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Theorem wlkiswwlksupgr2 29906

Description: A walk as word corresponds to the sequence of vertices in a walk in a pseudograph. This variant of wlkiswwlks2 29904 does not require 𝐺 to be a simple pseudograph, but it requires the Axiom of Choice (ac6 10517) 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 29904). (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 2734	. . 3 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
2		eqid 2734	. . 3 ⊢ (Edg‘𝐺) = (Edg‘𝐺)
3	1, 2	iswwlks 29865	. 2 ⊢ (𝑃 ∈ (WWalks‘𝐺) ↔ (𝑃 ≠ ∅ ∧ 𝑃 ∈ Word (Vtx‘𝐺) ∧ ∀𝑖 ∈ (0..^((♯‘𝑃) − 1)){(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ∈ (Edg‘𝐺)))
4		edgval 29080	. . . . . . . . . . . . 13 ⊢ (Edg‘𝐺) = ran (iEdg‘𝐺)
5	4	eleq2i 2830	. . . . . . . . . . . 12 ⊢ ({(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ∈ (Edg‘𝐺) ↔ {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ∈ ran (iEdg‘𝐺))
6		upgruhgr 29133	. . . . . . . . . . . . . . 15 ⊢ (𝐺 ∈ UPGraph → 𝐺 ∈ UHGraph)
7		eqid 2734	. . . . . . . . . . . . . . . 16 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺)
8	7	uhgrfun 29097	. . . . . . . . . . . . . . 15 ⊢ (𝐺 ∈ UHGraph → Fun (iEdg‘𝐺))
9	6, 8	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝐺 ∈ UPGraph → Fun (iEdg‘𝐺))
10	9	adantl 481	. . . . . . . . . . . . 13 ⊢ (((𝑃 ≠ ∅ ∧ 𝑃 ∈ Word (Vtx‘𝐺)) ∧ 𝐺 ∈ UPGraph) → Fun (iEdg‘𝐺))
11		elrnrexdm 7108	. . . . . . . . . . . . . 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 2741	. . . . . . . . . . . . . . 15 ⊢ (((iEdg‘𝐺)‘𝑥) = {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ↔ {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} = ((iEdg‘𝐺)‘𝑥))
13	12	rexbii 3091	. . . . . . . . . . . . . 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	imbitrrdi 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 242	. . . . . . . . . . 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 3166	. . . . . . . . . 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 412	. . . . . . . . 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 1116	. . . . . . 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 407	. . . . . 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 7463	. . . . . . 7 ⊢ (0..^((♯‘𝑃) − 1)) ∈ V
23		fvex 6919	. . . . . . . 8 ⊢ (iEdg‘𝐺) ∈ V
24	23	dmex 7931	. . . . . . 7 ⊢ dom (iEdg‘𝐺) ∈ V
25		fveqeq2 6915	. . . . . . 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 10517	. . . . . 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 14552	. . . . . . . . . 10 ⊢ (𝑓:(0..^((♯‘𝑃) − 1))⟶dom (iEdg‘𝐺) → 𝑓 ∈ Word dom (iEdg‘𝐺))
29	28	adantr 480	. . . . . . . . 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 481	. . . . . . . 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 14585	. . . . . . . . . . . . . . 15 ⊢ (𝑃 ∈ Word (Vtx‘𝐺) → (𝑃 ≠ ∅ ↔ (♯‘𝑃) ∈ ℕ))
32	31	biimpac 478	. . . . . . . . . . . . . 14 ⊢ ((𝑃 ≠ ∅ ∧ 𝑃 ∈ Word (Vtx‘𝐺)) → (♯‘𝑃) ∈ ℕ)
33		wrdf 14553	. . . . . . . . . . . . . . . 16 ⊢ (𝑃 ∈ Word (Vtx‘𝐺) → 𝑃:(0..^(♯‘𝑃))⟶(Vtx‘𝐺))
34		nnz 12631	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((♯‘𝑃) ∈ ℕ → (♯‘𝑃) ∈ ℤ)
35		fzoval 13696	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((♯‘𝑃) ∈ ℤ → (0..^(♯‘𝑃)) = (0...((♯‘𝑃) − 1)))
36	34, 35	syl 17	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((♯‘𝑃) ∈ ℕ → (0..^(♯‘𝑃)) = (0...((♯‘𝑃) − 1)))
37	36	adantr 480	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((♯‘𝑃) ∈ ℕ ∧ 𝑓:(0..^((♯‘𝑃) − 1))⟶dom (iEdg‘𝐺)) → (0..^(♯‘𝑃)) = (0...((♯‘𝑃) − 1)))
38		nnm1nn0 12564	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((♯‘𝑃) ∈ ℕ → ((♯‘𝑃) − 1) ∈ ℕ₀)
39		fnfzo0hash 14485	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((♯‘𝑃) − 1) ∈ ℕ₀ ∧ 𝑓:(0..^((♯‘𝑃) − 1))⟶dom (iEdg‘𝐺)) → (♯‘𝑓) = ((♯‘𝑃) − 1))
40	38, 39	sylan 580	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((♯‘𝑃) ∈ ℕ ∧ 𝑓:(0..^((♯‘𝑃) − 1))⟶dom (iEdg‘𝐺)) → (♯‘𝑓) = ((♯‘𝑃) − 1))
41	40	eqcomd 2740	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((♯‘𝑃) ∈ ℕ ∧ 𝑓:(0..^((♯‘𝑃) − 1))⟶dom (iEdg‘𝐺)) → ((♯‘𝑃) − 1) = (♯‘𝑓))
42	41	oveq2d 7446	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((♯‘𝑃) ∈ ℕ ∧ 𝑓:(0..^((♯‘𝑃) − 1))⟶dom (iEdg‘𝐺)) → (0...((♯‘𝑃) − 1)) = (0...(♯‘𝑓)))
43	37, 42	eqtrd 2774	. . . . . . . . . . . . . . . . . . 19 ⊢ (((♯‘𝑃) ∈ ℕ ∧ 𝑓:(0..^((♯‘𝑃) − 1))⟶dom (iEdg‘𝐺)) → (0..^(♯‘𝑃)) = (0...(♯‘𝑓)))
44	43	feq2d 6722	. . . . . . . . . . . . . . . . . 18 ⊢ (((♯‘𝑃) ∈ ℕ ∧ 𝑓:(0..^((♯‘𝑃) − 1))⟶dom (iEdg‘𝐺)) → (𝑃:(0..^(♯‘𝑃))⟶(Vtx‘𝐺) ↔ 𝑃:(0...(♯‘𝑓))⟶(Vtx‘𝐺)))
45	44	biimpcd 249	. . . . . . . . . . . . . . . . 17 ⊢ (𝑃:(0..^(♯‘𝑃))⟶(Vtx‘𝐺) → (((♯‘𝑃) ∈ ℕ ∧ 𝑓:(0..^((♯‘𝑃) − 1))⟶dom (iEdg‘𝐺)) → 𝑃:(0...(♯‘𝑓))⟶(Vtx‘𝐺)))
46	45	expd 415	. . . . . . . . . . . . . . . 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 481	. . . . . . . . . . . . . 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 1131	. . . . . . . . . . . 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 481	. . . . . . . . . . 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 480	. . . . . . . . 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 407	. . . . . . . 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 484	. . . . . . . . . 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 580	. . . . . . . . . . . . . . . 16 ⊢ (((𝑃 ≠ ∅ ∧ 𝑃 ∈ Word (Vtx‘𝐺)) ∧ 𝑓:(0..^((♯‘𝑃) − 1))⟶dom (iEdg‘𝐺)) → (♯‘𝑓) = ((♯‘𝑃) − 1))
57	56	oveq2d 7446	. . . . . . . . . . . . . . 15 ⊢ (((𝑃 ≠ ∅ ∧ 𝑃 ∈ Word (Vtx‘𝐺)) ∧ 𝑓:(0..^((♯‘𝑃) − 1))⟶dom (iEdg‘𝐺)) → (0..^(♯‘𝑓)) = (0..^((♯‘𝑃) − 1)))
58	57	ex 412	. . . . . . . . . . . . . 14 ⊢ ((𝑃 ≠ ∅ ∧ 𝑃 ∈ Word (Vtx‘𝐺)) → (𝑓:(0..^((♯‘𝑃) − 1))⟶dom (iEdg‘𝐺) → (0..^(♯‘𝑓)) = (0..^((♯‘𝑃) − 1))))
59	58	3adant3 1131	. . . . . . . . . . . . 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 481	. . . . . . . . . . . 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 406	. . . . . . . . . . 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 480	. . . . . . . . . 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 3327	. . . . . . . . 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 466	. . . . . . . 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 1127	. . . . . . 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 412	. . . . . 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 1914	. . . . 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 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))}))
69	1, 7	upgriswlk 29673	. . . . . 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 480	. . . . 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 1918	. . . 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 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)
73	72	ex 412	. 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 242	1 ⊢ (𝐺 ∈ UPGraph → (𝑃 ∈ (WWalks‘𝐺) → ∃𝑓 𝑓(Walks‘𝐺)𝑃))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1086 = wceq 1536 ∃wex 1775 ∈ wcel 2105 ≠ wne 2937 ∀wral 3058 ∃wrex 3067 ∅c0 4338 {cpr 4632 class class class wbr 5147 dom cdm 5688 ran crn 5689 Fun wfun 6556 ⟶wf 6558 ‘cfv 6562 (class class class)co 7430 0cc0 11152 1c1 11153 + caddc 11155 − cmin 11489 ℕcn 12263 ℕ₀cn0 12523 ℤcz 12610 ...cfz 13543 ..^cfzo 13690 ♯chash 14365 Word cword 14548 Vtxcvtx 29027 iEdgciedg 29028 Edgcedg 29078 UHGraphcuhgr 29087 UPGraphcupgr 29111 Walkscwlks 29628 WWalkscwwlks 29854

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-rep 5284 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 ax-ac2 10500 ax-cnex 11208 ax-resscn 11209 ax-1cn 11210 ax-icn 11211 ax-addcl 11212 ax-addrcl 11213 ax-mulcl 11214 ax-mulrcl 11215 ax-mulcom 11216 ax-addass 11217 ax-mulass 11218 ax-distr 11219 ax-i2m1 11220 ax-1ne0 11221 ax-1rid 11222 ax-rnegex 11223 ax-rrecex 11224 ax-cnre 11225 ax-pre-lttri 11226 ax-pre-lttrn 11227 ax-pre-ltadd 11228 ax-pre-mulgt0 11229

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-ifp 1063 df-3or 1087 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3377 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-pss 3982 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-int 4951 df-iun 4997 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5582 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5640 df-se 5641 df-we 5642 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-pred 6322 df-ord 6388 df-on 6389 df-lim 6390 df-suc 6391 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-isom 6571 df-riota 7387 df-ov 7433 df-oprab 7434 df-mpo 7435 df-om 7887 df-1st 8012 df-2nd 8013 df-frecs 8304 df-wrecs 8335 df-recs 8409 df-rdg 8448 df-1o 8504 df-2o 8505 df-oadd 8508 df-er 8743 df-map 8866 df-pm 8867 df-en 8984 df-dom 8985 df-sdom 8986 df-fin 8987 df-dju 9938 df-card 9976 df-ac 10153 df-pnf 11294 df-mnf 11295 df-xr 11296 df-ltxr 11297 df-le 11298 df-sub 11491 df-neg 11492 df-nn 12264 df-2 12326 df-n0 12524 df-xnn0 12597 df-z 12611 df-uz 12876 df-fz 13544 df-fzo 13691 df-hash 14366 df-word 14549 df-edg 29079 df-uhgr 29089 df-upgr 29113 df-wlks 29631 df-wwlks 29859

This theorem is referenced by: wlkiswwlkupgr 29907 wlklnwwlklnupgr2 29914

W3C validator