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Theorem wlkiswwlksupgr2 29131

Description: A walk as word corresponds to the sequence of vertices in a walk in a pseudograph. This variant of wlkiswwlks2 29129 does not require 𝐺 to be a simple pseudograph, but it requires the Axiom of Choice (ac6 10475) 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 29129). (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 2733	. . 3 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
2		eqid 2733	. . 3 ⊢ (Edg‘𝐺) = (Edg‘𝐺)
3	1, 2	iswwlks 29090	. 2 ⊢ (𝑃 ∈ (WWalks‘𝐺) ↔ (𝑃 ≠ ∅ ∧ 𝑃 ∈ Word (Vtx‘𝐺) ∧ ∀𝑖 ∈ (0..^((♯‘𝑃) − 1)){(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ∈ (Edg‘𝐺)))
4		edgval 28309	. . . . . . . . . . . . 13 ⊢ (Edg‘𝐺) = ran (iEdg‘𝐺)
5	4	eleq2i 2826	. . . . . . . . . . . 12 ⊢ ({(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ∈ (Edg‘𝐺) ↔ {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ∈ ran (iEdg‘𝐺))
6		upgruhgr 28362	. . . . . . . . . . . . . . 15 ⊢ (𝐺 ∈ UPGraph → 𝐺 ∈ UHGraph)
7		eqid 2733	. . . . . . . . . . . . . . . 16 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺)
8	7	uhgrfun 28326	. . . . . . . . . . . . . . 15 ⊢ (𝐺 ∈ UHGraph → Fun (iEdg‘𝐺))
9	6, 8	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝐺 ∈ UPGraph → Fun (iEdg‘𝐺))
10	9	adantl 483	. . . . . . . . . . . . 13 ⊢ (((𝑃 ≠ ∅ ∧ 𝑃 ∈ Word (Vtx‘𝐺)) ∧ 𝐺 ∈ UPGraph) → Fun (iEdg‘𝐺))
11		elrnrexdm 7091	. . . . . . . . . . . . . 14 ⊢ (Fun (iEdg‘𝐺) → ({(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ∈ ran (iEdg‘𝐺) → ∃𝑥 ∈ dom (iEdg‘𝐺){(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} = ((iEdg‘𝐺)‘𝑥)))
12		eqcom 2740	. . . . . . . . . . . . . . 15 ⊢ (((iEdg‘𝐺)‘𝑥) = {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ↔ {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} = ((iEdg‘𝐺)‘𝑥))
13	12	rexbii 3095	. . . . . . . . . . . . . 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))
14	11, 13	syl6ibr 252	. . . . . . . . . . . . 13 ⊢ (Fun (iEdg‘𝐺) → ({(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ∈ ran (iEdg‘𝐺) → ∃𝑥 ∈ dom (iEdg‘𝐺)((iEdg‘𝐺)‘𝑥) = {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))}))
15	10, 14	syl 17	. . . . . . . . . . . 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))}))
16	5, 15	biimtrid 241	. . . . . . . . . . 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))}))
17	16	ralimdv 3170	. . . . . . . . . 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))}))
18	17	ex 414	. . . . . . . . 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))})))
19	18	com23 86	. . . . . . . 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))})))
20	19	3impia 1118	. . . . . . 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))}))
21	20	impcom 409	. . . . . 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))})
22		ovex 7442	. . . . . . 7 ⊢ (0..^((♯‘𝑃) − 1)) ∈ V
23		fvex 6905	. . . . . . . 8 ⊢ (iEdg‘𝐺) ∈ V
24	23	dmex 7902	. . . . . . 7 ⊢ dom (iEdg‘𝐺) ∈ V
25		fveqeq2 6901	. . . . . . 7 ⊢ (𝑥 = (𝑓‘𝑖) → (((iEdg‘𝐺)‘𝑥) = {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ↔ ((iEdg‘𝐺)‘(𝑓‘𝑖)) = {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))}))
26	22, 24, 25	ac6 10475	. . . . . 6 ⊢ (∀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	21, 26	syl 17	. . . . 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 14468	. . . . . . . . . 10 ⊢ (𝑓:(0..^((♯‘𝑃) − 1))⟶dom (iEdg‘𝐺) → 𝑓 ∈ Word dom (iEdg‘𝐺))
29	28	adantr 482	. . . . . . . . 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 483	. . . . . . . 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 14501	. . . . . . . . . . . . . . 15 ⊢ (𝑃 ∈ Word (Vtx‘𝐺) → (𝑃 ≠ ∅ ↔ (♯‘𝑃) ∈ ℕ))
32	31	biimpac 480	. . . . . . . . . . . . . 14 ⊢ ((𝑃 ≠ ∅ ∧ 𝑃 ∈ Word (Vtx‘𝐺)) → (♯‘𝑃) ∈ ℕ)
33		wrdf 14469	. . . . . . . . . . . . . . . 16 ⊢ (𝑃 ∈ Word (Vtx‘𝐺) → 𝑃:(0..^(♯‘𝑃))⟶(Vtx‘𝐺))
34		nnz 12579	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((♯‘𝑃) ∈ ℕ → (♯‘𝑃) ∈ ℤ)
35		fzoval 13633	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((♯‘𝑃) ∈ ℤ → (0..^(♯‘𝑃)) = (0...((♯‘𝑃) − 1)))
36	34, 35	syl 17	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((♯‘𝑃) ∈ ℕ → (0..^(♯‘𝑃)) = (0...((♯‘𝑃) − 1)))
37	36	adantr 482	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((♯‘𝑃) ∈ ℕ ∧ 𝑓:(0..^((♯‘𝑃) − 1))⟶dom (iEdg‘𝐺)) → (0..^(♯‘𝑃)) = (0...((♯‘𝑃) − 1)))
38		nnm1nn0 12513	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((♯‘𝑃) ∈ ℕ → ((♯‘𝑃) − 1) ∈ ℕ₀)
39		fnfzo0hash 14409	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((♯‘𝑃) − 1) ∈ ℕ₀ ∧ 𝑓:(0..^((♯‘𝑃) − 1))⟶dom (iEdg‘𝐺)) → (♯‘𝑓) = ((♯‘𝑃) − 1))
40	38, 39	sylan 581	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((♯‘𝑃) ∈ ℕ ∧ 𝑓:(0..^((♯‘𝑃) − 1))⟶dom (iEdg‘𝐺)) → (♯‘𝑓) = ((♯‘𝑃) − 1))
41	40	eqcomd 2739	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((♯‘𝑃) ∈ ℕ ∧ 𝑓:(0..^((♯‘𝑃) − 1))⟶dom (iEdg‘𝐺)) → ((♯‘𝑃) − 1) = (♯‘𝑓))
42	41	oveq2d 7425	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((♯‘𝑃) ∈ ℕ ∧ 𝑓:(0..^((♯‘𝑃) − 1))⟶dom (iEdg‘𝐺)) → (0...((♯‘𝑃) − 1)) = (0...(♯‘𝑓)))
43	37, 42	eqtrd 2773	. . . . . . . . . . . . . . . . . . 19 ⊢ (((♯‘𝑃) ∈ ℕ ∧ 𝑓:(0..^((♯‘𝑃) − 1))⟶dom (iEdg‘𝐺)) → (0..^(♯‘𝑃)) = (0...(♯‘𝑓)))
44	43	feq2d 6704	. . . . . . . . . . . . . . . . . 18 ⊢ (((♯‘𝑃) ∈ ℕ ∧ 𝑓:(0..^((♯‘𝑃) − 1))⟶dom (iEdg‘𝐺)) → (𝑃:(0..^(♯‘𝑃))⟶(Vtx‘𝐺) ↔ 𝑃:(0...(♯‘𝑓))⟶(Vtx‘𝐺)))
45	44	biimpcd 248	. . . . . . . . . . . . . . . . 17 ⊢ (𝑃:(0..^(♯‘𝑃))⟶(Vtx‘𝐺) → (((♯‘𝑃) ∈ ℕ ∧ 𝑓:(0..^((♯‘𝑃) − 1))⟶dom (iEdg‘𝐺)) → 𝑃:(0...(♯‘𝑓))⟶(Vtx‘𝐺)))
46	45	expd 417	. . . . . . . . . . . . . . . 16 ⊢ (𝑃:(0..^(♯‘𝑃))⟶(Vtx‘𝐺) → ((♯‘𝑃) ∈ ℕ → (𝑓:(0..^((♯‘𝑃) − 1))⟶dom (iEdg‘𝐺) → 𝑃:(0...(♯‘𝑓))⟶(Vtx‘𝐺))))
47	33, 46	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝑃 ∈ Word (Vtx‘𝐺) → ((♯‘𝑃) ∈ ℕ → (𝑓:(0..^((♯‘𝑃) − 1))⟶dom (iEdg‘𝐺) → 𝑃:(0...(♯‘𝑓))⟶(Vtx‘𝐺))))
48	47	adantl 483	. . . . . . . . . . . . . 14 ⊢ ((𝑃 ≠ ∅ ∧ 𝑃 ∈ Word (Vtx‘𝐺)) → ((♯‘𝑃) ∈ ℕ → (𝑓:(0..^((♯‘𝑃) − 1))⟶dom (iEdg‘𝐺) → 𝑃:(0...(♯‘𝑓))⟶(Vtx‘𝐺))))
49	32, 48	mpd 15	. . . . . . . . . . . . 13 ⊢ ((𝑃 ≠ ∅ ∧ 𝑃 ∈ Word (Vtx‘𝐺)) → (𝑓:(0..^((♯‘𝑃) − 1))⟶dom (iEdg‘𝐺) → 𝑃:(0...(♯‘𝑓))⟶(Vtx‘𝐺)))
50	49	3adant3 1133	. . . . . . . . . . . 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 483	. . . . . . . . . . 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 32	. . . . . . . . . 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 482	. . . . . . . . 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 409	. . . . . . . 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 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))}) → ∀𝑖 ∈ (0..^((♯‘𝑃) − 1))((iEdg‘𝐺)‘(𝑓‘𝑖)) = {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))})
56	32, 40	sylan 581	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑃 ≠ ∅ ∧ 𝑃 ∈ Word (Vtx‘𝐺)) ∧ 𝑓:(0..^((♯‘𝑃) − 1))⟶dom (iEdg‘𝐺)) → (♯‘𝑓) = ((♯‘𝑃) − 1))
57	56	oveq2d 7425	. . . . . . . . . . . . . . . 16 ⊢ (((𝑃 ≠ ∅ ∧ 𝑃 ∈ Word (Vtx‘𝐺)) ∧ 𝑓:(0..^((♯‘𝑃) − 1))⟶dom (iEdg‘𝐺)) → (0..^(♯‘𝑓)) = (0..^((♯‘𝑃) − 1)))
58	57	ex 414	. . . . . . . . . . . . . . 15 ⊢ ((𝑃 ≠ ∅ ∧ 𝑃 ∈ Word (Vtx‘𝐺)) → (𝑓:(0..^((♯‘𝑃) − 1))⟶dom (iEdg‘𝐺) → (0..^(♯‘𝑓)) = (0..^((♯‘𝑃) − 1))))
59	58	3adant3 1133	. . . . . . . . . . . . . 14 ⊢ ((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 483	. . . . . . . . . . . . 13 ⊢ ((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 408	. . . . . . . . . . . 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)))
62	61	adantr 482	. . . . . . . . . . 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)) ∧ ∀𝑖 ∈ (0..^((♯‘𝑃) − 1))((iEdg‘𝐺)‘(𝑓‘𝑖)) = {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))}) → (0..^(♯‘𝑓)) = (0..^((♯‘𝑃) − 1)))
63	62	raleqdv 3326	. . . . . . . . . 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..^(♯‘𝑓))((iEdg‘𝐺)‘(𝑓‘𝑖)) = {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ↔ ∀𝑖 ∈ (0..^((♯‘𝑃) − 1))((iEdg‘𝐺)‘(𝑓‘𝑖)) = {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))}))
64	55, 63	mpbird 257	. . . . . . . . 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))})
65	64	anasss 468	. . . . . . . 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))})
66	30, 54, 65	3jca 1129	. . . . . . 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))}))
67	66	ex 414	. . . . . 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))})))
68	67	eximdv 1921	. . . . 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))})))
69	27, 68	mpd 15	. . . 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))}))
70	1, 7	upgriswlk 28898	. . . . . 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))})))
71	70	adantr 482	. . . . 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))})))
72	71	exbidv 1925	. . . 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))})))
73	69, 72	mpbird 257	. . 3 ⊢ ((𝐺 ∈ UPGraph ∧ (𝑃 ≠ ∅ ∧ 𝑃 ∈ Word (Vtx‘𝐺) ∧ ∀𝑖 ∈ (0..^((♯‘𝑃) − 1)){(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ∈ (Edg‘𝐺))) → ∃𝑓 𝑓(Walks‘𝐺)𝑃)
74	73	ex 414	. 2 ⊢ (𝐺 ∈ UPGraph → ((𝑃 ≠ ∅ ∧ 𝑃 ∈ Word (Vtx‘𝐺) ∧ ∀𝑖 ∈ (0..^((♯‘𝑃) − 1)){(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ∈ (Edg‘𝐺)) → ∃𝑓 𝑓(Walks‘𝐺)𝑃))
75	3, 74	biimtrid 241	1 ⊢ (𝐺 ∈ UPGraph → (𝑃 ∈ (WWalks‘𝐺) → ∃𝑓 𝑓(Walks‘𝐺)𝑃))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 ∧ w3a 1088 = wceq 1542 ∃wex 1782 ∈ wcel 2107 ≠ wne 2941 ∀wral 3062 ∃wrex 3071 ∅c0 4323 {cpr 4631 class class class wbr 5149 dom cdm 5677 ran crn 5678 Fun wfun 6538 ⟶wf 6540 ‘cfv 6544 (class class class)co 7409 0cc0 11110 1c1 11111 + caddc 11113 − cmin 11444 ℕcn 12212 ℕ₀cn0 12472 ℤcz 12558 ...cfz 13484 ..^cfzo 13627 ♯chash 14290 Word cword 14464 Vtxcvtx 28256 iEdgciedg 28257 Edgcedg 28307 UHGraphcuhgr 28316 UPGraphcupgr 28340 Walkscwlks 28853 WWalkscwwlks 29079

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 ax-ac2 10458 ax-cnex 11166 ax-resscn 11167 ax-1cn 11168 ax-icn 11169 ax-addcl 11170 ax-addrcl 11171 ax-mulcl 11172 ax-mulrcl 11173 ax-mulcom 11174 ax-addass 11175 ax-mulass 11176 ax-distr 11177 ax-i2m1 11178 ax-1ne0 11179 ax-1rid 11180 ax-rnegex 11181 ax-rrecex 11182 ax-cnre 11183 ax-pre-lttri 11184 ax-pre-lttrn 11185 ax-pre-ltadd 11186 ax-pre-mulgt0 11187

This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-ifp 1063 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-int 4952 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-se 5633 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-isom 6553 df-riota 7365 df-ov 7412 df-oprab 7413 df-mpo 7414 df-om 7856 df-1st 7975 df-2nd 7976 df-frecs 8266 df-wrecs 8297 df-recs 8371 df-rdg 8410 df-1o 8466 df-2o 8467 df-oadd 8470 df-er 8703 df-map 8822 df-pm 8823 df-en 8940 df-dom 8941 df-sdom 8942 df-fin 8943 df-dju 9896 df-card 9934 df-ac 10111 df-pnf 11250 df-mnf 11251 df-xr 11252 df-ltxr 11253 df-le 11254 df-sub 11446 df-neg 11447 df-nn 12213 df-2 12275 df-n0 12473 df-xnn0 12545 df-z 12559 df-uz 12823 df-fz 13485 df-fzo 13628 df-hash 14291 df-word 14465 df-edg 28308 df-uhgr 28318 df-upgr 28342 df-wlks 28856 df-wwlks 29084

This theorem is referenced by: wlkiswwlkupgr 29132 wlklnwwlklnupgr2 29139

W3C validator