Theorem wlkiswwlks2 29961

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 2739	. . . 4 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
2	1	wwlkbp 29927	. . 3 ⊢ (𝑃 ∈ (WWalks‘𝐺) → (𝐺 ∈ V ∧ 𝑃 ∈ Word (Vtx‘𝐺)))
3		eqid 2739	. . . . 5 ⊢ (Edg‘𝐺) = (Edg‘𝐺)
4	1, 3	iswwlks 29922	. . . 4 ⊢ (𝑃 ∈ (WWalks‘𝐺) ↔ (𝑃 ≠ ∅ ∧ 𝑃 ∈ Word (Vtx‘𝐺) ∧ ∀𝑖 ∈ (0..^((♯‘𝑃) − 1)){(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ∈ (Edg‘𝐺)))
5		ovex 7389	. . . . . . . . . . . . . . 15 ⊢ (0..^((♯‘𝑃) − 1)) ∈ V
6		mptexg 7165	. . . . . . . . . . . . . . 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 778	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑃 ≠ ∅ ∧ 𝑃 ∈ Word (Vtx‘𝐺)) ∧ ((𝐺 ∈ V ∧ 𝑃 ∈ Word (Vtx‘𝐺)) ∧ 𝐺 ∈ USPGraph)) → 𝐺 ∈ USPGraph)
9		simplr 774	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑃 ≠ ∅ ∧ 𝑃 ∈ Word (Vtx‘𝐺)) ∧ ((𝐺 ∈ V ∧ 𝑃 ∈ Word (Vtx‘𝐺)) ∧ 𝐺 ∈ USPGraph)) → 𝑃 ∈ Word (Vtx‘𝐺))
10		hashge1 14342	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑃 ∈ Word (Vtx‘𝐺) ∧ 𝑃 ≠ ∅) → 1 ≤ (♯‘𝑃))
11	10	ancoms 459	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑃 ≠ ∅ ∧ 𝑃 ∈ Word (Vtx‘𝐺)) → 1 ≤ (♯‘𝑃))
12	11	adantr 481	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑃 ≠ ∅ ∧ 𝑃 ∈ Word (Vtx‘𝐺)) ∧ ((𝐺 ∈ V ∧ 𝑃 ∈ Word (Vtx‘𝐺)) ∧ 𝐺 ∈ USPGraph)) → 1 ≤ (♯‘𝑃))
13	8, 9, 12	3jca 1134	. . . . . . . . . . . . . . . . 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 481	. . . . . . . . . . . . . . . 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 29136	. . . . . . . . . . . . . . . . . . . 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 2825	. . . . . . . . . . . . . . . . . 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 3162	. . . . . . . . . . . . . . . . 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 230	. . . . . . . . . . . . . . . 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 2739	. . . . . . . . . . . . . . . . 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 2739	. . . . . . . . . . . . . . . . 17 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺)
22	20, 21	wlkiswwlks2lem6 29960	. . . . . . . . . . . . . . . 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 2827	. . . . . . . . . . . . . . . . . 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 6827	. . . . . . . . . . . . . . . . . . . 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 7372	. . . . . . . . . . . . . . . . . . 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 6639	. . . . . . . . . . . . . . . . . 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 7372	. . . . . . . . . . . . . . . . . . 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 6826	. . . . . . . . . . . . . . . . . . . 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 6835	. . . . . . . . . . . . . . . . . . 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 3313	. . . . . . . . . . . . . . . . . 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 1444	. . . . . . . . . . . . . . . . 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 341	. . . . . . . . . . . . . . . 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 482	. . . . . . . . . . . . . . 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 258	. . . . . . . . . . . . . 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 3533	. . . . . . . . . . . . 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 413	. . . . . . . . . . . 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 1123	. . . . . . . . . 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 416	. . . . . . . . 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 408	. . . . . . . 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 407	. . . . . . 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 29265	. . . . . . . . . 10 ⊢ (𝐺 ∈ USPGraph → 𝐺 ∈ UPGraph)
44	1, 21	upgriswlk 29727	. . . . . . . . . 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 482	. . . . . . . 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 1928	. . . . . . 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 258	. . . . . 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 413	. . . . 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 413	. . . 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 243	. . 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 207   ∧ wa 396   ∧ w3a 1092   = wceq 1547  ∃wex 1786   ∈ wcel 2119   ≠ wne 2934  ∀wral 3053  Vcvv 3431  ∅c0 4261  {cpr 4557   class class class wbr 5072   ↦ cmpt 5153  ^◡ccnv 5617  dom cdm 5618  ran crn 5619  ⟶wf 6481  ‘cfv 6485  (class class class)co 7356  0cc0 11029  1c1 11030   + caddc 11032   ≤ cle 11171   − cmin 11368  ...cfz 13452  ..^cfzo 13599  ♯chash 14283  Word cword 14466  Vtxcvtx 29083  iEdgciedg 29084  Edgcedg 29134  UPGraphcupgr 29167  USPGraphcuspgr 29235  Walkscwlks 29683  WWalkscwwlks 29911

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-rep 5199  ax-sep 5218  ax-nul 5228  ax-pow 5294  ax-pr 5362  ax-un 7678  ax-cnex 11085  ax-resscn 11086  ax-1cn 11087  ax-icn 11088  ax-addcl 11089  ax-addrcl 11090  ax-mulcl 11091  ax-mulrcl 11092  ax-mulcom 11093  ax-addass 11094  ax-mulass 11095  ax-distr 11096  ax-i2m1 11097  ax-1ne0 11098  ax-1rid 11099  ax-rnegex 11100  ax-rrecex 11101  ax-cnre 11102  ax-pre-lttri 11103  ax-pre-lttrn 11104  ax-pre-ltadd 11105  ax-pre-mulgt0 11106

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-ifp 1069  df-3or 1093  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-nel 3039  df-ral 3054  df-rex 3064  df-reu 3345  df-rab 3392  df-v 3433  df-sbc 3724  df-csb 3832  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3903  df-nul 4262  df-if 4455  df-pw 4531  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-int 4878  df-iun 4923  df-br 5073  df-opab 5135  df-mpt 5154  df-tr 5180  df-id 5513  df-eprel 5518  df-po 5526  df-so 5527  df-fr 5571  df-we 5573  df-xp 5624  df-rel 5625  df-cnv 5626  df-co 5627  df-dm 5628  df-rn 5629  df-res 5630  df-ima 5631  df-pred 6252  df-ord 6313  df-on 6314  df-lim 6315  df-suc 6316  df-iota 6441  df-fun 6487  df-fn 6488  df-f 6489  df-f1 6490  df-fo 6491  df-f1o 6492  df-fv 6493  df-riota 7313  df-ov 7359  df-oprab 7360  df-mpo 7361  df-om 7807  df-1st 7931  df-2nd 7932  df-frecs 8221  df-wrecs 8252  df-recs 8301  df-rdg 8339  df-1o 8395  df-2o 8396  df-oadd 8399  df-er 8633  df-map 8765  df-pm 8766  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-dju 9816  df-card 9854  df-pnf 11172  df-mnf 11173  df-xr 11174  df-ltxr 11175  df-le 11176  df-sub 11370  df-neg 11371  df-nn 12166  df-2 12235  df-n0 12429  df-xnn0 12502  df-z 12516  df-uz 12780  df-fz 13453  df-fzo 13600  df-hash 14284  df-word 14467  df-edg 29135  df-uhgr 29145  df-upgr 29169  df-uspgr 29237  df-wlks 29686  df-wwlks 29916

This theorem is referenced by:  wlkiswwlks  29962  wlklnwwlkln2  29969