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-pred 6304  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-riota 7369  df-ov 7415  df-oprab 7416  df-mpo 7417  df-om 7864  df-1st 7987  df-2nd 7988  df-frecs 8279  df-wrecs 8310  df-recs 8359  df-rdg 8398  df-1o 8454  df-2o 8455  df-oadd 8458  df-er 8695  df-map 8827  df-pm 8828  df-en 8945  df-dom 8946  df-sdom 8947  df-fin 8948  df-dju 9888  df-card 9926  df-pnf 11246  df-mnf 11247  df-xr 11248  df-ltxr 11249  df-le 11250  df-sub 11444  df-neg 11445  df-nn 12235  df-2 12304  df-n0 12506  df-xnn0 12579  df-z 12593  df-uz 12864  df-fz 13537  df-fzo 13685  df-hash 14369  df-word 14553  df-edg 29379  df-uhgr 29389  df-upgr 29413  df-uspgr 29481  df-wlks 29930  df-wwlks 30160

This theorem is referenced by:  wlkiswwlks  30206  wlklnwwlkln2  30213