Theorem upgriswlkdc 16601

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

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

Proof of Theorem upgriswlkdc

Step	Hyp	Ref	Expression
1		upgriswlk.v	. . 3 ⊢ 𝑉 = (Vtx‘𝐺)
2		upgriswlk.i	. . 3 ⊢ 𝐼 = (iEdg‘𝐺)
3	1, 2	iswlkg 16570	. 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		ifpdc 992	. . . . . . . . 9 ⊢ (if-((𝑃‘𝑘) = (𝑃‘(𝑘 + 1)), (𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘)}, {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ⊆ (𝐼‘(𝐹‘𝑘))) → DECID (𝑃‘𝑘) = (𝑃‘(𝑘 + 1)))
5	4	adantl 277	. . . . . . . 8 ⊢ ((((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)))) → DECID (𝑃‘𝑘) = (𝑃‘(𝑘 + 1)))
6		df-ifp 991	. . . . . . . . . 10 ⊢ (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)))))
7		dfsn2 3723	. . . . . . . . . . . . . . . 16 ⊢ {(𝑃‘𝑘)} = {(𝑃‘𝑘), (𝑃‘𝑘)}
8		preq2 3789	. . . . . . . . . . . . . . . 16 ⊢ ((𝑃‘𝑘) = (𝑃‘(𝑘 + 1)) → {(𝑃‘𝑘), (𝑃‘𝑘)} = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))})
9	7, 8	eqtrid 2283	. . . . . . . . . . . . . . 15 ⊢ ((𝑃‘𝑘) = (𝑃‘(𝑘 + 1)) → {(𝑃‘𝑘)} = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))})
10	9	eqeq2d 2250	. . . . . . . . . . . . . 14 ⊢ ((𝑃‘𝑘) = (𝑃‘(𝑘 + 1)) → ((𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘)} ↔ (𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))}))
11	10	biimpa 296	. . . . . . . . . . . . 13 ⊢ (((𝑃‘𝑘) = (𝑃‘(𝑘 + 1)) ∧ (𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘)}) → (𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))})
12	11	a1d 22	. . . . . . . . . . . 12 ⊢ (((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))}))
13		eqid 2238	. . . . . . . . . . . . . . . . 17 ⊢ (Edg‘𝐺) = (Edg‘𝐺)
14	2, 13	upgredginwlk 16597	. . . . . . . . . . . . . . . 16 ⊢ ((𝐺 ∈ UPGraph ∧ 𝐹 ∈ Word dom 𝐼) → (𝑘 ∈ (0..^(♯‘𝐹)) → (𝐼‘(𝐹‘𝑘)) ∈ (Edg‘𝐺)))
15	14	adantrr 483	. . . . . . . . . . . . . . 15 ⊢ ((𝐺 ∈ UPGraph ∧ (𝐹 ∈ Word dom 𝐼 ∧ 𝑃:(0...(♯‘𝐹))⟶𝑉)) → (𝑘 ∈ (0..^(♯‘𝐹)) → (𝐼‘(𝐹‘𝑘)) ∈ (Edg‘𝐺)))
16	15	imp 124	. . . . . . . . . . . . . 14 ⊢ (((𝐺 ∈ UPGraph ∧ (𝐹 ∈ Word dom 𝐼 ∧ 𝑃:(0...(♯‘𝐹))⟶𝑉)) ∧ 𝑘 ∈ (0..^(♯‘𝐹))) → (𝐼‘(𝐹‘𝑘)) ∈ (Edg‘𝐺))
17		simp-4l 547	. . . . . . . . . . . . . . . . 17 ⊢ (((((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)
18		simplr 533	. . . . . . . . . . . . . . . . 17 ⊢ (((((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))
19		simprr 537	. . . . . . . . . . . . . . . . 17 ⊢ (((((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		simprr 537	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐺 ∈ UPGraph ∧ (𝐹 ∈ Word dom 𝐼 ∧ 𝑃:(0...(♯‘𝐹))⟶𝑉)) → 𝑃:(0...(♯‘𝐹))⟶𝑉)
21	20	ad5ant12 522	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((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:(0...(♯‘𝐹))⟶𝑉)
22		elfzofz 10570	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑘 ∈ (0..^(♯‘𝐹)) → 𝑘 ∈ (0...(♯‘𝐹)))
23	22	ad3antlr 497	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((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)))) → 𝑘 ∈ (0...(♯‘𝐹)))
24	21, 23	ffvelcdmd 5844	. . . . . . . . . . . . . . . . . 18 ⊢ (((((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)
25	24	elexd 2835	. . . . . . . . . . . . . . . . 17 ⊢ (((((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)
26		fzofzp1 10645	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑘 ∈ (0..^(♯‘𝐹)) → (𝑘 + 1) ∈ (0...(♯‘𝐹)))
27	26	ad3antlr 497	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((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)))) → (𝑘 + 1) ∈ (0...(♯‘𝐹)))
28	21, 27	ffvelcdmd 5844	. . . . . . . . . . . . . . . . . 18 ⊢ (((((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 + 1)) ∈ 𝑉)
29	28	elexd 2835	. . . . . . . . . . . . . . . . 17 ⊢ (((((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 + 1)) ∈ V)
30		neqne 2428	. . . . . . . . . . . . . . . . . 18 ⊢ (¬ (𝑃‘𝑘) = (𝑃‘(𝑘 + 1)) → (𝑃‘𝑘) ≠ (𝑃‘(𝑘 + 1)))
31	30	ad2antrl 494	. . . . . . . . . . . . . . . . 17 ⊢ (((((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)))
32	1, 13	upgredgpr 16390	. . . . . . . . . . . . . . . . 17 ⊢ (((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)))
33	17, 18, 19, 25, 29, 31, 32	syl33anc 1293	. . . . . . . . . . . . . . . 16 ⊢ (((((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)))
34	33	eqcomd 2244	. . . . . . . . . . . . . . 15 ⊢ (((((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))})
35	34	exp31 364	. . . . . . . . . . . . . 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)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))})))
36	16, 35	mpd 13	. . . . . . . . . . . . 13 ⊢ (((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))}))
37	36	com12 30	. . . . . . . . . . . 12 ⊢ ((¬ (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))}))
38	12, 37	jaoi 728	. . . . . . . . . . 11 ⊢ ((((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))}))
39	38	com12 30	. . . . . . . . . 10 ⊢ (((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))}))
40	6, 39	biimtrid 152	. . . . . . . . 9 ⊢ (((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))}))
41	40	imp 124	. . . . . . . 8 ⊢ ((((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))})
42	5, 41	jca 306	. . . . . . 7 ⊢ ((((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)))) → (DECID (𝑃‘𝑘) = (𝑃‘(𝑘 + 1)) ∧ (𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))}))
43	42	ex 115	. . . . . 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))) → (DECID (𝑃‘𝑘) = (𝑃‘(𝑘 + 1)) ∧ (𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))})))
44		ifpprsnssdc 3820	. . . . . . 7 ⊢ (((𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ∧ DECID (𝑃‘𝑘) = (𝑃‘(𝑘 + 1))) → if-((𝑃‘𝑘) = (𝑃‘(𝑘 + 1)), (𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘)}, {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ⊆ (𝐼‘(𝐹‘𝑘))))
45	44	ancoms 268	. . . . . 6 ⊢ ((DECID (𝑃‘𝑘) = (𝑃‘(𝑘 + 1)) ∧ (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))))
46	43, 45	impbid1 142	. . . . 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))) ↔ (DECID (𝑃‘𝑘) = (𝑃‘(𝑘 + 1)) ∧ (𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))})))
47	46	ralbidva 2546	. . . 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))(DECID (𝑃‘𝑘) = (𝑃‘(𝑘 + 1)) ∧ (𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))})))
48	47	pm5.32da 456	. . 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))(DECID (𝑃‘𝑘) = (𝑃‘(𝑘 + 1)) ∧ (𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))}))))
49		df-3an 1011	. . 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)))))
50		df-3an 1011	. . 3 ⊢ ((𝐹 ∈ Word dom 𝐼 ∧ 𝑃:(0...(♯‘𝐹))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(♯‘𝐹))(DECID (𝑃‘𝑘) = (𝑃‘(𝑘 + 1)) ∧ (𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))})) ↔ ((𝐹 ∈ Word dom 𝐼 ∧ 𝑃:(0...(♯‘𝐹))⟶𝑉) ∧ ∀𝑘 ∈ (0..^(♯‘𝐹))(DECID (𝑃‘𝑘) = (𝑃‘(𝑘 + 1)) ∧ (𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))})))
51	48, 49, 50	3bitr4g 223	. 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))(DECID (𝑃‘𝑘) = (𝑃‘(𝑘 + 1)) ∧ (𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))}))))
52	3, 51	bitrd 188	1 ⊢ (𝐺 ∈ UPGraph → (𝐹(Walks‘𝐺)𝑃 ↔ (𝐹 ∈ Word dom 𝐼 ∧ 𝑃:(0...(♯‘𝐹))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(♯‘𝐹))(DECID (𝑃‘𝑘) = (𝑃‘(𝑘 + 1)) ∧ (𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))}))))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 104   ↔ wb 105   ∨ wo 720  DECID wdc 846  if-wif 990   ∧ w3a 1009   = wceq 1402   ∈ wcel 2209   ≠ wne 2420  ∀wral 2528  Vcvv 2821   ⊆ wss 3220  {csn 3709  {cpr 3710   class class class wbr 4130  dom cdm 4774  ⟶wf 5373  ‘cfv 5377  (class class class)co 6085  0cc0 8179  1c1 8180   + caddc 8182  ...cfz 10411  ..^cfzo 10549  ♯chash 11214  Word cword 11304  Vtxcvtx 16253  iEdgciedg 16254  Edgcedg 16298  UPGraphcupgr 16332  Walkscwlks 16558

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4246  ax-sep 4249  ax-nul 4259  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-iinf 4735  ax-cnex 8270  ax-resscn 8271  ax-1cn 8272  ax-1re 8273  ax-icn 8274  ax-addcl 8275  ax-addrcl 8276  ax-mulcl 8277  ax-addcom 8279  ax-mulcom 8280  ax-addass 8281  ax-mulass 8282  ax-distr 8283  ax-i2m1 8284  ax-0lt1 8285  ax-1rid 8286  ax-0id 8287  ax-rnegex 8288  ax-cnre 8290  ax-pre-ltirr 8291  ax-pre-ltwlin 8292  ax-pre-lttrn 8293  ax-pre-apti 8294  ax-pre-ltadd 8295

This proof depends on definitions:  df-bi 117  df-dc 847  df-ifp 991  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3639  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4131  df-opab 4193  df-mpt 4194  df-tr 4230  df-id 4438  df-iord 4511  df-on 4513  df-ilim 4514  df-suc 4516  df-iom 4738  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-riota 6038  df-ov 6088  df-oprab 6089  df-mpo 6090  df-1st 6374  df-2nd 6375  df-recs 6576  df-frec 6662  df-1o 6687  df-2o 6688  df-er 6807  df-map 6924  df-en 7023  df-dom 7024  df-fin 7025  df-pnf 8362  df-mnf 8363  df-xr 8364  df-ltxr 8365  df-le 8366  df-sub 8499  df-neg 8500  df-inn 9305  df-2 9363  df-3 9364  df-4 9365  df-5 9366  df-6 9367  df-7 9368  df-8 9369  df-9 9370  df-n0 9564  df-z 9645  df-dec 9778  df-uz 9922  df-fz 10412  df-fzo 10550  df-ihash 11215  df-word 11305  df-ndx 13355  df-slot 13356  df-base 13358  df-edgf 16246  df-vtx 16255  df-iedg 16256  df-edg 16299  df-uhgrm 16310  df-upgren 16334  df-wlks 16559

This theorem is used by:  upgrwlkedg  16602  upgrwlkcompim  16603  upgrwlkvtxedg  16605  upgr2wlkdc  16618