1wlkiswwlksupgr2 - Mathbox for Alexander van der Vekens

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Theorem 1wlkiswwlksupgr2 41074

Description: A walk as word corresponds to the sequence of vertices in a walk in a pseudograph. This variant of 1wlkiswwlks2 41072 does not require 𝐺 to be a simple pseudograph, but it requires the Axiom of Choice (ac6 9185) for its proof. Notice that only the existence of a function 𝑓 can be proven, but, in general, it cannot be "constructed" (as in 1wlkiswwlks2 41072). (Contributed by Alexander van der Vekens, 21-Jul-2018.) (Revised by AV, 10-Apr-2021.)

**Assertion**
Ref	Expression
1wlkiswwlksupgr2	⊢ (𝐺 ∈ UPGraph → (𝑃 ∈ (WWalkS‘𝐺) → ∃𝑓 𝑓(1Walks‘𝐺)𝑃))

Distinct variable groups: 𝑓,𝐺 𝑃,𝑓

Proof of Theorem 1wlkiswwlksupgr2 Dummy variables 𝑖 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2610	. . 3 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
2		eqid 2610	. . 3 ⊢ (Edg‘𝐺) = (Edg‘𝐺)
3	1, 2	iswwlks 41039	. 2 ⊢ (𝑃 ∈ (WWalkS‘𝐺) ↔ (𝑃 ≠ ∅ ∧ 𝑃 ∈ Word (Vtx‘𝐺) ∧ ∀𝑖 ∈ (0..^((#‘𝑃) − 1)){(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ∈ (Edg‘𝐺)))
4		edgaval 25794	. . . . . . . . . . . . . 14 ⊢ (𝐺 ∈ UPGraph → (Edg‘𝐺) = ran (iEdg‘𝐺))
5	4	eleq2d 2673	. . . . . . . . . . . . 13 ⊢ (𝐺 ∈ UPGraph → ({(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ∈ (Edg‘𝐺) ↔ {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ∈ ran (iEdg‘𝐺)))
6	5	adantl 481	. . . . . . . . . . . 12 ⊢ (((𝑃 ≠ ∅ ∧ 𝑃 ∈ Word (Vtx‘𝐺)) ∧ 𝐺 ∈ UPGraph ) → ({(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ∈ (Edg‘𝐺) ↔ {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ∈ ran (iEdg‘𝐺)))
7		upgruhgr 25768	. . . . . . . . . . . . . . 15 ⊢ (𝐺 ∈ UPGraph → 𝐺 ∈ UHGraph )
8		eqid 2610	. . . . . . . . . . . . . . . 16 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺)
9	8	uhgrfun 25732	. . . . . . . . . . . . . . 15 ⊢ (𝐺 ∈ UHGraph → Fun (iEdg‘𝐺))
10	7, 9	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝐺 ∈ UPGraph → Fun (iEdg‘𝐺))
11	10	adantl 481	. . . . . . . . . . . . 13 ⊢ (((𝑃 ≠ ∅ ∧ 𝑃 ∈ Word (Vtx‘𝐺)) ∧ 𝐺 ∈ UPGraph ) → Fun (iEdg‘𝐺))
12		elrnrexdm 6271	. . . . . . . . . . . . . 14 ⊢ (Fun (iEdg‘𝐺) → ({(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ∈ ran (iEdg‘𝐺) → ∃𝑥 ∈ dom (iEdg‘𝐺){(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} = ((iEdg‘𝐺)‘𝑥)))
13		eqcom 2617	. . . . . . . . . . . . . . 15 ⊢ (((iEdg‘𝐺)‘𝑥) = {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ↔ {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} = ((iEdg‘𝐺)‘𝑥))
14	13	rexbii 3023	. . . . . . . . . . . . . 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))
15	12, 14	syl6ibr 241	. . . . . . . . . . . . 13 ⊢ (Fun (iEdg‘𝐺) → ({(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ∈ ran (iEdg‘𝐺) → ∃𝑥 ∈ dom (iEdg‘𝐺)((iEdg‘𝐺)‘𝑥) = {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))}))
16	11, 15	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))}))
17	6, 16	sylbid 229	. . . . . . . . . . 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))}))
18	17	ralimdv 2946	. . . . . . . . . 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))}))
19	18	ex 449	. . . . . . . . 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))})))
20	19	com23 84	. . . . . . . 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))})))
21	20	3impia 1253	. . . . . . 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))}))
22	21	impcom 445	. . . . . 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))})
23		ovex 6577	. . . . . . 7 ⊢ (0..^((#‘𝑃) − 1)) ∈ V
24		fvex 6113	. . . . . . . 8 ⊢ (iEdg‘𝐺) ∈ V
25	24	dmex 6991	. . . . . . 7 ⊢ dom (iEdg‘𝐺) ∈ V
26		fveq2 6103	. . . . . . . 8 ⊢ (𝑥 = (𝑓‘𝑖) → ((iEdg‘𝐺)‘𝑥) = ((iEdg‘𝐺)‘(𝑓‘𝑖)))
27	26	eqeq1d 2612	. . . . . . 7 ⊢ (𝑥 = (𝑓‘𝑖) → (((iEdg‘𝐺)‘𝑥) = {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ↔ ((iEdg‘𝐺)‘(𝑓‘𝑖)) = {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))}))
28	23, 25, 27	ac6 9185	. . . . . 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))}))
29	22, 28	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))}))
30		iswrdi 13164	. . . . . . . . . 10 ⊢ (𝑓:(0..^((#‘𝑃) − 1))⟶dom (iEdg‘𝐺) → 𝑓 ∈ Word dom (iEdg‘𝐺))
31	30	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))
32	31	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))
33		wrdfin 13178	. . . . . . . . . . . . . . . 16 ⊢ (𝑃 ∈ Word (Vtx‘𝐺) → 𝑃 ∈ Fin)
34		hashnncl 13018	. . . . . . . . . . . . . . . . 17 ⊢ (𝑃 ∈ Fin → ((#‘𝑃) ∈ ℕ ↔ 𝑃 ≠ ∅))
35	34	bicomd 212	. . . . . . . . . . . . . . . 16 ⊢ (𝑃 ∈ Fin → (𝑃 ≠ ∅ ↔ (#‘𝑃) ∈ ℕ))
36	33, 35	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝑃 ∈ Word (Vtx‘𝐺) → (𝑃 ≠ ∅ ↔ (#‘𝑃) ∈ ℕ))
37	36	biimpac 502	. . . . . . . . . . . . . 14 ⊢ ((𝑃 ≠ ∅ ∧ 𝑃 ∈ Word (Vtx‘𝐺)) → (#‘𝑃) ∈ ℕ)
38		wrdf 13165	. . . . . . . . . . . . . . . 16 ⊢ (𝑃 ∈ Word (Vtx‘𝐺) → 𝑃:(0..^(#‘𝑃))⟶(Vtx‘𝐺))
39		nnz 11276	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((#‘𝑃) ∈ ℕ → (#‘𝑃) ∈ ℤ)
40		fzoval 12340	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((#‘𝑃) ∈ ℤ → (0..^(#‘𝑃)) = (0...((#‘𝑃) − 1)))
41	39, 40	syl 17	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((#‘𝑃) ∈ ℕ → (0..^(#‘𝑃)) = (0...((#‘𝑃) − 1)))
42	41	adantr 480	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((#‘𝑃) ∈ ℕ ∧ 𝑓:(0..^((#‘𝑃) − 1))⟶dom (iEdg‘𝐺)) → (0..^(#‘𝑃)) = (0...((#‘𝑃) − 1)))
43		nnm1nn0 11211	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((#‘𝑃) ∈ ℕ → ((#‘𝑃) − 1) ∈ ℕ₀)
44		fnfzo0hash 13091	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((#‘𝑃) − 1) ∈ ℕ₀ ∧ 𝑓:(0..^((#‘𝑃) − 1))⟶dom (iEdg‘𝐺)) → (#‘𝑓) = ((#‘𝑃) − 1))
45	43, 44	sylan 487	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((#‘𝑃) ∈ ℕ ∧ 𝑓:(0..^((#‘𝑃) − 1))⟶dom (iEdg‘𝐺)) → (#‘𝑓) = ((#‘𝑃) − 1))
46	45	eqcomd 2616	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((#‘𝑃) ∈ ℕ ∧ 𝑓:(0..^((#‘𝑃) − 1))⟶dom (iEdg‘𝐺)) → ((#‘𝑃) − 1) = (#‘𝑓))
47	46	oveq2d 6565	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((#‘𝑃) ∈ ℕ ∧ 𝑓:(0..^((#‘𝑃) − 1))⟶dom (iEdg‘𝐺)) → (0...((#‘𝑃) − 1)) = (0...(#‘𝑓)))
48	42, 47	eqtrd 2644	. . . . . . . . . . . . . . . . . . 19 ⊢ (((#‘𝑃) ∈ ℕ ∧ 𝑓:(0..^((#‘𝑃) − 1))⟶dom (iEdg‘𝐺)) → (0..^(#‘𝑃)) = (0...(#‘𝑓)))
49	48	feq2d 5944	. . . . . . . . . . . . . . . . . 18 ⊢ (((#‘𝑃) ∈ ℕ ∧ 𝑓:(0..^((#‘𝑃) − 1))⟶dom (iEdg‘𝐺)) → (𝑃:(0..^(#‘𝑃))⟶(Vtx‘𝐺) ↔ 𝑃:(0...(#‘𝑓))⟶(Vtx‘𝐺)))
50	49	biimpcd 238	. . . . . . . . . . . . . . . . 17 ⊢ (𝑃:(0..^(#‘𝑃))⟶(Vtx‘𝐺) → (((#‘𝑃) ∈ ℕ ∧ 𝑓:(0..^((#‘𝑃) − 1))⟶dom (iEdg‘𝐺)) → 𝑃:(0...(#‘𝑓))⟶(Vtx‘𝐺)))
51	50	expd 451	. . . . . . . . . . . . . . . 16 ⊢ (𝑃:(0..^(#‘𝑃))⟶(Vtx‘𝐺) → ((#‘𝑃) ∈ ℕ → (𝑓:(0..^((#‘𝑃) − 1))⟶dom (iEdg‘𝐺) → 𝑃:(0...(#‘𝑓))⟶(Vtx‘𝐺))))
52	38, 51	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝑃 ∈ Word (Vtx‘𝐺) → ((#‘𝑃) ∈ ℕ → (𝑓:(0..^((#‘𝑃) − 1))⟶dom (iEdg‘𝐺) → 𝑃:(0...(#‘𝑓))⟶(Vtx‘𝐺))))
53	52	adantl 481	. . . . . . . . . . . . . 14 ⊢ ((𝑃 ≠ ∅ ∧ 𝑃 ∈ Word (Vtx‘𝐺)) → ((#‘𝑃) ∈ ℕ → (𝑓:(0..^((#‘𝑃) − 1))⟶dom (iEdg‘𝐺) → 𝑃:(0...(#‘𝑓))⟶(Vtx‘𝐺))))
54	37, 53	mpd 15	. . . . . . . . . . . . 13 ⊢ ((𝑃 ≠ ∅ ∧ 𝑃 ∈ Word (Vtx‘𝐺)) → (𝑓:(0..^((#‘𝑃) − 1))⟶dom (iEdg‘𝐺) → 𝑃:(0...(#‘𝑓))⟶(Vtx‘𝐺)))
55	54	3adant3 1074	. . . . . . . . . . . 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)))
56	55	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)))
57	56	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)))
58	57	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)))
59	58	impcom 445	. . . . . . . 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))
60		simpr 476	. . . . . . . . . 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))})
61	37, 45	sylan 487	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑃 ≠ ∅ ∧ 𝑃 ∈ Word (Vtx‘𝐺)) ∧ 𝑓:(0..^((#‘𝑃) − 1))⟶dom (iEdg‘𝐺)) → (#‘𝑓) = ((#‘𝑃) − 1))
62	61	oveq2d 6565	. . . . . . . . . . . . . . . 16 ⊢ (((𝑃 ≠ ∅ ∧ 𝑃 ∈ Word (Vtx‘𝐺)) ∧ 𝑓:(0..^((#‘𝑃) − 1))⟶dom (iEdg‘𝐺)) → (0..^(#‘𝑓)) = (0..^((#‘𝑃) − 1)))
63	62	ex 449	. . . . . . . . . . . . . . 15 ⊢ ((𝑃 ≠ ∅ ∧ 𝑃 ∈ Word (Vtx‘𝐺)) → (𝑓:(0..^((#‘𝑃) − 1))⟶dom (iEdg‘𝐺) → (0..^(#‘𝑓)) = (0..^((#‘𝑃) − 1))))
64	63	3adant3 1074	. . . . . . . . . . . . . 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))))
65	64	adantl 481	. . . . . . . . . . . . 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))))
66	65	imp 444	. . . . . . . . . . . 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)))
67	66	adantr 480	. . . . . . . . . . 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)))
68	67	raleqdv 3121	. . . . . . . . . 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))}))
69	60, 68	mpbird 246	. . . . . . . . 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))})
70	69	anasss 677	. . . . . . . 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))})
71	32, 59, 70	3jca 1235	. . . . . . 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))}))
72	71	ex 449	. . . . . 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))})))
73	72	eximdv 1833	. . . . 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))})))
74	29, 73	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))}))
75		simpl 472	. . . . . 6 ⊢ ((𝐺 ∈ UPGraph ∧ (𝑃 ≠ ∅ ∧ 𝑃 ∈ Word (Vtx‘𝐺) ∧ ∀𝑖 ∈ (0..^((#‘𝑃) − 1)){(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ∈ (Edg‘𝐺))) → 𝐺 ∈ UPGraph )
76		vex 3176	. . . . . . 7 ⊢ 𝑓 ∈ V
77	76	a1i 11	. . . . . 6 ⊢ ((𝐺 ∈ UPGraph ∧ (𝑃 ≠ ∅ ∧ 𝑃 ∈ Word (Vtx‘𝐺) ∧ ∀𝑖 ∈ (0..^((#‘𝑃) − 1)){(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ∈ (Edg‘𝐺))) → 𝑓 ∈ V)
78		simpr2 1061	. . . . . 6 ⊢ ((𝐺 ∈ UPGraph ∧ (𝑃 ≠ ∅ ∧ 𝑃 ∈ Word (Vtx‘𝐺) ∧ ∀𝑖 ∈ (0..^((#‘𝑃) − 1)){(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ∈ (Edg‘𝐺))) → 𝑃 ∈ Word (Vtx‘𝐺))
79	1, 8	upgriswlk 40849	. . . . . 6 ⊢ ((𝐺 ∈ UPGraph ∧ 𝑓 ∈ V ∧ 𝑃 ∈ Word (Vtx‘𝐺)) → (𝑓(1Walks‘𝐺)𝑃 ↔ (𝑓 ∈ Word dom (iEdg‘𝐺) ∧ 𝑃:(0...(#‘𝑓))⟶(Vtx‘𝐺) ∧ ∀𝑖 ∈ (0..^(#‘𝑓))((iEdg‘𝐺)‘(𝑓‘𝑖)) = {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))})))
80	75, 77, 78, 79	syl3anc 1318	. . . . 5 ⊢ ((𝐺 ∈ UPGraph ∧ (𝑃 ≠ ∅ ∧ 𝑃 ∈ Word (Vtx‘𝐺) ∧ ∀𝑖 ∈ (0..^((#‘𝑃) − 1)){(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ∈ (Edg‘𝐺))) → (𝑓(1Walks‘𝐺)𝑃 ↔ (𝑓 ∈ Word dom (iEdg‘𝐺) ∧ 𝑃:(0...(#‘𝑓))⟶(Vtx‘𝐺) ∧ ∀𝑖 ∈ (0..^(#‘𝑓))((iEdg‘𝐺)‘(𝑓‘𝑖)) = {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))})))
81	80	exbidv 1837	. . . 4 ⊢ ((𝐺 ∈ UPGraph ∧ (𝑃 ≠ ∅ ∧ 𝑃 ∈ Word (Vtx‘𝐺) ∧ ∀𝑖 ∈ (0..^((#‘𝑃) − 1)){(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ∈ (Edg‘𝐺))) → (∃𝑓 𝑓(1Walks‘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))})))
82	74, 81	mpbird 246	. . 3 ⊢ ((𝐺 ∈ UPGraph ∧ (𝑃 ≠ ∅ ∧ 𝑃 ∈ Word (Vtx‘𝐺) ∧ ∀𝑖 ∈ (0..^((#‘𝑃) − 1)){(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ∈ (Edg‘𝐺))) → ∃𝑓 𝑓(1Walks‘𝐺)𝑃)
83	82	ex 449	. 2 ⊢ (𝐺 ∈ UPGraph → ((𝑃 ≠ ∅ ∧ 𝑃 ∈ Word (Vtx‘𝐺) ∧ ∀𝑖 ∈ (0..^((#‘𝑃) − 1)){(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ∈ (Edg‘𝐺)) → ∃𝑓 𝑓(1Walks‘𝐺)𝑃))
84	3, 83	syl5bi 231	1 ⊢ (𝐺 ∈ UPGraph → (𝑃 ∈ (WWalkS‘𝐺) → ∃𝑓 𝑓(1Walks‘𝐺)𝑃))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 ∧ w3a 1031 = wceq 1475 ∃wex 1695 ∈ wcel 1977 ≠ wne 2780 ∀wral 2896 ∃wrex 2897 Vcvv 3173 ∅c0 3874 {cpr 4127 class class class wbr 4583 dom cdm 5038 ran crn 5039 Fun wfun 5798 ⟶wf 5800 ‘cfv 5804 (class class class)co 6549 Fincfn 7841 0cc0 9815 1c1 9816 + caddc 9818 − cmin 10145 ℕcn 10897 ℕ₀cn0 11169 ℤcz 11254 ...cfz 12197 ..^cfzo 12334 #chash 12979 Word cword 13146 Vtxcvtx 25673 iEdgciedg 25674 UHGraph cuhgr 25722 UPGraph cupgr 25747 Edgcedga 25792 1Walksc1wlks 40796 WWalkScwwlks 41028

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-ac2 9168 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-ifp 1007 df-3or 1032 df-3an 1033 df-tru 1478 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-rmo 2904 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-se 4998 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-isom 5813 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-2o 7448 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-ac 8822 df-cda 8873 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-uhgr 25724 df-upgr 25749 df-edga 25793 df-1wlks 40800 df-wlks 40801 df-wwlks 41033

This theorem is referenced by: 1wlkiswwlkupgr 41075 1wlklnwwlklnupgr2 41082

W3C validator