Theorem upgriswlk 29322

Description: Properties of a pair of functions to be a walk in a pseudograph. (Contributed by AV, 2-Jan-2021.) (Revised by AV, 28-Oct-2021.)

Hypotheses

Ref	Expression
upgriswlk.v	⊢ 𝑉 = (Vtx‘𝐺)
upgriswlk.i	⊢ 𝐼 = (iEdg‘𝐺)

Assertion

Ref	Expression
upgriswlk	⊢ (𝐺 ∈ UPGraph → (𝐹(Walks‘𝐺)𝑃 ↔ (𝐹 ∈ Word dom 𝐼 ∧ 𝑃:(0...(♯‘𝐹))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(♯‘𝐹))(𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))})))

Distinct variable groups:   𝑘,𝐺   𝑘,𝐹   𝑘,𝐼   𝑃,𝑘   𝑘,𝑉

Proof of Theorem upgriswlk

Step	Hyp	Ref	Expression
1		upgriswlk.v	. . 3 ⊢ 𝑉 = (Vtx‘𝐺)
2		upgriswlk.i	. . 3 ⊢ 𝐼 = (iEdg‘𝐺)
3	1, 2	iswlkg 29294	. 2 ⊢ (𝐺 ∈ UPGraph → (𝐹(Walks‘𝐺)𝑃 ↔ (𝐹 ∈ Word dom 𝐼 ∧ 𝑃:(0...(♯‘𝐹))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(♯‘𝐹))if-((𝑃‘𝑘) = (𝑃‘(𝑘 + 1)), (𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘)}, {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ⊆ (𝐼‘(𝐹‘𝑘))))))
4		df-ifp 1060	. . . . . . 7 ⊢ (if-((𝑃‘𝑘) = (𝑃‘(𝑘 + 1)), (𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘)}, {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ⊆ (𝐼‘(𝐹‘𝑘))) ↔ (((𝑃‘𝑘) = (𝑃‘(𝑘 + 1)) ∧ (𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘)}) ∨ (¬ (𝑃‘𝑘) = (𝑃‘(𝑘 + 1)) ∧ {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ⊆ (𝐼‘(𝐹‘𝑘)))))
5		dfsn2 4633	. . . . . . . . . . . . 13 ⊢ {(𝑃‘𝑘)} = {(𝑃‘𝑘), (𝑃‘𝑘)}
6		preq2 4730	. . . . . . . . . . . . 13 ⊢ ((𝑃‘𝑘) = (𝑃‘(𝑘 + 1)) → {(𝑃‘𝑘), (𝑃‘𝑘)} = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))})
7	5, 6	eqtrid 2776	. . . . . . . . . . . 12 ⊢ ((𝑃‘𝑘) = (𝑃‘(𝑘 + 1)) → {(𝑃‘𝑘)} = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))})
8	7	eqeq2d 2735	. . . . . . . . . . 11 ⊢ ((𝑃‘𝑘) = (𝑃‘(𝑘 + 1)) → ((𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘)} ↔ (𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))}))
9	8	biimpa 476	. . . . . . . . . 10 ⊢ (((𝑃‘𝑘) = (𝑃‘(𝑘 + 1)) ∧ (𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘)}) → (𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))})
10	9	a1d 25	. . . . . . . . 9 ⊢ (((𝑃‘𝑘) = (𝑃‘(𝑘 + 1)) ∧ (𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘)}) → (((𝐺 ∈ UPGraph ∧ (𝐹 ∈ Word dom 𝐼 ∧ 𝑃:(0...(♯‘𝐹))⟶𝑉)) ∧ 𝑘 ∈ (0..^(♯‘𝐹))) → (𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))}))
11		eqid 2724	. . . . . . . . . . . . . 14 ⊢ (Edg‘𝐺) = (Edg‘𝐺)
12	2, 11	upgredginwlk 29317	. . . . . . . . . . . . 13 ⊢ ((𝐺 ∈ UPGraph ∧ 𝐹 ∈ Word dom 𝐼) → (𝑘 ∈ (0..^(♯‘𝐹)) → (𝐼‘(𝐹‘𝑘)) ∈ (Edg‘𝐺)))
13	12	adantrr 714	. . . . . . . . . . . 12 ⊢ ((𝐺 ∈ UPGraph ∧ (𝐹 ∈ Word dom 𝐼 ∧ 𝑃:(0...(♯‘𝐹))⟶𝑉)) → (𝑘 ∈ (0..^(♯‘𝐹)) → (𝐼‘(𝐹‘𝑘)) ∈ (Edg‘𝐺)))
14	13	imp 406	. . . . . . . . . . 11 ⊢ (((𝐺 ∈ UPGraph ∧ (𝐹 ∈ Word dom 𝐼 ∧ 𝑃:(0...(♯‘𝐹))⟶𝑉)) ∧ 𝑘 ∈ (0..^(♯‘𝐹))) → (𝐼‘(𝐹‘𝑘)) ∈ (Edg‘𝐺))
15		simp-4l 780	. . . . . . . . . . . . . 14 ⊢ (((((𝐺 ∈ UPGraph ∧ (𝐹 ∈ Word dom 𝐼 ∧ 𝑃:(0...(♯‘𝐹))⟶𝑉)) ∧ 𝑘 ∈ (0..^(♯‘𝐹))) ∧ (𝐼‘(𝐹‘𝑘)) ∈ (Edg‘𝐺)) ∧ (¬ (𝑃‘𝑘) = (𝑃‘(𝑘 + 1)) ∧ {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ⊆ (𝐼‘(𝐹‘𝑘)))) → 𝐺 ∈ UPGraph)
16		simpr 484	. . . . . . . . . . . . . . 15 ⊢ ((((𝐺 ∈ UPGraph ∧ (𝐹 ∈ Word dom 𝐼 ∧ 𝑃:(0...(♯‘𝐹))⟶𝑉)) ∧ 𝑘 ∈ (0..^(♯‘𝐹))) ∧ (𝐼‘(𝐹‘𝑘)) ∈ (Edg‘𝐺)) → (𝐼‘(𝐹‘𝑘)) ∈ (Edg‘𝐺))
17	16	adantr 480	. . . . . . . . . . . . . 14 ⊢ (((((𝐺 ∈ UPGraph ∧ (𝐹 ∈ Word dom 𝐼 ∧ 𝑃:(0...(♯‘𝐹))⟶𝑉)) ∧ 𝑘 ∈ (0..^(♯‘𝐹))) ∧ (𝐼‘(𝐹‘𝑘)) ∈ (Edg‘𝐺)) ∧ (¬ (𝑃‘𝑘) = (𝑃‘(𝑘 + 1)) ∧ {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ⊆ (𝐼‘(𝐹‘𝑘)))) → (𝐼‘(𝐹‘𝑘)) ∈ (Edg‘𝐺))
18		simpr 484	. . . . . . . . . . . . . . 15 ⊢ ((¬ (𝑃‘𝑘) = (𝑃‘(𝑘 + 1)) ∧ {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ⊆ (𝐼‘(𝐹‘𝑘))) → {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ⊆ (𝐼‘(𝐹‘𝑘)))
19	18	adantl 481	. . . . . . . . . . . . . 14 ⊢ (((((𝐺 ∈ UPGraph ∧ (𝐹 ∈ Word dom 𝐼 ∧ 𝑃:(0...(♯‘𝐹))⟶𝑉)) ∧ 𝑘 ∈ (0..^(♯‘𝐹))) ∧ (𝐼‘(𝐹‘𝑘)) ∈ (Edg‘𝐺)) ∧ (¬ (𝑃‘𝑘) = (𝑃‘(𝑘 + 1)) ∧ {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ⊆ (𝐼‘(𝐹‘𝑘)))) → {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ⊆ (𝐼‘(𝐹‘𝑘)))
20		fvexd 6896	. . . . . . . . . . . . . . . . 17 ⊢ (¬ (𝑃‘𝑘) = (𝑃‘(𝑘 + 1)) → (𝑃‘𝑘) ∈ V)
21		fvexd 6896	. . . . . . . . . . . . . . . . 17 ⊢ (¬ (𝑃‘𝑘) = (𝑃‘(𝑘 + 1)) → (𝑃‘(𝑘 + 1)) ∈ V)
22		neqne 2940	. . . . . . . . . . . . . . . . 17 ⊢ (¬ (𝑃‘𝑘) = (𝑃‘(𝑘 + 1)) → (𝑃‘𝑘) ≠ (𝑃‘(𝑘 + 1)))
23	20, 21, 22	3jca 1125	. . . . . . . . . . . . . . . 16 ⊢ (¬ (𝑃‘𝑘) = (𝑃‘(𝑘 + 1)) → ((𝑃‘𝑘) ∈ V ∧ (𝑃‘(𝑘 + 1)) ∈ V ∧ (𝑃‘𝑘) ≠ (𝑃‘(𝑘 + 1))))
24	23	adantr 480	. . . . . . . . . . . . . . 15 ⊢ ((¬ (𝑃‘𝑘) = (𝑃‘(𝑘 + 1)) ∧ {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ⊆ (𝐼‘(𝐹‘𝑘))) → ((𝑃‘𝑘) ∈ V ∧ (𝑃‘(𝑘 + 1)) ∈ V ∧ (𝑃‘𝑘) ≠ (𝑃‘(𝑘 + 1))))
25	24	adantl 481	. . . . . . . . . . . . . 14 ⊢ (((((𝐺 ∈ UPGraph ∧ (𝐹 ∈ Word dom 𝐼 ∧ 𝑃:(0...(♯‘𝐹))⟶𝑉)) ∧ 𝑘 ∈ (0..^(♯‘𝐹))) ∧ (𝐼‘(𝐹‘𝑘)) ∈ (Edg‘𝐺)) ∧ (¬ (𝑃‘𝑘) = (𝑃‘(𝑘 + 1)) ∧ {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ⊆ (𝐼‘(𝐹‘𝑘)))) → ((𝑃‘𝑘) ∈ V ∧ (𝑃‘(𝑘 + 1)) ∈ V ∧ (𝑃‘𝑘) ≠ (𝑃‘(𝑘 + 1))))
26	1, 11	upgredgpr 28826	. . . . . . . . . . . . . 14 ⊢ (((𝐺 ∈ UPGraph ∧ (𝐼‘(𝐹‘𝑘)) ∈ (Edg‘𝐺) ∧ {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ⊆ (𝐼‘(𝐹‘𝑘))) ∧ ((𝑃‘𝑘) ∈ V ∧ (𝑃‘(𝑘 + 1)) ∈ V ∧ (𝑃‘𝑘) ≠ (𝑃‘(𝑘 + 1)))) → {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} = (𝐼‘(𝐹‘𝑘)))
27	15, 17, 19, 25, 26	syl31anc 1370	. . . . . . . . . . . . 13 ⊢ (((((𝐺 ∈ UPGraph ∧ (𝐹 ∈ Word dom 𝐼 ∧ 𝑃:(0...(♯‘𝐹))⟶𝑉)) ∧ 𝑘 ∈ (0..^(♯‘𝐹))) ∧ (𝐼‘(𝐹‘𝑘)) ∈ (Edg‘𝐺)) ∧ (¬ (𝑃‘𝑘) = (𝑃‘(𝑘 + 1)) ∧ {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ⊆ (𝐼‘(𝐹‘𝑘)))) → {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} = (𝐼‘(𝐹‘𝑘)))
28	27	eqcomd 2730	. . . . . . . . . . . 12 ⊢ (((((𝐺 ∈ UPGraph ∧ (𝐹 ∈ Word dom 𝐼 ∧ 𝑃:(0...(♯‘𝐹))⟶𝑉)) ∧ 𝑘 ∈ (0..^(♯‘𝐹))) ∧ (𝐼‘(𝐹‘𝑘)) ∈ (Edg‘𝐺)) ∧ (¬ (𝑃‘𝑘) = (𝑃‘(𝑘 + 1)) ∧ {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ⊆ (𝐼‘(𝐹‘𝑘)))) → (𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))})
29	28	exp31 419	. . . . . . . . . . 11 ⊢ (((𝐺 ∈ UPGraph ∧ (𝐹 ∈ Word dom 𝐼 ∧ 𝑃:(0...(♯‘𝐹))⟶𝑉)) ∧ 𝑘 ∈ (0..^(♯‘𝐹))) → ((𝐼‘(𝐹‘𝑘)) ∈ (Edg‘𝐺) → ((¬ (𝑃‘𝑘) = (𝑃‘(𝑘 + 1)) ∧ {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ⊆ (𝐼‘(𝐹‘𝑘))) → (𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))})))
30	14, 29	mpd 15	. . . . . . . . . 10 ⊢ (((𝐺 ∈ UPGraph ∧ (𝐹 ∈ Word dom 𝐼 ∧ 𝑃:(0...(♯‘𝐹))⟶𝑉)) ∧ 𝑘 ∈ (0..^(♯‘𝐹))) → ((¬ (𝑃‘𝑘) = (𝑃‘(𝑘 + 1)) ∧ {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ⊆ (𝐼‘(𝐹‘𝑘))) → (𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))}))
31	30	com12 32	. . . . . . . . 9 ⊢ ((¬ (𝑃‘𝑘) = (𝑃‘(𝑘 + 1)) ∧ {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ⊆ (𝐼‘(𝐹‘𝑘))) → (((𝐺 ∈ UPGraph ∧ (𝐹 ∈ Word dom 𝐼 ∧ 𝑃:(0...(♯‘𝐹))⟶𝑉)) ∧ 𝑘 ∈ (0..^(♯‘𝐹))) → (𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))}))
32	10, 31	jaoi 854	. . . . . . . 8 ⊢ ((((𝑃‘𝑘) = (𝑃‘(𝑘 + 1)) ∧ (𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘)}) ∨ (¬ (𝑃‘𝑘) = (𝑃‘(𝑘 + 1)) ∧ {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ⊆ (𝐼‘(𝐹‘𝑘)))) → (((𝐺 ∈ UPGraph ∧ (𝐹 ∈ Word dom 𝐼 ∧ 𝑃:(0...(♯‘𝐹))⟶𝑉)) ∧ 𝑘 ∈ (0..^(♯‘𝐹))) → (𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))}))
33	32	com12 32	. . . . . . 7 ⊢ (((𝐺 ∈ UPGraph ∧ (𝐹 ∈ Word dom 𝐼 ∧ 𝑃:(0...(♯‘𝐹))⟶𝑉)) ∧ 𝑘 ∈ (0..^(♯‘𝐹))) → ((((𝑃‘𝑘) = (𝑃‘(𝑘 + 1)) ∧ (𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘)}) ∨ (¬ (𝑃‘𝑘) = (𝑃‘(𝑘 + 1)) ∧ {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ⊆ (𝐼‘(𝐹‘𝑘)))) → (𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))}))
34	4, 33	biimtrid 241	. . . . . 6 ⊢ (((𝐺 ∈ UPGraph ∧ (𝐹 ∈ Word dom 𝐼 ∧ 𝑃:(0...(♯‘𝐹))⟶𝑉)) ∧ 𝑘 ∈ (0..^(♯‘𝐹))) → (if-((𝑃‘𝑘) = (𝑃‘(𝑘 + 1)), (𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘)}, {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ⊆ (𝐼‘(𝐹‘𝑘))) → (𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))}))
35		ifpprsnss 4760	. . . . . 6 ⊢ ((𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} → if-((𝑃‘𝑘) = (𝑃‘(𝑘 + 1)), (𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘)}, {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ⊆ (𝐼‘(𝐹‘𝑘))))
36	34, 35	impbid1 224	. . . . 5 ⊢ (((𝐺 ∈ UPGraph ∧ (𝐹 ∈ Word dom 𝐼 ∧ 𝑃:(0...(♯‘𝐹))⟶𝑉)) ∧ 𝑘 ∈ (0..^(♯‘𝐹))) → (if-((𝑃‘𝑘) = (𝑃‘(𝑘 + 1)), (𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘)}, {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ⊆ (𝐼‘(𝐹‘𝑘))) ↔ (𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))}))
37	36	ralbidva 3167	. . . 4 ⊢ ((𝐺 ∈ UPGraph ∧ (𝐹 ∈ Word dom 𝐼 ∧ 𝑃:(0...(♯‘𝐹))⟶𝑉)) → (∀𝑘 ∈ (0..^(♯‘𝐹))if-((𝑃‘𝑘) = (𝑃‘(𝑘 + 1)), (𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘)}, {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ⊆ (𝐼‘(𝐹‘𝑘))) ↔ ∀𝑘 ∈ (0..^(♯‘𝐹))(𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))}))
38	37	pm5.32da 578	. . 3 ⊢ (𝐺 ∈ UPGraph → (((𝐹 ∈ Word dom 𝐼 ∧ 𝑃:(0...(♯‘𝐹))⟶𝑉) ∧ ∀𝑘 ∈ (0..^(♯‘𝐹))if-((𝑃‘𝑘) = (𝑃‘(𝑘 + 1)), (𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘)}, {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ⊆ (𝐼‘(𝐹‘𝑘)))) ↔ ((𝐹 ∈ Word dom 𝐼 ∧ 𝑃:(0...(♯‘𝐹))⟶𝑉) ∧ ∀𝑘 ∈ (0..^(♯‘𝐹))(𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))})))
39		df-3an 1086	. . 3 ⊢ ((𝐹 ∈ Word dom 𝐼 ∧ 𝑃:(0...(♯‘𝐹))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(♯‘𝐹))if-((𝑃‘𝑘) = (𝑃‘(𝑘 + 1)), (𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘)}, {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ⊆ (𝐼‘(𝐹‘𝑘)))) ↔ ((𝐹 ∈ Word dom 𝐼 ∧ 𝑃:(0...(♯‘𝐹))⟶𝑉) ∧ ∀𝑘 ∈ (0..^(♯‘𝐹))if-((𝑃‘𝑘) = (𝑃‘(𝑘 + 1)), (𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘)}, {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ⊆ (𝐼‘(𝐹‘𝑘)))))
40		df-3an 1086	. . 3 ⊢ ((𝐹 ∈ Word dom 𝐼 ∧ 𝑃:(0...(♯‘𝐹))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(♯‘𝐹))(𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))}) ↔ ((𝐹 ∈ Word dom 𝐼 ∧ 𝑃:(0...(♯‘𝐹))⟶𝑉) ∧ ∀𝑘 ∈ (0..^(♯‘𝐹))(𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))}))
41	38, 39, 40	3bitr4g 314	. 2 ⊢ (𝐺 ∈ UPGraph → ((𝐹 ∈ Word dom 𝐼 ∧ 𝑃:(0...(♯‘𝐹))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(♯‘𝐹))if-((𝑃‘𝑘) = (𝑃‘(𝑘 + 1)), (𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘)}, {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ⊆ (𝐼‘(𝐹‘𝑘)))) ↔ (𝐹 ∈ Word dom 𝐼 ∧ 𝑃:(0...(♯‘𝐹))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(♯‘𝐹))(𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))})))
42	3, 41	bitrd 279	1 ⊢ (𝐺 ∈ UPGraph → (𝐹(Walks‘𝐺)𝑃 ↔ (𝐹 ∈ Word dom 𝐼 ∧ 𝑃:(0...(♯‘𝐹))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(♯‘𝐹))(𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))})))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 395   ∨ wo 844  if-wif 1059   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098   ≠ wne 2932  ∀wral 3053  Vcvv 3466   ⊆ wss 3940  {csn 4620  {cpr 4622   class class class wbr 5138  dom cdm 5666  ⟶wf 6529  ‘cfv 6533  (class class class)co 7401  0cc0 11105  1c1 11106   + caddc 11108  ...cfz 13480  ..^cfzo 13623  ♯chash 14286  Word cword 14460  Vtxcvtx 28680  iEdgciedg 28681  Edgcedg 28731  UPGraphcupgr 28764  Walkscwlks 29277

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-rep 5275  ax-sep 5289  ax-nul 5296  ax-pow 5353  ax-pr 5417  ax-un 7718  ax-cnex 11161  ax-resscn 11162  ax-1cn 11163  ax-icn 11164  ax-addcl 11165  ax-addrcl 11166  ax-mulcl 11167  ax-mulrcl 11168  ax-mulcom 11169  ax-addass 11170  ax-mulass 11171  ax-distr 11172  ax-i2m1 11173  ax-1ne0 11174  ax-1rid 11175  ax-rnegex 11176  ax-rrecex 11177  ax-cnre 11178  ax-pre-lttri 11179  ax-pre-lttrn 11180  ax-pre-ltadd 11181  ax-pre-mulgt0 11182

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-ifp 1060  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-nel 3039  df-ral 3054  df-rex 3063  df-reu 3369  df-rab 3425  df-v 3468  df-sbc 3770  df-csb 3886  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-pss 3959  df-nul 4315  df-if 4521  df-pw 4596  df-sn 4621  df-pr 4623  df-op 4627  df-uni 4900  df-int 4941  df-iun 4989  df-br 5139  df-opab 5201  df-mpt 5222  df-tr 5256  df-id 5564  df-eprel 5570  df-po 5578  df-so 5579  df-fr 5621  df-we 5623  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-pred 6290  df-ord 6357  df-on 6358  df-lim 6359  df-suc 6360  df-iota 6485  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-riota 7357  df-ov 7404  df-oprab 7405  df-mpo 7406  df-om 7849  df-1st 7968  df-2nd 7969  df-frecs 8261  df-wrecs 8292  df-recs 8366  df-rdg 8405  df-1o 8461  df-2o 8462  df-oadd 8465  df-er 8698  df-map 8817  df-pm 8818  df-en 8935  df-dom 8936  df-sdom 8937  df-fin 8938  df-dju 9891  df-card 9929  df-pnf 11246  df-mnf 11247  df-xr 11248  df-ltxr 11249  df-le 11250  df-sub 11442  df-neg 11443  df-nn 12209  df-2 12271  df-n0 12469  df-xnn0 12541  df-z 12555  df-uz 12819  df-fz 13481  df-fzo 13624  df-hash 14287  df-word 14461  df-edg 28732  df-uhgr 28742  df-upgr 28766  df-wlks 29280

This theorem is referenced by:  upgrwlkedg  29323  upgrwlkcompim  29324  upgrwlkvtxedg  29326  upgr2wlk  29349  upgrtrls  29382  upgristrl  29383  upgrwlkdvde  29418  usgr2wlkneq  29437  isclwlkupgr  29459  uspgrn2crct  29486  wlkiswwlks1  29545  wlkiswwlks2  29553  wlkiswwlksupgr2  29555  wlk2v2e  29834  upgriseupth  29884