Theorem wlkiswwlks2 29943

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 2736	. . . 4 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
2	1	wwlkbp 29909	. . 3 ⊢ (𝑃 ∈ (WWalks‘𝐺) → (𝐺 ∈ V ∧ 𝑃 ∈ Word (Vtx‘𝐺)))
3		eqid 2736	. . . . 5 ⊢ (Edg‘𝐺) = (Edg‘𝐺)
4	1, 3	iswwlks 29904	. . . 4 ⊢ (𝑃 ∈ (WWalks‘𝐺) ↔ (𝑃 ≠ ∅ ∧ 𝑃 ∈ Word (Vtx‘𝐺) ∧ ∀𝑖 ∈ (0..^((♯‘𝑃) − 1)){(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ∈ (Edg‘𝐺)))
5		ovex 7400	. . . . . . . . . . . . . . 15 ⊢ (0..^((♯‘𝑃) − 1)) ∈ V
6		mptexg 7176	. . . . . . . . . . . . . . 15 ⊢ ((0..^((♯‘𝑃) − 1)) ∈ V → (𝑥 ∈ (0..^((♯‘𝑃) − 1)) ↦ (◡(iEdg‘𝐺)‘{(𝑃‘𝑥), (𝑃‘(𝑥 + 1))})) ∈ V)
7	5, 6	mp1i 13	. . . . . . . . . . . . . 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 773	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑃 ≠ ∅ ∧ 𝑃 ∈ Word (Vtx‘𝐺)) ∧ ((𝐺 ∈ V ∧ 𝑃 ∈ Word (Vtx‘𝐺)) ∧ 𝐺 ∈ USPGraph)) → 𝐺 ∈ USPGraph)
9		simplr 769	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑃 ≠ ∅ ∧ 𝑃 ∈ Word (Vtx‘𝐺)) ∧ ((𝐺 ∈ V ∧ 𝑃 ∈ Word (Vtx‘𝐺)) ∧ 𝐺 ∈ USPGraph)) → 𝑃 ∈ Word (Vtx‘𝐺))
10		hashge1 14351	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑃 ∈ Word (Vtx‘𝐺) ∧ 𝑃 ≠ ∅) → 1 ≤ (♯‘𝑃))
11	10	ancoms 458	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑃 ≠ ∅ ∧ 𝑃 ∈ Word (Vtx‘𝐺)) → 1 ≤ (♯‘𝑃))
12	11	adantr 480	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑃 ≠ ∅ ∧ 𝑃 ∈ Word (Vtx‘𝐺)) ∧ ((𝐺 ∈ V ∧ 𝑃 ∈ Word (Vtx‘𝐺)) ∧ 𝐺 ∈ USPGraph)) → 1 ≤ (♯‘𝑃))
13	8, 9, 12	3jca 1129	. . . . . . . . . . . . . . . . 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 480	. . . . . . . . . . . . . . . 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 29118	. . . . . . . . . . . . . . . . . . . 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 2822	. . . . . . . . . . . . . . . . . 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 3160	. . . . . . . . . . . . . . . . 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 229	. . . . . . . . . . . . . . . 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 2736	. . . . . . . . . . . . . . . . 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 2736	. . . . . . . . . . . . . . . . 17 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺)
22	20, 21	wlkiswwlks2lem6 29942	. . . . . . . . . . . . . . . 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 61	. . . . . . . . . . . . . . 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 2824	. . . . . . . . . . . . . . . . . 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 6840	. . . . . . . . . . . . . . . . . . . 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 7383	. . . . . . . . . . . . . . . . . . 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 6652	. . . . . . . . . . . . . . . . . 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 7383	. . . . . . . . . . . . . . . . . . 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 6839	. . . . . . . . . . . . . . . . . . . 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 6848	. . . . . . . . . . . . . . . . . . 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 3311	. . . . . . . . . . . . . . . . . 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 1439	. . . . . . . . . . . . . . . . 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 340	. . . . . . . . . . . . . . . 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 481	. . . . . . . . . . . . . . 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 257	. . . . . . . . . . . . . 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 3537	. . . . . . . . . . . . 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 412	. . . . . . . . . . . 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 86	. . . . . . . . . . 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 1118	. . . . . . . . . 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 415	. . . . . . . . 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 407	. . . . . . . 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 406	. . . . . . 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 29247	. . . . . . . . . 10 ⊢ (𝐺 ∈ USPGraph → 𝐺 ∈ UPGraph)
44	1, 21	upgriswlk 29709	. . . . . . . . . 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 17	. . . . . . . . 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 481	. . . . . . . 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 1923	. . . . . . 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 257	. . . . . 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 412	. . . . 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 412	. . . 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 242	. . 3 ⊢ ((𝐺 ∈ V ∧ 𝑃 ∈ Word (Vtx‘𝐺)) → (𝑃 ∈ (WWalks‘𝐺) → (𝐺 ∈ USPGraph → ∃𝑓 𝑓(Walks‘𝐺)𝑃)))
52	2, 51	mpcom 38	. 2 ⊢ (𝑃 ∈ (WWalks‘𝐺) → (𝐺 ∈ USPGraph → ∃𝑓 𝑓(Walks‘𝐺)𝑃))
53	52	com12 32	1 ⊢ (𝐺 ∈ USPGraph → (𝑃 ∈ (WWalks‘𝐺) → ∃𝑓 𝑓(Walks‘𝐺)𝑃))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1542  ∃wex 1781   ∈ wcel 2114   ≠ wne 2932  ∀wral 3051  Vcvv 3429  ∅c0 4273  {cpr 4569   class class class wbr 5085   ↦ cmpt 5166  ^◡ccnv 5630  dom cdm 5631  ran crn 5632  ⟶wf 6494  ‘cfv 6498  (class class class)co 7367  0cc0 11038  1c1 11039   + caddc 11041   ≤ cle 11180   − cmin 11377  ...cfz 13461  ..^cfzo 13608  ♯chash 14292  Word cword 14475  Vtxcvtx 29065  iEdgciedg 29066  Edgcedg 29116  UPGraphcupgr 29149  USPGraphcuspgr 29217  Walkscwlks 29665  WWalkscwwlks 29893

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 2708  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375  ax-un 7689  ax-cnex 11094  ax-resscn 11095  ax-1cn 11096  ax-icn 11097  ax-addcl 11098  ax-addrcl 11099  ax-mulcl 11100  ax-mulrcl 11101  ax-mulcom 11102  ax-addass 11103  ax-mulass 11104  ax-distr 11105  ax-i2m1 11106  ax-1ne0 11107  ax-1rid 11108  ax-rnegex 11109  ax-rrecex 11110  ax-cnre 11111  ax-pre-lttri 11112  ax-pre-lttrn 11113  ax-pre-ltadd 11114  ax-pre-mulgt0 11115

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 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3062  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-pss 3909  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-int 4890  df-iun 4935  df-br 5086  df-opab 5148  df-mpt 5167  df-tr 5193  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6265  df-ord 6326  df-on 6327  df-lim 6328  df-suc 6329  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506  df-riota 7324  df-ov 7370  df-oprab 7371  df-mpo 7372  df-om 7818  df-1st 7942  df-2nd 7943  df-frecs 8231  df-wrecs 8262  df-recs 8311  df-rdg 8349  df-1o 8405  df-2o 8406  df-oadd 8409  df-er 8643  df-map 8775  df-pm 8776  df-en 8894  df-dom 8895  df-sdom 8896  df-fin 8897  df-dju 9825  df-card 9863  df-pnf 11181  df-mnf 11182  df-xr 11183  df-ltxr 11184  df-le 11185  df-sub 11379  df-neg 11380  df-nn 12175  df-2 12244  df-n0 12438  df-xnn0 12511  df-z 12525  df-uz 12789  df-fz 13462  df-fzo 13609  df-hash 14293  df-word 14476  df-edg 29117  df-uhgr 29127  df-upgr 29151  df-uspgr 29219  df-wlks 29668  df-wwlks 29898

This theorem is referenced by:  wlkiswwlks  29944  wlklnwwlkln2  29951