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Theorem wlkiswwlksupgr2 30475

Description: A walk as word corresponds to the sequence of vertices in a walk in a pseudograph. This variant of wlkiswwlks2 30473 does not require 𝐺 to be a simple pseudograph, but it requires the Axiom of Choice (ac6 10565) for its proof. Notice that only the existence of a function 𝑓 can be proven, but, in general, it cannot be "constructed" (as in wlkiswwlks2 30473). (Contributed by Alexander van der Vekens, 21-Jul-2018.) (Revised by AV, 10-Apr-2021.)

**Assertion**
Ref	Expression
wlkiswwlksupgr2	⊢ (𝐺 ∈ UPGraph → (𝑃 ∈ (WWalks‘𝐺) → ∃𝑓 𝑓(Walks‘𝐺)𝑃))

Distinct variable groups: 𝑓,𝐺 𝑃,𝑓

Proof of Theorem wlkiswwlksupgr2 Dummy variables 𝑖 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2760	. . 3 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
2		eqid 2760	. . 3 ⊢ (Edg‘𝐺) = (Edg‘𝐺)
3	1, 2	iswwlks 30434	. 2 ⊢ (𝑃 ∈ (WWalks‘𝐺) ↔ (𝑃 ≠ ∅ ∧ 𝑃 ∈ Word (Vtx‘𝐺) ∧ ∀𝑖 ∈ (0..^((♯‘𝑃) − 1)){(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ∈ (Edg‘𝐺)))
4		ovex 7457	. . . . . 6 ⊢ (0..^((♯‘𝑃) − 1)) ∈ V
5		fvex 6903	. . . . . . 7 ⊢ (iEdg‘𝐺) ∈ V
6	5	dmex 7925	. . . . . 6 ⊢ dom (iEdg‘𝐺) ∈ V
7		edgval 29636	. . . . . . . . . . . . 13 ⊢ (Edg‘𝐺) = ran (iEdg‘𝐺)
8	7	eleq2i 2852	. . . . . . . . . . . 12 ⊢ ({(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ∈ (Edg‘𝐺) ↔ {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ∈ ran (iEdg‘𝐺))
9		upgruhgr 29689	. . . . . . . . . . . . . . 15 ⊢ (𝐺 ∈ UPGraph → 𝐺 ∈ UHGraph)
10		eqid 2760	. . . . . . . . . . . . . . . 16 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺)
11	10	uhgrfun 29653	. . . . . . . . . . . . . . 15 ⊢ (𝐺 ∈ UHGraph → Fun (iEdg‘𝐺))
12	9, 11	syl 18	. . . . . . . . . . . . . 14 ⊢ (𝐺 ∈ UPGraph → Fun (iEdg‘𝐺))
13	12	adantl 487	. . . . . . . . . . . . 13 ⊢ (((𝑃 ≠ ∅ ∧ 𝑃 ∈ Word (Vtx‘𝐺)) ∧ 𝐺 ∈ UPGraph) → Fun (iEdg‘𝐺))
14		elrnrexdm 7094	. . . . . . . . . . . . . 14 ⊢ (Fun (iEdg‘𝐺) → ({(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ∈ ran (iEdg‘𝐺) → ∃𝑥 ∈ dom (iEdg‘𝐺){(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} = ((iEdg‘𝐺)‘𝑥)))
15		eqcom 2767	. . . . . . . . . . . . . . 15 ⊢ (((iEdg‘𝐺)‘𝑥) = {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ↔ {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} = ((iEdg‘𝐺)‘𝑥))
16	15	rexbii 3109	. . . . . . . . . . . . . 14 ⊢ (∃𝑥 ∈ dom (iEdg‘𝐺)((iEdg‘𝐺)‘𝑥) = {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ↔ ∃𝑥 ∈ dom (iEdg‘𝐺){(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} = ((iEdg‘𝐺)‘𝑥))
17	14, 16	imbitrrdi 255	. . . . . . . . . . . . 13 ⊢ (Fun (iEdg‘𝐺) → ({(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ∈ ran (iEdg‘𝐺) → ∃𝑥 ∈ dom (iEdg‘𝐺)((iEdg‘𝐺)‘𝑥) = {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))}))
18	13, 17	syl 18	. . . . . . . . . . . 12 ⊢ (((𝑃 ≠ ∅ ∧ 𝑃 ∈ Word (Vtx‘𝐺)) ∧ 𝐺 ∈ UPGraph) → ({(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ∈ ran (iEdg‘𝐺) → ∃𝑥 ∈ dom (iEdg‘𝐺)((iEdg‘𝐺)‘𝑥) = {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))}))
19	8, 18	biimtrid 245	. . . . . . . . . . 11 ⊢ (((𝑃 ≠ ∅ ∧ 𝑃 ∈ Word (Vtx‘𝐺)) ∧ 𝐺 ∈ UPGraph) → ({(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ∈ (Edg‘𝐺) → ∃𝑥 ∈ dom (iEdg‘𝐺)((iEdg‘𝐺)‘𝑥) = {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))}))
20	19	ralimdv 3176	. . . . . . . . . 10 ⊢ (((𝑃 ≠ ∅ ∧ 𝑃 ∈ Word (Vtx‘𝐺)) ∧ 𝐺 ∈ UPGraph) → (∀𝑖 ∈ (0..^((♯‘𝑃) − 1)){(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ∈ (Edg‘𝐺) → ∀𝑖 ∈ (0..^((♯‘𝑃) − 1))∃𝑥 ∈ dom (iEdg‘𝐺)((iEdg‘𝐺)‘𝑥) = {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))}))
21	20	ex 418	. . . . . . . . 9 ⊢ ((𝑃 ≠ ∅ ∧ 𝑃 ∈ Word (Vtx‘𝐺)) → (𝐺 ∈ UPGraph → (∀𝑖 ∈ (0..^((♯‘𝑃) − 1)){(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ∈ (Edg‘𝐺) → ∀𝑖 ∈ (0..^((♯‘𝑃) − 1))∃𝑥 ∈ dom (iEdg‘𝐺)((iEdg‘𝐺)‘𝑥) = {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))})))
22	21	com23 87	. . . . . . . 8 ⊢ ((𝑃 ≠ ∅ ∧ 𝑃 ∈ Word (Vtx‘𝐺)) → (∀𝑖 ∈ (0..^((♯‘𝑃) − 1)){(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ∈ (Edg‘𝐺) → (𝐺 ∈ UPGraph → ∀𝑖 ∈ (0..^((♯‘𝑃) − 1))∃𝑥 ∈ dom (iEdg‘𝐺)((iEdg‘𝐺)‘𝑥) = {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))})))
23	22	3impia 1135	. . . . . . 7 ⊢ ((𝑃 ≠ ∅ ∧ 𝑃 ∈ Word (Vtx‘𝐺) ∧ ∀𝑖 ∈ (0..^((♯‘𝑃) − 1)){(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ∈ (Edg‘𝐺)) → (𝐺 ∈ UPGraph → ∀𝑖 ∈ (0..^((♯‘𝑃) − 1))∃𝑥 ∈ dom (iEdg‘𝐺)((iEdg‘𝐺)‘𝑥) = {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))}))
24	23	impcom 413	. . . . . 6 ⊢ ((𝐺 ∈ UPGraph ∧ (𝑃 ≠ ∅ ∧ 𝑃 ∈ Word (Vtx‘𝐺) ∧ ∀𝑖 ∈ (0..^((♯‘𝑃) − 1)){(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ∈ (Edg‘𝐺))) → ∀𝑖 ∈ (0..^((♯‘𝑃) − 1))∃𝑥 ∈ dom (iEdg‘𝐺)((iEdg‘𝐺)‘𝑥) = {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))})
25		fveqeq2 6899	. . . . . . 7 ⊢ (𝑥 = (𝑓‘𝑖) → (((iEdg‘𝐺)‘𝑥) = {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ↔ ((iEdg‘𝐺)‘(𝑓‘𝑖)) = {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))}))
26	25	ac6 10565	. . . . . 6 ⊢ (((0..^((♯‘𝑃) − 1)) ∈ V ∧ dom (iEdg‘𝐺) ∈ V ∧ ∀𝑖 ∈ (0..^((♯‘𝑃) − 1))∃𝑥 ∈ dom (iEdg‘𝐺)((iEdg‘𝐺)‘𝑥) = {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))}) → ∃𝑓(𝑓:(0..^((♯‘𝑃) − 1))⟶dom (iEdg‘𝐺) ∧ ∀𝑖 ∈ (0..^((♯‘𝑃) − 1))((iEdg‘𝐺)‘(𝑓‘𝑖)) = {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))}))
27	4, 6, 24, 26	mp3an12i 1494	. . . . 5 ⊢ ((𝐺 ∈ UPGraph ∧ (𝑃 ≠ ∅ ∧ 𝑃 ∈ Word (Vtx‘𝐺) ∧ ∀𝑖 ∈ (0..^((♯‘𝑃) − 1)){(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ∈ (Edg‘𝐺))) → ∃𝑓(𝑓:(0..^((♯‘𝑃) − 1))⟶dom (iEdg‘𝐺) ∧ ∀𝑖 ∈ (0..^((♯‘𝑃) − 1))((iEdg‘𝐺)‘(𝑓‘𝑖)) = {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))}))
28		iswrdi 14669	. . . . . . . . . 10 ⊢ (𝑓:(0..^((♯‘𝑃) − 1))⟶dom (iEdg‘𝐺) → 𝑓 ∈ Word dom (iEdg‘𝐺))
29	28	adantr 486	. . . . . . . . 9 ⊢ ((𝑓:(0..^((♯‘𝑃) − 1))⟶dom (iEdg‘𝐺) ∧ ∀𝑖 ∈ (0..^((♯‘𝑃) − 1))((iEdg‘𝐺)‘(𝑓‘𝑖)) = {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))}) → 𝑓 ∈ Word dom (iEdg‘𝐺))
30	29	adantl 487	. . . . . . . 8 ⊢ (((𝐺 ∈ UPGraph ∧ (𝑃 ≠ ∅ ∧ 𝑃 ∈ Word (Vtx‘𝐺) ∧ ∀𝑖 ∈ (0..^((♯‘𝑃) − 1)){(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ∈ (Edg‘𝐺))) ∧ (𝑓:(0..^((♯‘𝑃) − 1))⟶dom (iEdg‘𝐺) ∧ ∀𝑖 ∈ (0..^((♯‘𝑃) − 1))((iEdg‘𝐺)‘(𝑓‘𝑖)) = {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))})) → 𝑓 ∈ Word dom (iEdg‘𝐺))
31		len0nnbi 14703	. . . . . . . . . . . . . . 15 ⊢ (𝑃 ∈ Word (Vtx‘𝐺) → (𝑃 ≠ ∅ ↔ (♯‘𝑃) ∈ ℕ))
32	31	biimpac 484	. . . . . . . . . . . . . 14 ⊢ ((𝑃 ≠ ∅ ∧ 𝑃 ∈ Word (Vtx‘𝐺)) → (♯‘𝑃) ∈ ℕ)
33		wrdf 14670	. . . . . . . . . . . . . . . 16 ⊢ (𝑃 ∈ Word (Vtx‘𝐺) → 𝑃:(0..^(♯‘𝑃))⟶(Vtx‘𝐺))
34		nnz 12721	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((♯‘𝑃) ∈ ℕ → (♯‘𝑃) ∈ ℤ)
35		fzoval 13801	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((♯‘𝑃) ∈ ℤ → (0..^(♯‘𝑃)) = (0...((♯‘𝑃) − 1)))
36	34, 35	syl 18	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((♯‘𝑃) ∈ ℕ → (0..^(♯‘𝑃)) = (0...((♯‘𝑃) − 1)))
37	36	adantr 486	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((♯‘𝑃) ∈ ℕ ∧ 𝑓:(0..^((♯‘𝑃) − 1))⟶dom (iEdg‘𝐺)) → (0..^(♯‘𝑃)) = (0...((♯‘𝑃) − 1)))
38		nnm1nn0 12654	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((♯‘𝑃) ∈ ℕ → ((♯‘𝑃) − 1) ∈ ℕ₀)
39		fnfzo0hash 14602	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((♯‘𝑃) − 1) ∈ ℕ₀ ∧ 𝑓:(0..^((♯‘𝑃) − 1))⟶dom (iEdg‘𝐺)) → (♯‘𝑓) = ((♯‘𝑃) − 1))
40	38, 39	sylan 592	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((♯‘𝑃) ∈ ℕ ∧ 𝑓:(0..^((♯‘𝑃) − 1))⟶dom (iEdg‘𝐺)) → (♯‘𝑓) = ((♯‘𝑃) − 1))
41	40	eqcomd 2766	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((♯‘𝑃) ∈ ℕ ∧ 𝑓:(0..^((♯‘𝑃) − 1))⟶dom (iEdg‘𝐺)) → ((♯‘𝑃) − 1) = (♯‘𝑓))
42	41	oveq2d 7440	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((♯‘𝑃) ∈ ℕ ∧ 𝑓:(0..^((♯‘𝑃) − 1))⟶dom (iEdg‘𝐺)) → (0...((♯‘𝑃) − 1)) = (0...(♯‘𝑓)))
43	37, 42	eqtrd 2795	. . . . . . . . . . . . . . . . . . 19 ⊢ (((♯‘𝑃) ∈ ℕ ∧ 𝑓:(0..^((♯‘𝑃) − 1))⟶dom (iEdg‘𝐺)) → (0..^(♯‘𝑃)) = (0...(♯‘𝑓)))
44	43	feq2d 6698	. . . . . . . . . . . . . . . . . 18 ⊢ (((♯‘𝑃) ∈ ℕ ∧ 𝑓:(0..^((♯‘𝑃) − 1))⟶dom (iEdg‘𝐺)) → (𝑃:(0..^(♯‘𝑃))⟶(Vtx‘𝐺) ↔ 𝑃:(0...(♯‘𝑓))⟶(Vtx‘𝐺)))
45	44	biimpcd 252	. . . . . . . . . . . . . . . . 17 ⊢ (𝑃:(0..^(♯‘𝑃))⟶(Vtx‘𝐺) → (((♯‘𝑃) ∈ ℕ ∧ 𝑓:(0..^((♯‘𝑃) − 1))⟶dom (iEdg‘𝐺)) → 𝑃:(0...(♯‘𝑓))⟶(Vtx‘𝐺)))
46	45	expd 421	. . . . . . . . . . . . . . . 16 ⊢ (𝑃:(0..^(♯‘𝑃))⟶(Vtx‘𝐺) → ((♯‘𝑃) ∈ ℕ → (𝑓:(0..^((♯‘𝑃) − 1))⟶dom (iEdg‘𝐺) → 𝑃:(0...(♯‘𝑓))⟶(Vtx‘𝐺))))
47	33, 46	syl 18	. . . . . . . . . . . . . . 15 ⊢ (𝑃 ∈ Word (Vtx‘𝐺) → ((♯‘𝑃) ∈ ℕ → (𝑓:(0..^((♯‘𝑃) − 1))⟶dom (iEdg‘𝐺) → 𝑃:(0...(♯‘𝑓))⟶(Vtx‘𝐺))))
48	47	adantl 487	. . . . . . . . . . . . . 14 ⊢ ((𝑃 ≠ ∅ ∧ 𝑃 ∈ Word (Vtx‘𝐺)) → ((♯‘𝑃) ∈ ℕ → (𝑓:(0..^((♯‘𝑃) − 1))⟶dom (iEdg‘𝐺) → 𝑃:(0...(♯‘𝑓))⟶(Vtx‘𝐺))))
49	32, 48	mpd 16	. . . . . . . . . . . . 13 ⊢ ((𝑃 ≠ ∅ ∧ 𝑃 ∈ Word (Vtx‘𝐺)) → (𝑓:(0..^((♯‘𝑃) − 1))⟶dom (iEdg‘𝐺) → 𝑃:(0...(♯‘𝑓))⟶(Vtx‘𝐺)))
50	49	3adant3 1150	. . . . . . . . . . . 12 ⊢ ((𝑃 ≠ ∅ ∧ 𝑃 ∈ Word (Vtx‘𝐺) ∧ ∀𝑖 ∈ (0..^((♯‘𝑃) − 1)){(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ∈ (Edg‘𝐺)) → (𝑓:(0..^((♯‘𝑃) − 1))⟶dom (iEdg‘𝐺) → 𝑃:(0...(♯‘𝑓))⟶(Vtx‘𝐺)))
51	50	adantl 487	. . . . . . . . . . 11 ⊢ ((𝐺 ∈ UPGraph ∧ (𝑃 ≠ ∅ ∧ 𝑃 ∈ Word (Vtx‘𝐺) ∧ ∀𝑖 ∈ (0..^((♯‘𝑃) − 1)){(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ∈ (Edg‘𝐺))) → (𝑓:(0..^((♯‘𝑃) − 1))⟶dom (iEdg‘𝐺) → 𝑃:(0...(♯‘𝑓))⟶(Vtx‘𝐺)))
52	51	com12 33	. . . . . . . . . 10 ⊢ (𝑓:(0..^((♯‘𝑃) − 1))⟶dom (iEdg‘𝐺) → ((𝐺 ∈ UPGraph ∧ (𝑃 ≠ ∅ ∧ 𝑃 ∈ Word (Vtx‘𝐺) ∧ ∀𝑖 ∈ (0..^((♯‘𝑃) − 1)){(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ∈ (Edg‘𝐺))) → 𝑃:(0...(♯‘𝑓))⟶(Vtx‘𝐺)))
53	52	adantr 486	. . . . . . . . 9 ⊢ ((𝑓:(0..^((♯‘𝑃) − 1))⟶dom (iEdg‘𝐺) ∧ ∀𝑖 ∈ (0..^((♯‘𝑃) − 1))((iEdg‘𝐺)‘(𝑓‘𝑖)) = {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))}) → ((𝐺 ∈ UPGraph ∧ (𝑃 ≠ ∅ ∧ 𝑃 ∈ Word (Vtx‘𝐺) ∧ ∀𝑖 ∈ (0..^((♯‘𝑃) − 1)){(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ∈ (Edg‘𝐺))) → 𝑃:(0...(♯‘𝑓))⟶(Vtx‘𝐺)))
54	53	impcom 413	. . . . . . . 8 ⊢ (((𝐺 ∈ UPGraph ∧ (𝑃 ≠ ∅ ∧ 𝑃 ∈ Word (Vtx‘𝐺) ∧ ∀𝑖 ∈ (0..^((♯‘𝑃) − 1)){(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ∈ (Edg‘𝐺))) ∧ (𝑓:(0..^((♯‘𝑃) − 1))⟶dom (iEdg‘𝐺) ∧ ∀𝑖 ∈ (0..^((♯‘𝑃) − 1))((iEdg‘𝐺)‘(𝑓‘𝑖)) = {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))})) → 𝑃:(0...(♯‘𝑓))⟶(Vtx‘𝐺))
55		simpr 490	. . . . . . . . . 10 ⊢ ((((𝐺 ∈ UPGraph ∧ (𝑃 ≠ ∅ ∧ 𝑃 ∈ Word (Vtx‘𝐺) ∧ ∀𝑖 ∈ (0..^((♯‘𝑃) − 1)){(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ∈ (Edg‘𝐺))) ∧ 𝑓:(0..^((♯‘𝑃) − 1))⟶dom (iEdg‘𝐺)) ∧ ∀𝑖 ∈ (0..^((♯‘𝑃) − 1))((iEdg‘𝐺)‘(𝑓‘𝑖)) = {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))}) → ∀𝑖 ∈ (0..^((♯‘𝑃) − 1))((iEdg‘𝐺)‘(𝑓‘𝑖)) = {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))})
56	32, 40	sylan 592	. . . . . . . . . . . . . . . 16 ⊢ (((𝑃 ≠ ∅ ∧ 𝑃 ∈ Word (Vtx‘𝐺)) ∧ 𝑓:(0..^((♯‘𝑃) − 1))⟶dom (iEdg‘𝐺)) → (♯‘𝑓) = ((♯‘𝑃) − 1))
57	56	oveq2d 7440	. . . . . . . . . . . . . . 15 ⊢ (((𝑃 ≠ ∅ ∧ 𝑃 ∈ Word (Vtx‘𝐺)) ∧ 𝑓:(0..^((♯‘𝑃) − 1))⟶dom (iEdg‘𝐺)) → (0..^(♯‘𝑓)) = (0..^((♯‘𝑃) − 1)))
58	57	ex 418	. . . . . . . . . . . . . 14 ⊢ ((𝑃 ≠ ∅ ∧ 𝑃 ∈ Word (Vtx‘𝐺)) → (𝑓:(0..^((♯‘𝑃) − 1))⟶dom (iEdg‘𝐺) → (0..^(♯‘𝑓)) = (0..^((♯‘𝑃) − 1))))
59	58	3adant3 1150	. . . . . . . . . . . . 13 ⊢ ((𝑃 ≠ ∅ ∧ 𝑃 ∈ Word (Vtx‘𝐺) ∧ ∀𝑖 ∈ (0..^((♯‘𝑃) − 1)){(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ∈ (Edg‘𝐺)) → (𝑓:(0..^((♯‘𝑃) − 1))⟶dom (iEdg‘𝐺) → (0..^(♯‘𝑓)) = (0..^((♯‘𝑃) − 1))))
60	59	adantl 487	. . . . . . . . . . . 12 ⊢ ((𝐺 ∈ UPGraph ∧ (𝑃 ≠ ∅ ∧ 𝑃 ∈ Word (Vtx‘𝐺) ∧ ∀𝑖 ∈ (0..^((♯‘𝑃) − 1)){(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ∈ (Edg‘𝐺))) → (𝑓:(0..^((♯‘𝑃) − 1))⟶dom (iEdg‘𝐺) → (0..^(♯‘𝑓)) = (0..^((♯‘𝑃) − 1))))
61	60	imp 412	. . . . . . . . . . 11 ⊢ (((𝐺 ∈ UPGraph ∧ (𝑃 ≠ ∅ ∧ 𝑃 ∈ Word (Vtx‘𝐺) ∧ ∀𝑖 ∈ (0..^((♯‘𝑃) − 1)){(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ∈ (Edg‘𝐺))) ∧ 𝑓:(0..^((♯‘𝑃) − 1))⟶dom (iEdg‘𝐺)) → (0..^(♯‘𝑓)) = (0..^((♯‘𝑃) − 1)))
62	61	adantr 486	. . . . . . . . . 10 ⊢ ((((𝐺 ∈ UPGraph ∧ (𝑃 ≠ ∅ ∧ 𝑃 ∈ Word (Vtx‘𝐺) ∧ ∀𝑖 ∈ (0..^((♯‘𝑃) − 1)){(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ∈ (Edg‘𝐺))) ∧ 𝑓:(0..^((♯‘𝑃) − 1))⟶dom (iEdg‘𝐺)) ∧ ∀𝑖 ∈ (0..^((♯‘𝑃) − 1))((iEdg‘𝐺)‘(𝑓‘𝑖)) = {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))}) → (0..^(♯‘𝑓)) = (0..^((♯‘𝑃) − 1)))
63	55, 62	raleqtrrdv 3323	. . . . . . . . 9 ⊢ ((((𝐺 ∈ UPGraph ∧ (𝑃 ≠ ∅ ∧ 𝑃 ∈ Word (Vtx‘𝐺) ∧ ∀𝑖 ∈ (0..^((♯‘𝑃) − 1)){(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ∈ (Edg‘𝐺))) ∧ 𝑓:(0..^((♯‘𝑃) − 1))⟶dom (iEdg‘𝐺)) ∧ ∀𝑖 ∈ (0..^((♯‘𝑃) − 1))((iEdg‘𝐺)‘(𝑓‘𝑖)) = {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))}) → ∀𝑖 ∈ (0..^(♯‘𝑓))((iEdg‘𝐺)‘(𝑓‘𝑖)) = {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))})
64	63	anasss 472	. . . . . . . 8 ⊢ (((𝐺 ∈ UPGraph ∧ (𝑃 ≠ ∅ ∧ 𝑃 ∈ Word (Vtx‘𝐺) ∧ ∀𝑖 ∈ (0..^((♯‘𝑃) − 1)){(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ∈ (Edg‘𝐺))) ∧ (𝑓:(0..^((♯‘𝑃) − 1))⟶dom (iEdg‘𝐺) ∧ ∀𝑖 ∈ (0..^((♯‘𝑃) − 1))((iEdg‘𝐺)‘(𝑓‘𝑖)) = {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))})) → ∀𝑖 ∈ (0..^(♯‘𝑓))((iEdg‘𝐺)‘(𝑓‘𝑖)) = {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))})
65	30, 54, 64	3jca 1146	. . . . . . 7 ⊢ (((𝐺 ∈ UPGraph ∧ (𝑃 ≠ ∅ ∧ 𝑃 ∈ Word (Vtx‘𝐺) ∧ ∀𝑖 ∈ (0..^((♯‘𝑃) − 1)){(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ∈ (Edg‘𝐺))) ∧ (𝑓:(0..^((♯‘𝑃) − 1))⟶dom (iEdg‘𝐺) ∧ ∀𝑖 ∈ (0..^((♯‘𝑃) − 1))((iEdg‘𝐺)‘(𝑓‘𝑖)) = {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))})) → (𝑓 ∈ Word dom (iEdg‘𝐺) ∧ 𝑃:(0...(♯‘𝑓))⟶(Vtx‘𝐺) ∧ ∀𝑖 ∈ (0..^(♯‘𝑓))((iEdg‘𝐺)‘(𝑓‘𝑖)) = {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))}))
66	65	ex 418	. . . . . 6 ⊢ ((𝐺 ∈ UPGraph ∧ (𝑃 ≠ ∅ ∧ 𝑃 ∈ Word (Vtx‘𝐺) ∧ ∀𝑖 ∈ (0..^((♯‘𝑃) − 1)){(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ∈ (Edg‘𝐺))) → ((𝑓:(0..^((♯‘𝑃) − 1))⟶dom (iEdg‘𝐺) ∧ ∀𝑖 ∈ (0..^((♯‘𝑃) − 1))((iEdg‘𝐺)‘(𝑓‘𝑖)) = {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))}) → (𝑓 ∈ Word dom (iEdg‘𝐺) ∧ 𝑃:(0...(♯‘𝑓))⟶(Vtx‘𝐺) ∧ ∀𝑖 ∈ (0..^(♯‘𝑓))((iEdg‘𝐺)‘(𝑓‘𝑖)) = {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))})))
67	66	eximdv 1949	. . . . 5 ⊢ ((𝐺 ∈ UPGraph ∧ (𝑃 ≠ ∅ ∧ 𝑃 ∈ Word (Vtx‘𝐺) ∧ ∀𝑖 ∈ (0..^((♯‘𝑃) − 1)){(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ∈ (Edg‘𝐺))) → (∃𝑓(𝑓:(0..^((♯‘𝑃) − 1))⟶dom (iEdg‘𝐺) ∧ ∀𝑖 ∈ (0..^((♯‘𝑃) − 1))((iEdg‘𝐺)‘(𝑓‘𝑖)) = {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))}) → ∃𝑓(𝑓 ∈ Word dom (iEdg‘𝐺) ∧ 𝑃:(0...(♯‘𝑓))⟶(Vtx‘𝐺) ∧ ∀𝑖 ∈ (0..^(♯‘𝑓))((iEdg‘𝐺)‘(𝑓‘𝑖)) = {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))})))
68	27, 67	mpd 16	. . . 4 ⊢ ((𝐺 ∈ UPGraph ∧ (𝑃 ≠ ∅ ∧ 𝑃 ∈ Word (Vtx‘𝐺) ∧ ∀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))}))
69	1, 10	upgriswlk 30230	. . . . . 6 ⊢ (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))})))
70	69	adantr 486	. . . . 5 ⊢ ((𝐺 ∈ UPGraph ∧ (𝑃 ≠ ∅ ∧ 𝑃 ∈ Word (Vtx‘𝐺) ∧ ∀𝑖 ∈ (0..^((♯‘𝑃) − 1)){(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ∈ (Edg‘𝐺))) → (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))})))
71	70	exbidv 1953	. . . 4 ⊢ ((𝐺 ∈ UPGraph ∧ (𝑃 ≠ ∅ ∧ 𝑃 ∈ Word (Vtx‘𝐺) ∧ ∀𝑖 ∈ (0..^((♯‘𝑃) − 1)){(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ∈ (Edg‘𝐺))) → (∃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))})))
72	68, 71	mpbird 260	. . 3 ⊢ ((𝐺 ∈ UPGraph ∧ (𝑃 ≠ ∅ ∧ 𝑃 ∈ Word (Vtx‘𝐺) ∧ ∀𝑖 ∈ (0..^((♯‘𝑃) − 1)){(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ∈ (Edg‘𝐺))) → ∃𝑓 𝑓(Walks‘𝐺)𝑃)
73	72	ex 418	. 2 ⊢ (𝐺 ∈ UPGraph → ((𝑃 ≠ ∅ ∧ 𝑃 ∈ Word (Vtx‘𝐺) ∧ ∀𝑖 ∈ (0..^((♯‘𝑃) − 1)){(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ∈ (Edg‘𝐺)) → ∃𝑓 𝑓(Walks‘𝐺)𝑃))
74	3, 73	biimtrid 245	1 ⊢ (𝐺 ∈ UPGraph → (𝑃 ∈ (WWalks‘𝐺) → ∃𝑓 𝑓(Walks‘𝐺)𝑃))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 401 ∧ w3a 1103 = wceq 1569 ∃wex 1811 ∈ wcel 2144 ≠ wne 2955 ∀wral 3076 ∃wrex 3086 Vcvv 3450 ∅c0 4277 {cpr 4586 class class class wbr 5102 dom cdm 5651 ran crn 5652 Fun wfun 6537 ⟶wf 6539 ‘cfv 6543 (class class class)co 7424 0cc0 11207 1c1 11208 + caddc 11210 − cmin 11548 ℕcn 12342 ℕ₀cn0 12613 ℤcz 12700 ...cfz 13646 ..^cfzo 13795 ♯chash 14481 Word cword 14665 Vtxcvtx 29583 iEdgciedg 29584 Edgcedg 29634 UHGraphcuhgr 29643 UPGraphcupgr 29667 Walkscwlks 30186 WWalkscwwlks 30423

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1827 ax-4 1841 ax-5 1942 ax-6 1999 ax-7 2040 ax-8 2146 ax-9 2154 ax-10 2177 ax-11 2193 ax-12 2212 ax-ext 2732 ax-rep 5231 ax-sep 5248 ax-nul 5259 ax-pow 5327 ax-pr 5391 ax-un 7755 ax-ac2 10548 ax-cnex 11263 ax-resscn 11264 ax-1cn 11265 ax-icn 11266 ax-addcl 11267 ax-addrcl 11268 ax-mulcl 11269 ax-mulrcl 11270 ax-mulcom 11271 ax-addass 11272 ax-mulass 11273 ax-distr 11274 ax-i2m1 11275 ax-1ne0 11276 ax-1rid 11277 ax-rnegex 11278 ax-rrecex 11279 ax-cnre 11280 ax-pre-lttri 11281 ax-pre-lttrn 11282 ax-pre-ltadd 11283 ax-pre-mulgt0 11284

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-ifp 1079 df-3or 1104 df-3an 1105 df-tru 1572 df-fal 1582 df-ex 1812 df-nf 1816 df-sb 2099 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-nel 3062 df-ral 3077 df-rex 3087 df-rmo 3365 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3739 df-csb 3848 df-ss 3897 df-pss 4013 df-dif 4046 df-in 4090 df-un 4150 df-nul 4278 df-if 4482 df-pw 4558 df-sn 4585 df-pr 4587 df-op 4835 df-uni 4867 df-int 4907 df-iun 4952 df-br 5103 df-opab 5167 df-mpt 5186 df-tr 5212 df-id 5546 df-eprel 5551 df-po 5559 df-so 5560 df-fr 5604 df-se 5605 df-we 5606 df-xp 5657 df-rel 5658 df-cnv 5659 df-co 5660 df-dm 5661 df-rn 5662 df-res 5663 df-ima 5664 df-pred 6307 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6497 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-isom 6552 df-riota 7381 df-ov 7427 df-oprab 7428 df-mpo 7429 df-om 7882 df-1st 8005 df-2nd 8006 df-frecs 8307 df-wrecs 8338 df-recs 8387 df-rdg 8426 df-1o 8484 df-2o 8485 df-oadd 8488 df-er 8725 df-map 8857 df-pm 8858 df-en 8982 df-dom 8983 df-sdom 8984 df-fin 8985 df-dju 9989 df-card 10027 df-ac 10202 df-pnf 11352 df-mnf 11353 df-xr 11354 df-ltxr 11355 df-le 11356 df-sub 11550 df-neg 11551 df-nn 12343 df-2 12412 df-n0 12614 df-xnn0 12687 df-z 12701 df-uz 12973 df-fz 13647 df-fzo 13796 df-hash 14482 df-word 14666 df-edg 29635 df-uhgr 29645 df-upgr 29669 df-wlks 30189 df-wwlks 30428

This theorem is used by: wlkiswwlkupgr 30476 wlklnwwlklnupgr2 30483

W3C validator