Theorem wlkiswwlk2 26225

Description: A walk as word corresponds to the sequence of vertices in a walk in an undirected simple graph. (Contributed by Alexander van der Vekens, 21-Jul-2018.)

Assertion

Ref	Expression
wlkiswwlk2	⊢ (𝑉 USGrph 𝐸 → (𝑃 ∈ (𝑉 WWalks 𝐸) → ∃𝑓 𝑓(𝑉 Walks 𝐸)𝑃))

Distinct variable groups:   𝑓,𝐸   𝑃,𝑓   𝑓,𝑉

Proof of Theorem wlkiswwlk2

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

Step	Hyp	Ref	Expression
1		wwlkprop 26213	. . 3 ⊢ (𝑃 ∈ (𝑉 WWalks 𝐸) → (𝑉 ∈ V ∧ 𝐸 ∈ V ∧ 𝑃 ∈ Word 𝑉))
2		iswwlk 26211	. . . . 5 ⊢ ((𝑉 ∈ V ∧ 𝐸 ∈ V) → (𝑃 ∈ (𝑉 WWalks 𝐸) ↔ (𝑃 ≠ ∅ ∧ 𝑃 ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((#‘𝑃) − 1)){(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ∈ ran 𝐸)))
3	2	3adant3 1074	. . . 4 ⊢ ((𝑉 ∈ V ∧ 𝐸 ∈ V ∧ 𝑃 ∈ Word 𝑉) → (𝑃 ∈ (𝑉 WWalks 𝐸) ↔ (𝑃 ≠ ∅ ∧ 𝑃 ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((#‘𝑃) − 1)){(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ∈ ran 𝐸)))
4		ovex 6577	. . . . . . . . . . . . . . 15 ⊢ (0..^((#‘𝑃) − 1)) ∈ V
5		mptexg 6389	. . . . . . . . . . . . . . 15 ⊢ ((0..^((#‘𝑃) − 1)) ∈ V → (𝑥 ∈ (0..^((#‘𝑃) − 1)) ↦ (◡𝐸‘{(𝑃‘𝑥), (𝑃‘(𝑥 + 1))})) ∈ V)
6	4, 5	mp1i 13	. . . . . . . . . . . . . 14 ⊢ (((𝑃 ≠ ∅ ∧ 𝑃 ∈ Word 𝑉) ∧ ((𝑉 ∈ V ∧ 𝐸 ∈ V ∧ 𝑃 ∈ Word 𝑉) ∧ 𝑉 USGrph 𝐸)) → (𝑥 ∈ (0..^((#‘𝑃) − 1)) ↦ (◡𝐸‘{(𝑃‘𝑥), (𝑃‘(𝑥 + 1))})) ∈ V)
7		simpr 476	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑉 ∈ V ∧ 𝐸 ∈ V ∧ 𝑃 ∈ Word 𝑉) ∧ 𝑉 USGrph 𝐸) → 𝑉 USGrph 𝐸)
8	7	adantl 481	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑃 ≠ ∅ ∧ 𝑃 ∈ Word 𝑉) ∧ ((𝑉 ∈ V ∧ 𝐸 ∈ V ∧ 𝑃 ∈ Word 𝑉) ∧ 𝑉 USGrph 𝐸)) → 𝑉 USGrph 𝐸)
9		simpr 476	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑃 ≠ ∅ ∧ 𝑃 ∈ Word 𝑉) → 𝑃 ∈ Word 𝑉)
10	9	adantr 480	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑃 ≠ ∅ ∧ 𝑃 ∈ Word 𝑉) ∧ ((𝑉 ∈ V ∧ 𝐸 ∈ V ∧ 𝑃 ∈ Word 𝑉) ∧ 𝑉 USGrph 𝐸)) → 𝑃 ∈ Word 𝑉)
11		hashge1 13039	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑃 ∈ Word 𝑉 ∧ 𝑃 ≠ ∅) → 1 ≤ (#‘𝑃))
12	11	ancoms 468	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑃 ≠ ∅ ∧ 𝑃 ∈ Word 𝑉) → 1 ≤ (#‘𝑃))
13	12	adantr 480	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑃 ≠ ∅ ∧ 𝑃 ∈ Word 𝑉) ∧ ((𝑉 ∈ V ∧ 𝐸 ∈ V ∧ 𝑃 ∈ Word 𝑉) ∧ 𝑉 USGrph 𝐸)) → 1 ≤ (#‘𝑃))
14	8, 10, 13	3jca 1235	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑃 ≠ ∅ ∧ 𝑃 ∈ Word 𝑉) ∧ ((𝑉 ∈ V ∧ 𝐸 ∈ V ∧ 𝑃 ∈ Word 𝑉) ∧ 𝑉 USGrph 𝐸)) → (𝑉 USGrph 𝐸 ∧ 𝑃 ∈ Word 𝑉 ∧ 1 ≤ (#‘𝑃)))
15	14	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑃 ≠ ∅ ∧ 𝑃 ∈ Word 𝑉) ∧ ((𝑉 ∈ V ∧ 𝐸 ∈ V ∧ 𝑃 ∈ Word 𝑉) ∧ 𝑉 USGrph 𝐸)) ∧ 𝑓 = (𝑥 ∈ (0..^((#‘𝑃) − 1)) ↦ (◡𝐸‘{(𝑃‘𝑥), (𝑃‘(𝑥 + 1))}))) → (𝑉 USGrph 𝐸 ∧ 𝑃 ∈ Word 𝑉 ∧ 1 ≤ (#‘𝑃)))
16		eqid 2610	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ (0..^((#‘𝑃) − 1)) ↦ (◡𝐸‘{(𝑃‘𝑥), (𝑃‘(𝑥 + 1))})) = (𝑥 ∈ (0..^((#‘𝑃) − 1)) ↦ (◡𝐸‘{(𝑃‘𝑥), (𝑃‘(𝑥 + 1))}))
17	16	wlkiswwlk2lem6 26224	. . . . . . . . . . . . . . . 16 ⊢ ((𝑉 USGrph 𝐸 ∧ 𝑃 ∈ Word 𝑉 ∧ 1 ≤ (#‘𝑃)) → (∀𝑖 ∈ (0..^((#‘𝑃) − 1)){(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ∈ ran 𝐸 → ((𝑥 ∈ (0..^((#‘𝑃) − 1)) ↦ (◡𝐸‘{(𝑃‘𝑥), (𝑃‘(𝑥 + 1))})) ∈ Word dom 𝐸 ∧ 𝑃:(0...(#‘(𝑥 ∈ (0..^((#‘𝑃) − 1)) ↦ (◡𝐸‘{(𝑃‘𝑥), (𝑃‘(𝑥 + 1))}))))⟶𝑉 ∧ ∀𝑖 ∈ (0..^(#‘(𝑥 ∈ (0..^((#‘𝑃) − 1)) ↦ (◡𝐸‘{(𝑃‘𝑥), (𝑃‘(𝑥 + 1))}))))(𝐸‘((𝑥 ∈ (0..^((#‘𝑃) − 1)) ↦ (◡𝐸‘{(𝑃‘𝑥), (𝑃‘(𝑥 + 1))}))‘𝑖)) = {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))})))
18	15, 17	syl 17	. . . . . . . . . . . . . . 15 ⊢ ((((𝑃 ≠ ∅ ∧ 𝑃 ∈ Word 𝑉) ∧ ((𝑉 ∈ V ∧ 𝐸 ∈ V ∧ 𝑃 ∈ Word 𝑉) ∧ 𝑉 USGrph 𝐸)) ∧ 𝑓 = (𝑥 ∈ (0..^((#‘𝑃) − 1)) ↦ (◡𝐸‘{(𝑃‘𝑥), (𝑃‘(𝑥 + 1))}))) → (∀𝑖 ∈ (0..^((#‘𝑃) − 1)){(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ∈ ran 𝐸 → ((𝑥 ∈ (0..^((#‘𝑃) − 1)) ↦ (◡𝐸‘{(𝑃‘𝑥), (𝑃‘(𝑥 + 1))})) ∈ Word dom 𝐸 ∧ 𝑃:(0...(#‘(𝑥 ∈ (0..^((#‘𝑃) − 1)) ↦ (◡𝐸‘{(𝑃‘𝑥), (𝑃‘(𝑥 + 1))}))))⟶𝑉 ∧ ∀𝑖 ∈ (0..^(#‘(𝑥 ∈ (0..^((#‘𝑃) − 1)) ↦ (◡𝐸‘{(𝑃‘𝑥), (𝑃‘(𝑥 + 1))}))))(𝐸‘((𝑥 ∈ (0..^((#‘𝑃) − 1)) ↦ (◡𝐸‘{(𝑃‘𝑥), (𝑃‘(𝑥 + 1))}))‘𝑖)) = {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))})))
19		eleq1 2676	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑓 = (𝑥 ∈ (0..^((#‘𝑃) − 1)) ↦ (◡𝐸‘{(𝑃‘𝑥), (𝑃‘(𝑥 + 1))})) → (𝑓 ∈ Word dom 𝐸 ↔ (𝑥 ∈ (0..^((#‘𝑃) − 1)) ↦ (◡𝐸‘{(𝑃‘𝑥), (𝑃‘(𝑥 + 1))})) ∈ Word dom 𝐸))
20		fveq2 6103	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑓 = (𝑥 ∈ (0..^((#‘𝑃) − 1)) ↦ (◡𝐸‘{(𝑃‘𝑥), (𝑃‘(𝑥 + 1))})) → (#‘𝑓) = (#‘(𝑥 ∈ (0..^((#‘𝑃) − 1)) ↦ (◡𝐸‘{(𝑃‘𝑥), (𝑃‘(𝑥 + 1))}))))
21	20	oveq2d 6565	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑓 = (𝑥 ∈ (0..^((#‘𝑃) − 1)) ↦ (◡𝐸‘{(𝑃‘𝑥), (𝑃‘(𝑥 + 1))})) → (0...(#‘𝑓)) = (0...(#‘(𝑥 ∈ (0..^((#‘𝑃) − 1)) ↦ (◡𝐸‘{(𝑃‘𝑥), (𝑃‘(𝑥 + 1))})))))
22	21	feq2d 5944	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑓 = (𝑥 ∈ (0..^((#‘𝑃) − 1)) ↦ (◡𝐸‘{(𝑃‘𝑥), (𝑃‘(𝑥 + 1))})) → (𝑃:(0...(#‘𝑓))⟶𝑉 ↔ 𝑃:(0...(#‘(𝑥 ∈ (0..^((#‘𝑃) − 1)) ↦ (◡𝐸‘{(𝑃‘𝑥), (𝑃‘(𝑥 + 1))}))))⟶𝑉))
23	20	oveq2d 6565	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑓 = (𝑥 ∈ (0..^((#‘𝑃) − 1)) ↦ (◡𝐸‘{(𝑃‘𝑥), (𝑃‘(𝑥 + 1))})) → (0..^(#‘𝑓)) = (0..^(#‘(𝑥 ∈ (0..^((#‘𝑃) − 1)) ↦ (◡𝐸‘{(𝑃‘𝑥), (𝑃‘(𝑥 + 1))})))))
24		fveq1 6102	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑓 = (𝑥 ∈ (0..^((#‘𝑃) − 1)) ↦ (◡𝐸‘{(𝑃‘𝑥), (𝑃‘(𝑥 + 1))})) → (𝑓‘𝑖) = ((𝑥 ∈ (0..^((#‘𝑃) − 1)) ↦ (◡𝐸‘{(𝑃‘𝑥), (𝑃‘(𝑥 + 1))}))‘𝑖))
25	24	fveq2d 6107	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑓 = (𝑥 ∈ (0..^((#‘𝑃) − 1)) ↦ (◡𝐸‘{(𝑃‘𝑥), (𝑃‘(𝑥 + 1))})) → (𝐸‘(𝑓‘𝑖)) = (𝐸‘((𝑥 ∈ (0..^((#‘𝑃) − 1)) ↦ (◡𝐸‘{(𝑃‘𝑥), (𝑃‘(𝑥 + 1))}))‘𝑖)))
26	25	eqeq1d 2612	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑓 = (𝑥 ∈ (0..^((#‘𝑃) − 1)) ↦ (◡𝐸‘{(𝑃‘𝑥), (𝑃‘(𝑥 + 1))})) → ((𝐸‘(𝑓‘𝑖)) = {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ↔ (𝐸‘((𝑥 ∈ (0..^((#‘𝑃) − 1)) ↦ (◡𝐸‘{(𝑃‘𝑥), (𝑃‘(𝑥 + 1))}))‘𝑖)) = {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))}))
27	23, 26	raleqbidv 3129	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑓 = (𝑥 ∈ (0..^((#‘𝑃) − 1)) ↦ (◡𝐸‘{(𝑃‘𝑥), (𝑃‘(𝑥 + 1))})) → (∀𝑖 ∈ (0..^(#‘𝑓))(𝐸‘(𝑓‘𝑖)) = {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ↔ ∀𝑖 ∈ (0..^(#‘(𝑥 ∈ (0..^((#‘𝑃) − 1)) ↦ (◡𝐸‘{(𝑃‘𝑥), (𝑃‘(𝑥 + 1))}))))(𝐸‘((𝑥 ∈ (0..^((#‘𝑃) − 1)) ↦ (◡𝐸‘{(𝑃‘𝑥), (𝑃‘(𝑥 + 1))}))‘𝑖)) = {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))}))
28	19, 22, 27	3anbi123d 1391	. . . . . . . . . . . . . . . . 17 ⊢ (𝑓 = (𝑥 ∈ (0..^((#‘𝑃) − 1)) ↦ (◡𝐸‘{(𝑃‘𝑥), (𝑃‘(𝑥 + 1))})) → ((𝑓 ∈ Word dom 𝐸 ∧ 𝑃:(0...(#‘𝑓))⟶𝑉 ∧ ∀𝑖 ∈ (0..^(#‘𝑓))(𝐸‘(𝑓‘𝑖)) = {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))}) ↔ ((𝑥 ∈ (0..^((#‘𝑃) − 1)) ↦ (◡𝐸‘{(𝑃‘𝑥), (𝑃‘(𝑥 + 1))})) ∈ Word dom 𝐸 ∧ 𝑃:(0...(#‘(𝑥 ∈ (0..^((#‘𝑃) − 1)) ↦ (◡𝐸‘{(𝑃‘𝑥), (𝑃‘(𝑥 + 1))}))))⟶𝑉 ∧ ∀𝑖 ∈ (0..^(#‘(𝑥 ∈ (0..^((#‘𝑃) − 1)) ↦ (◡𝐸‘{(𝑃‘𝑥), (𝑃‘(𝑥 + 1))}))))(𝐸‘((𝑥 ∈ (0..^((#‘𝑃) − 1)) ↦ (◡𝐸‘{(𝑃‘𝑥), (𝑃‘(𝑥 + 1))}))‘𝑖)) = {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))})))
29	28	imbi2d 329	. . . . . . . . . . . . . . . 16 ⊢ (𝑓 = (𝑥 ∈ (0..^((#‘𝑃) − 1)) ↦ (◡𝐸‘{(𝑃‘𝑥), (𝑃‘(𝑥 + 1))})) → ((∀𝑖 ∈ (0..^((#‘𝑃) − 1)){(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ∈ ran 𝐸 → (𝑓 ∈ Word dom 𝐸 ∧ 𝑃:(0...(#‘𝑓))⟶𝑉 ∧ ∀𝑖 ∈ (0..^(#‘𝑓))(𝐸‘(𝑓‘𝑖)) = {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))})) ↔ (∀𝑖 ∈ (0..^((#‘𝑃) − 1)){(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ∈ ran 𝐸 → ((𝑥 ∈ (0..^((#‘𝑃) − 1)) ↦ (◡𝐸‘{(𝑃‘𝑥), (𝑃‘(𝑥 + 1))})) ∈ Word dom 𝐸 ∧ 𝑃:(0...(#‘(𝑥 ∈ (0..^((#‘𝑃) − 1)) ↦ (◡𝐸‘{(𝑃‘𝑥), (𝑃‘(𝑥 + 1))}))))⟶𝑉 ∧ ∀𝑖 ∈ (0..^(#‘(𝑥 ∈ (0..^((#‘𝑃) − 1)) ↦ (◡𝐸‘{(𝑃‘𝑥), (𝑃‘(𝑥 + 1))}))))(𝐸‘((𝑥 ∈ (0..^((#‘𝑃) − 1)) ↦ (◡𝐸‘{(𝑃‘𝑥), (𝑃‘(𝑥 + 1))}))‘𝑖)) = {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))}))))
30	29	adantl 481	. . . . . . . . . . . . . . 15 ⊢ ((((𝑃 ≠ ∅ ∧ 𝑃 ∈ Word 𝑉) ∧ ((𝑉 ∈ V ∧ 𝐸 ∈ V ∧ 𝑃 ∈ Word 𝑉) ∧ 𝑉 USGrph 𝐸)) ∧ 𝑓 = (𝑥 ∈ (0..^((#‘𝑃) − 1)) ↦ (◡𝐸‘{(𝑃‘𝑥), (𝑃‘(𝑥 + 1))}))) → ((∀𝑖 ∈ (0..^((#‘𝑃) − 1)){(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ∈ ran 𝐸 → (𝑓 ∈ Word dom 𝐸 ∧ 𝑃:(0...(#‘𝑓))⟶𝑉 ∧ ∀𝑖 ∈ (0..^(#‘𝑓))(𝐸‘(𝑓‘𝑖)) = {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))})) ↔ (∀𝑖 ∈ (0..^((#‘𝑃) − 1)){(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ∈ ran 𝐸 → ((𝑥 ∈ (0..^((#‘𝑃) − 1)) ↦ (◡𝐸‘{(𝑃‘𝑥), (𝑃‘(𝑥 + 1))})) ∈ Word dom 𝐸 ∧ 𝑃:(0...(#‘(𝑥 ∈ (0..^((#‘𝑃) − 1)) ↦ (◡𝐸‘{(𝑃‘𝑥), (𝑃‘(𝑥 + 1))}))))⟶𝑉 ∧ ∀𝑖 ∈ (0..^(#‘(𝑥 ∈ (0..^((#‘𝑃) − 1)) ↦ (◡𝐸‘{(𝑃‘𝑥), (𝑃‘(𝑥 + 1))}))))(𝐸‘((𝑥 ∈ (0..^((#‘𝑃) − 1)) ↦ (◡𝐸‘{(𝑃‘𝑥), (𝑃‘(𝑥 + 1))}))‘𝑖)) = {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))}))))
31	18, 30	mpbird 246	. . . . . . . . . . . . . 14 ⊢ ((((𝑃 ≠ ∅ ∧ 𝑃 ∈ Word 𝑉) ∧ ((𝑉 ∈ V ∧ 𝐸 ∈ V ∧ 𝑃 ∈ Word 𝑉) ∧ 𝑉 USGrph 𝐸)) ∧ 𝑓 = (𝑥 ∈ (0..^((#‘𝑃) − 1)) ↦ (◡𝐸‘{(𝑃‘𝑥), (𝑃‘(𝑥 + 1))}))) → (∀𝑖 ∈ (0..^((#‘𝑃) − 1)){(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ∈ ran 𝐸 → (𝑓 ∈ Word dom 𝐸 ∧ 𝑃:(0...(#‘𝑓))⟶𝑉 ∧ ∀𝑖 ∈ (0..^(#‘𝑓))(𝐸‘(𝑓‘𝑖)) = {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))})))
32	6, 31	spcimedv 3265	. . . . . . . . . . . . 13 ⊢ (((𝑃 ≠ ∅ ∧ 𝑃 ∈ Word 𝑉) ∧ ((𝑉 ∈ V ∧ 𝐸 ∈ V ∧ 𝑃 ∈ Word 𝑉) ∧ 𝑉 USGrph 𝐸)) → (∀𝑖 ∈ (0..^((#‘𝑃) − 1)){(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ∈ ran 𝐸 → ∃𝑓(𝑓 ∈ Word dom 𝐸 ∧ 𝑃:(0...(#‘𝑓))⟶𝑉 ∧ ∀𝑖 ∈ (0..^(#‘𝑓))(𝐸‘(𝑓‘𝑖)) = {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))})))
33	32	ex 449	. . . . . . . . . . . 12 ⊢ ((𝑃 ≠ ∅ ∧ 𝑃 ∈ Word 𝑉) → (((𝑉 ∈ V ∧ 𝐸 ∈ V ∧ 𝑃 ∈ Word 𝑉) ∧ 𝑉 USGrph 𝐸) → (∀𝑖 ∈ (0..^((#‘𝑃) − 1)){(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ∈ ran 𝐸 → ∃𝑓(𝑓 ∈ Word dom 𝐸 ∧ 𝑃:(0...(#‘𝑓))⟶𝑉 ∧ ∀𝑖 ∈ (0..^(#‘𝑓))(𝐸‘(𝑓‘𝑖)) = {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))}))))
34	33	com23 84	. . . . . . . . . . 11 ⊢ ((𝑃 ≠ ∅ ∧ 𝑃 ∈ Word 𝑉) → (∀𝑖 ∈ (0..^((#‘𝑃) − 1)){(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ∈ ran 𝐸 → (((𝑉 ∈ V ∧ 𝐸 ∈ V ∧ 𝑃 ∈ Word 𝑉) ∧ 𝑉 USGrph 𝐸) → ∃𝑓(𝑓 ∈ Word dom 𝐸 ∧ 𝑃:(0...(#‘𝑓))⟶𝑉 ∧ ∀𝑖 ∈ (0..^(#‘𝑓))(𝐸‘(𝑓‘𝑖)) = {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))}))))
35	34	3impia 1253	. . . . . . . . . 10 ⊢ ((𝑃 ≠ ∅ ∧ 𝑃 ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((#‘𝑃) − 1)){(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ∈ ran 𝐸) → (((𝑉 ∈ V ∧ 𝐸 ∈ V ∧ 𝑃 ∈ Word 𝑉) ∧ 𝑉 USGrph 𝐸) → ∃𝑓(𝑓 ∈ Word dom 𝐸 ∧ 𝑃:(0...(#‘𝑓))⟶𝑉 ∧ ∀𝑖 ∈ (0..^(#‘𝑓))(𝐸‘(𝑓‘𝑖)) = {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))})))
36	35	expd 451	. . . . . . . . 9 ⊢ ((𝑃 ≠ ∅ ∧ 𝑃 ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((#‘𝑃) − 1)){(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ∈ ran 𝐸) → ((𝑉 ∈ V ∧ 𝐸 ∈ V ∧ 𝑃 ∈ Word 𝑉) → (𝑉 USGrph 𝐸 → ∃𝑓(𝑓 ∈ Word dom 𝐸 ∧ 𝑃:(0...(#‘𝑓))⟶𝑉 ∧ ∀𝑖 ∈ (0..^(#‘𝑓))(𝐸‘(𝑓‘𝑖)) = {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))}))))
37	36	impcom 445	. . . . . . . 8 ⊢ (((𝑉 ∈ V ∧ 𝐸 ∈ V ∧ 𝑃 ∈ Word 𝑉) ∧ (𝑃 ≠ ∅ ∧ 𝑃 ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((#‘𝑃) − 1)){(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ∈ ran 𝐸)) → (𝑉 USGrph 𝐸 → ∃𝑓(𝑓 ∈ Word dom 𝐸 ∧ 𝑃:(0...(#‘𝑓))⟶𝑉 ∧ ∀𝑖 ∈ (0..^(#‘𝑓))(𝐸‘(𝑓‘𝑖)) = {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))})))
38	37	imp 444	. . . . . . 7 ⊢ ((((𝑉 ∈ V ∧ 𝐸 ∈ V ∧ 𝑃 ∈ Word 𝑉) ∧ (𝑃 ≠ ∅ ∧ 𝑃 ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((#‘𝑃) − 1)){(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ∈ ran 𝐸)) ∧ 𝑉 USGrph 𝐸) → ∃𝑓(𝑓 ∈ Word dom 𝐸 ∧ 𝑃:(0...(#‘𝑓))⟶𝑉 ∧ ∀𝑖 ∈ (0..^(#‘𝑓))(𝐸‘(𝑓‘𝑖)) = {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))}))
39		3simpa 1051	. . . . . . . . . . . 12 ⊢ ((𝑉 ∈ V ∧ 𝐸 ∈ V ∧ 𝑃 ∈ Word 𝑉) → (𝑉 ∈ V ∧ 𝐸 ∈ V))
40		vex 3176	. . . . . . . . . . . . . 14 ⊢ 𝑓 ∈ V
41	40	jctl 562	. . . . . . . . . . . . 13 ⊢ (𝑃 ∈ Word 𝑉 → (𝑓 ∈ V ∧ 𝑃 ∈ Word 𝑉))
42	41	3ad2ant3 1077	. . . . . . . . . . . 12 ⊢ ((𝑉 ∈ V ∧ 𝐸 ∈ V ∧ 𝑃 ∈ Word 𝑉) → (𝑓 ∈ V ∧ 𝑃 ∈ Word 𝑉))
43	39, 42	jca 553	. . . . . . . . . . 11 ⊢ ((𝑉 ∈ V ∧ 𝐸 ∈ V ∧ 𝑃 ∈ Word 𝑉) → ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝑓 ∈ V ∧ 𝑃 ∈ Word 𝑉)))
44	43	adantr 480	. . . . . . . . . 10 ⊢ (((𝑉 ∈ V ∧ 𝐸 ∈ V ∧ 𝑃 ∈ Word 𝑉) ∧ (𝑃 ≠ ∅ ∧ 𝑃 ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((#‘𝑃) − 1)){(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ∈ ran 𝐸)) → ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝑓 ∈ V ∧ 𝑃 ∈ Word 𝑉)))
45	44	adantr 480	. . . . . . . . 9 ⊢ ((((𝑉 ∈ V ∧ 𝐸 ∈ V ∧ 𝑃 ∈ Word 𝑉) ∧ (𝑃 ≠ ∅ ∧ 𝑃 ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((#‘𝑃) − 1)){(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ∈ ran 𝐸)) ∧ 𝑉 USGrph 𝐸) → ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝑓 ∈ V ∧ 𝑃 ∈ Word 𝑉)))
46		iswlk 26048	. . . . . . . . 9 ⊢ (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝑓 ∈ V ∧ 𝑃 ∈ Word 𝑉)) → (𝑓(𝑉 Walks 𝐸)𝑃 ↔ (𝑓 ∈ Word dom 𝐸 ∧ 𝑃:(0...(#‘𝑓))⟶𝑉 ∧ ∀𝑖 ∈ (0..^(#‘𝑓))(𝐸‘(𝑓‘𝑖)) = {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))})))
47	45, 46	syl 17	. . . . . . . 8 ⊢ ((((𝑉 ∈ V ∧ 𝐸 ∈ V ∧ 𝑃 ∈ Word 𝑉) ∧ (𝑃 ≠ ∅ ∧ 𝑃 ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((#‘𝑃) − 1)){(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ∈ ran 𝐸)) ∧ 𝑉 USGrph 𝐸) → (𝑓(𝑉 Walks 𝐸)𝑃 ↔ (𝑓 ∈ Word dom 𝐸 ∧ 𝑃:(0...(#‘𝑓))⟶𝑉 ∧ ∀𝑖 ∈ (0..^(#‘𝑓))(𝐸‘(𝑓‘𝑖)) = {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))})))
48	47	exbidv 1837	. . . . . . 7 ⊢ ((((𝑉 ∈ V ∧ 𝐸 ∈ V ∧ 𝑃 ∈ Word 𝑉) ∧ (𝑃 ≠ ∅ ∧ 𝑃 ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((#‘𝑃) − 1)){(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ∈ ran 𝐸)) ∧ 𝑉 USGrph 𝐸) → (∃𝑓 𝑓(𝑉 Walks 𝐸)𝑃 ↔ ∃𝑓(𝑓 ∈ Word dom 𝐸 ∧ 𝑃:(0...(#‘𝑓))⟶𝑉 ∧ ∀𝑖 ∈ (0..^(#‘𝑓))(𝐸‘(𝑓‘𝑖)) = {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))})))
49	38, 48	mpbird 246	. . . . . 6 ⊢ ((((𝑉 ∈ V ∧ 𝐸 ∈ V ∧ 𝑃 ∈ Word 𝑉) ∧ (𝑃 ≠ ∅ ∧ 𝑃 ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((#‘𝑃) − 1)){(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ∈ ran 𝐸)) ∧ 𝑉 USGrph 𝐸) → ∃𝑓 𝑓(𝑉 Walks 𝐸)𝑃)
50	49	ex 449	. . . . 5 ⊢ (((𝑉 ∈ V ∧ 𝐸 ∈ V ∧ 𝑃 ∈ Word 𝑉) ∧ (𝑃 ≠ ∅ ∧ 𝑃 ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((#‘𝑃) − 1)){(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ∈ ran 𝐸)) → (𝑉 USGrph 𝐸 → ∃𝑓 𝑓(𝑉 Walks 𝐸)𝑃))
51	50	ex 449	. . . 4 ⊢ ((𝑉 ∈ V ∧ 𝐸 ∈ V ∧ 𝑃 ∈ Word 𝑉) → ((𝑃 ≠ ∅ ∧ 𝑃 ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((#‘𝑃) − 1)){(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ∈ ran 𝐸) → (𝑉 USGrph 𝐸 → ∃𝑓 𝑓(𝑉 Walks 𝐸)𝑃)))
52	3, 51	sylbid 229	. . 3 ⊢ ((𝑉 ∈ V ∧ 𝐸 ∈ V ∧ 𝑃 ∈ Word 𝑉) → (𝑃 ∈ (𝑉 WWalks 𝐸) → (𝑉 USGrph 𝐸 → ∃𝑓 𝑓(𝑉 Walks 𝐸)𝑃)))
53	1, 52	mpcom 37	. 2 ⊢ (𝑃 ∈ (𝑉 WWalks 𝐸) → (𝑉 USGrph 𝐸 → ∃𝑓 𝑓(𝑉 Walks 𝐸)𝑃))
54	53	com12 32	1 ⊢ (𝑉 USGrph 𝐸 → (𝑃 ∈ (𝑉 WWalks 𝐸) → ∃𝑓 𝑓(𝑉 Walks 𝐸)𝑃))

Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475  ∃wex 1695   ∈ wcel 1977   ≠ wne 2780  ∀wral 2896  Vcvv 3173  ∅c0 3874  {cpr 4127   class class class wbr 4583   ↦ cmpt 4643  ^◡ccnv 5037  dom cdm 5038  ran crn 5039  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549  0cc0 9815  1c1 9816   + caddc 9818   ≤ cle 9954   − cmin 10145  ...cfz 12197  ..^cfzo 12334  #chash 12979  Word cword 13146   USGrph cusg 25859   Walks cwalk 26026   WWalks cwwlk 26205

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-fal 1481  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-er 7629  df-map 7746  df-pm 7747  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-card 8648  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-2 10956  df-n0 11170  df-z 11255  df-uz 11564  df-fz 12198  df-fzo 12335  df-hash 12980  df-word 13154  df-usgra 25862  df-wlk 26036  df-wwlk 26207

This theorem is referenced by:  wlkiswwlk  26226  wlklniswwlkn2  26228