Theorem upgrwlkupwlk 48125

Description: In a pseudograph, a walk is a simple walk. (Contributed by AV, 30-Dec-2020.) (Proof shortened by AV, 2-Jan-2021.)

Assertion

Ref	Expression
upgrwlkupwlk	⊢ ((𝐺 ∈ UPGraph ∧ 𝐹(Walks‘𝐺)𝑃) → 𝐹(UPWalks‘𝐺)𝑃)

Proof of Theorem upgrwlkupwlk

Dummy variable 𝑘 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		wlkv 29576	. . 3 ⊢ (𝐹(Walks‘𝐺)𝑃 → (𝐺 ∈ V ∧ 𝐹 ∈ V ∧ 𝑃 ∈ V))
2		eqid 2729	. . . . . . . . 9 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
3		eqid 2729	. . . . . . . . 9 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺)
4	2, 3	iswlk 29574	. . . . . . . 8 ⊢ ((𝐺 ∈ V ∧ 𝐹 ∈ V ∧ 𝑃 ∈ V) → (𝐹(Walks‘𝐺)𝑃 ↔ (𝐹 ∈ Word dom (iEdg‘𝐺) ∧ 𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺) ∧ ∀𝑘 ∈ (0..^(♯‘𝐹))if-((𝑃‘𝑘) = (𝑃‘(𝑘 + 1)), ((iEdg‘𝐺)‘(𝐹‘𝑘)) = {(𝑃‘𝑘)}, {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ⊆ ((iEdg‘𝐺)‘(𝐹‘𝑘))))))
5		simpr1 1195	. . . . . . . . . . 11 ⊢ ((((𝐺 ∈ V ∧ 𝐹 ∈ V ∧ 𝑃 ∈ V) ∧ 𝐺 ∈ UPGraph) ∧ (𝐹 ∈ Word dom (iEdg‘𝐺) ∧ 𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺) ∧ ∀𝑘 ∈ (0..^(♯‘𝐹))if-((𝑃‘𝑘) = (𝑃‘(𝑘 + 1)), ((iEdg‘𝐺)‘(𝐹‘𝑘)) = {(𝑃‘𝑘)}, {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ⊆ ((iEdg‘𝐺)‘(𝐹‘𝑘))))) → 𝐹 ∈ Word dom (iEdg‘𝐺))
6		simpr2 1196	. . . . . . . . . . 11 ⊢ ((((𝐺 ∈ V ∧ 𝐹 ∈ V ∧ 𝑃 ∈ V) ∧ 𝐺 ∈ UPGraph) ∧ (𝐹 ∈ Word dom (iEdg‘𝐺) ∧ 𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺) ∧ ∀𝑘 ∈ (0..^(♯‘𝐹))if-((𝑃‘𝑘) = (𝑃‘(𝑘 + 1)), ((iEdg‘𝐺)‘(𝐹‘𝑘)) = {(𝑃‘𝑘)}, {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ⊆ ((iEdg‘𝐺)‘(𝐹‘𝑘))))) → 𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺))
7		df-ifp 1063	. . . . . . . . . . . . . . . . 17 ⊢ (if-((𝑃‘𝑘) = (𝑃‘(𝑘 + 1)), ((iEdg‘𝐺)‘(𝐹‘𝑘)) = {(𝑃‘𝑘)}, {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ⊆ ((iEdg‘𝐺)‘(𝐹‘𝑘))) ↔ (((𝑃‘𝑘) = (𝑃‘(𝑘 + 1)) ∧ ((iEdg‘𝐺)‘(𝐹‘𝑘)) = {(𝑃‘𝑘)}) ∨ (¬ (𝑃‘𝑘) = (𝑃‘(𝑘 + 1)) ∧ {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ⊆ ((iEdg‘𝐺)‘(𝐹‘𝑘)))))
8		dfsn2 4592	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ {(𝑃‘𝑘)} = {(𝑃‘𝑘), (𝑃‘𝑘)}
9		preq2 4688	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑃‘𝑘) = (𝑃‘(𝑘 + 1)) → {(𝑃‘𝑘), (𝑃‘𝑘)} = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))})
10	8, 9	eqtrid 2776	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑃‘𝑘) = (𝑃‘(𝑘 + 1)) → {(𝑃‘𝑘)} = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))})
11	10	eqeq2d 2740	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑃‘𝑘) = (𝑃‘(𝑘 + 1)) → (((iEdg‘𝐺)‘(𝐹‘𝑘)) = {(𝑃‘𝑘)} ↔ ((iEdg‘𝐺)‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))}))
12	11	biimpa 476	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑃‘𝑘) = (𝑃‘(𝑘 + 1)) ∧ ((iEdg‘𝐺)‘(𝐹‘𝑘)) = {(𝑃‘𝑘)}) → ((iEdg‘𝐺)‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))})
13	12	a1d 25	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑃‘𝑘) = (𝑃‘(𝑘 + 1)) ∧ ((iEdg‘𝐺)‘(𝐹‘𝑘)) = {(𝑃‘𝑘)}) → ((((𝐹 ∈ Word dom (iEdg‘𝐺) ∧ 𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺)) ∧ ((𝐺 ∈ V ∧ 𝐹 ∈ V ∧ 𝑃 ∈ V) ∧ 𝐺 ∈ UPGraph)) ∧ 𝑘 ∈ (0..^(♯‘𝐹))) → ((iEdg‘𝐺)‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))}))
14		simpr 484	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝐺 ∈ V ∧ 𝐹 ∈ V ∧ 𝑃 ∈ V) ∧ 𝐺 ∈ UPGraph) → 𝐺 ∈ UPGraph)
15		simpl 482	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝐹 ∈ Word dom (iEdg‘𝐺) ∧ 𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺)) → 𝐹 ∈ Word dom (iEdg‘𝐺))
16		eqid 2729	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (Edg‘𝐺) = (Edg‘𝐺)
17	3, 16	upgredginwlk 29599	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝐺 ∈ UPGraph ∧ 𝐹 ∈ Word dom (iEdg‘𝐺)) → (𝑘 ∈ (0..^(♯‘𝐹)) → ((iEdg‘𝐺)‘(𝐹‘𝑘)) ∈ (Edg‘𝐺)))
18	14, 15, 17	syl2anr 597	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝐹 ∈ Word dom (iEdg‘𝐺) ∧ 𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺)) ∧ ((𝐺 ∈ V ∧ 𝐹 ∈ V ∧ 𝑃 ∈ V) ∧ 𝐺 ∈ UPGraph)) → (𝑘 ∈ (0..^(♯‘𝐹)) → ((iEdg‘𝐺)‘(𝐹‘𝑘)) ∈ (Edg‘𝐺)))
19	18	imp 406	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝐹 ∈ Word dom (iEdg‘𝐺) ∧ 𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺)) ∧ ((𝐺 ∈ V ∧ 𝐹 ∈ V ∧ 𝑃 ∈ V) ∧ 𝐺 ∈ UPGraph)) ∧ 𝑘 ∈ (0..^(♯‘𝐹))) → ((iEdg‘𝐺)‘(𝐹‘𝑘)) ∈ (Edg‘𝐺))
20		simprr 772	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((𝐹 ∈ Word dom (iEdg‘𝐺) ∧ 𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺)) ∧ ((𝐺 ∈ V ∧ 𝐹 ∈ V ∧ 𝑃 ∈ V) ∧ 𝐺 ∈ UPGraph)) → 𝐺 ∈ UPGraph)
21	20	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((((𝐹 ∈ Word dom (iEdg‘𝐺) ∧ 𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺)) ∧ ((𝐺 ∈ V ∧ 𝐹 ∈ V ∧ 𝑃 ∈ V) ∧ 𝐺 ∈ UPGraph)) ∧ 𝑘 ∈ (0..^(♯‘𝐹))) → 𝐺 ∈ UPGraph)
22	21	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((((𝐹 ∈ Word dom (iEdg‘𝐺) ∧ 𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺)) ∧ ((𝐺 ∈ V ∧ 𝐹 ∈ V ∧ 𝑃 ∈ V) ∧ 𝐺 ∈ UPGraph)) ∧ 𝑘 ∈ (0..^(♯‘𝐹))) ∧ ((iEdg‘𝐺)‘(𝐹‘𝑘)) ∈ (Edg‘𝐺)) → 𝐺 ∈ UPGraph)
23	22	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((((𝐹 ∈ Word dom (iEdg‘𝐺) ∧ 𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺)) ∧ ((𝐺 ∈ V ∧ 𝐹 ∈ V ∧ 𝑃 ∈ V) ∧ 𝐺 ∈ UPGraph)) ∧ 𝑘 ∈ (0..^(♯‘𝐹))) ∧ ((iEdg‘𝐺)‘(𝐹‘𝑘)) ∈ (Edg‘𝐺)) ∧ (¬ (𝑃‘𝑘) = (𝑃‘(𝑘 + 1)) ∧ {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ⊆ ((iEdg‘𝐺)‘(𝐹‘𝑘)))) → 𝐺 ∈ UPGraph)
24		simplr 768	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((((𝐹 ∈ Word dom (iEdg‘𝐺) ∧ 𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺)) ∧ ((𝐺 ∈ V ∧ 𝐹 ∈ V ∧ 𝑃 ∈ V) ∧ 𝐺 ∈ UPGraph)) ∧ 𝑘 ∈ (0..^(♯‘𝐹))) ∧ ((iEdg‘𝐺)‘(𝐹‘𝑘)) ∈ (Edg‘𝐺)) ∧ (¬ (𝑃‘𝑘) = (𝑃‘(𝑘 + 1)) ∧ {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ⊆ ((iEdg‘𝐺)‘(𝐹‘𝑘)))) → ((iEdg‘𝐺)‘(𝐹‘𝑘)) ∈ (Edg‘𝐺))
25		simprr 772	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((((𝐹 ∈ Word dom (iEdg‘𝐺) ∧ 𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺)) ∧ ((𝐺 ∈ V ∧ 𝐹 ∈ V ∧ 𝑃 ∈ V) ∧ 𝐺 ∈ UPGraph)) ∧ 𝑘 ∈ (0..^(♯‘𝐹))) ∧ ((iEdg‘𝐺)‘(𝐹‘𝑘)) ∈ (Edg‘𝐺)) ∧ (¬ (𝑃‘𝑘) = (𝑃‘(𝑘 + 1)) ∧ {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ⊆ ((iEdg‘𝐺)‘(𝐹‘𝑘)))) → {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ⊆ ((iEdg‘𝐺)‘(𝐹‘𝑘)))
26		df-ne 2926	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝑃‘𝑘) ≠ (𝑃‘(𝑘 + 1)) ↔ ¬ (𝑃‘𝑘) = (𝑃‘(𝑘 + 1)))
27		fvexd 6841	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝑃‘𝑘) ≠ (𝑃‘(𝑘 + 1)) → (𝑃‘𝑘) ∈ V)
28		fvexd 6841	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝑃‘𝑘) ≠ (𝑃‘(𝑘 + 1)) → (𝑃‘(𝑘 + 1)) ∈ V)
29		id 22	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝑃‘𝑘) ≠ (𝑃‘(𝑘 + 1)) → (𝑃‘𝑘) ≠ (𝑃‘(𝑘 + 1)))
30	27, 28, 29	3jca 1128	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝑃‘𝑘) ≠ (𝑃‘(𝑘 + 1)) → ((𝑃‘𝑘) ∈ V ∧ (𝑃‘(𝑘 + 1)) ∈ V ∧ (𝑃‘𝑘) ≠ (𝑃‘(𝑘 + 1))))
31	26, 30	sylbir 235	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (¬ (𝑃‘𝑘) = (𝑃‘(𝑘 + 1)) → ((𝑃‘𝑘) ∈ V ∧ (𝑃‘(𝑘 + 1)) ∈ V ∧ (𝑃‘𝑘) ≠ (𝑃‘(𝑘 + 1))))
32	31	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((¬ (𝑃‘𝑘) = (𝑃‘(𝑘 + 1)) ∧ {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ⊆ ((iEdg‘𝐺)‘(𝐹‘𝑘))) → ((𝑃‘𝑘) ∈ V ∧ (𝑃‘(𝑘 + 1)) ∈ V ∧ (𝑃‘𝑘) ≠ (𝑃‘(𝑘 + 1))))
33	32	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((((𝐹 ∈ Word dom (iEdg‘𝐺) ∧ 𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺)) ∧ ((𝐺 ∈ V ∧ 𝐹 ∈ V ∧ 𝑃 ∈ V) ∧ 𝐺 ∈ UPGraph)) ∧ 𝑘 ∈ (0..^(♯‘𝐹))) ∧ ((iEdg‘𝐺)‘(𝐹‘𝑘)) ∈ (Edg‘𝐺)) ∧ (¬ (𝑃‘𝑘) = (𝑃‘(𝑘 + 1)) ∧ {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ⊆ ((iEdg‘𝐺)‘(𝐹‘𝑘)))) → ((𝑃‘𝑘) ∈ V ∧ (𝑃‘(𝑘 + 1)) ∈ V ∧ (𝑃‘𝑘) ≠ (𝑃‘(𝑘 + 1))))
34	2, 16	upgredgpr 29105	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝐺 ∈ UPGraph ∧ ((iEdg‘𝐺)‘(𝐹‘𝑘)) ∈ (Edg‘𝐺) ∧ {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ⊆ ((iEdg‘𝐺)‘(𝐹‘𝑘))) ∧ ((𝑃‘𝑘) ∈ V ∧ (𝑃‘(𝑘 + 1)) ∈ V ∧ (𝑃‘𝑘) ≠ (𝑃‘(𝑘 + 1)))) → {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} = ((iEdg‘𝐺)‘(𝐹‘𝑘)))
35	23, 24, 25, 33, 34	syl31anc 1375	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((((𝐹 ∈ Word dom (iEdg‘𝐺) ∧ 𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺)) ∧ ((𝐺 ∈ V ∧ 𝐹 ∈ V ∧ 𝑃 ∈ V) ∧ 𝐺 ∈ UPGraph)) ∧ 𝑘 ∈ (0..^(♯‘𝐹))) ∧ ((iEdg‘𝐺)‘(𝐹‘𝑘)) ∈ (Edg‘𝐺)) ∧ (¬ (𝑃‘𝑘) = (𝑃‘(𝑘 + 1)) ∧ {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ⊆ ((iEdg‘𝐺)‘(𝐹‘𝑘)))) → {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} = ((iEdg‘𝐺)‘(𝐹‘𝑘)))
36	35	eqcomd 2735	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((((𝐹 ∈ Word dom (iEdg‘𝐺) ∧ 𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺)) ∧ ((𝐺 ∈ V ∧ 𝐹 ∈ V ∧ 𝑃 ∈ V) ∧ 𝐺 ∈ UPGraph)) ∧ 𝑘 ∈ (0..^(♯‘𝐹))) ∧ ((iEdg‘𝐺)‘(𝐹‘𝑘)) ∈ (Edg‘𝐺)) ∧ (¬ (𝑃‘𝑘) = (𝑃‘(𝑘 + 1)) ∧ {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ⊆ ((iEdg‘𝐺)‘(𝐹‘𝑘)))) → ((iEdg‘𝐺)‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))})
37	36	exp31 419	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝐹 ∈ Word dom (iEdg‘𝐺) ∧ 𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺)) ∧ ((𝐺 ∈ V ∧ 𝐹 ∈ V ∧ 𝑃 ∈ V) ∧ 𝐺 ∈ UPGraph)) ∧ 𝑘 ∈ (0..^(♯‘𝐹))) → (((iEdg‘𝐺)‘(𝐹‘𝑘)) ∈ (Edg‘𝐺) → ((¬ (𝑃‘𝑘) = (𝑃‘(𝑘 + 1)) ∧ {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ⊆ ((iEdg‘𝐺)‘(𝐹‘𝑘))) → ((iEdg‘𝐺)‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))})))
38	19, 37	mpd 15	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝐹 ∈ Word dom (iEdg‘𝐺) ∧ 𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺)) ∧ ((𝐺 ∈ V ∧ 𝐹 ∈ V ∧ 𝑃 ∈ V) ∧ 𝐺 ∈ UPGraph)) ∧ 𝑘 ∈ (0..^(♯‘𝐹))) → ((¬ (𝑃‘𝑘) = (𝑃‘(𝑘 + 1)) ∧ {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ⊆ ((iEdg‘𝐺)‘(𝐹‘𝑘))) → ((iEdg‘𝐺)‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))}))
39	38	com12 32	. . . . . . . . . . . . . . . . . . 19 ⊢ ((¬ (𝑃‘𝑘) = (𝑃‘(𝑘 + 1)) ∧ {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ⊆ ((iEdg‘𝐺)‘(𝐹‘𝑘))) → ((((𝐹 ∈ Word dom (iEdg‘𝐺) ∧ 𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺)) ∧ ((𝐺 ∈ V ∧ 𝐹 ∈ V ∧ 𝑃 ∈ V) ∧ 𝐺 ∈ UPGraph)) ∧ 𝑘 ∈ (0..^(♯‘𝐹))) → ((iEdg‘𝐺)‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))}))
40	13, 39	jaoi 857	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝑃‘𝑘) = (𝑃‘(𝑘 + 1)) ∧ ((iEdg‘𝐺)‘(𝐹‘𝑘)) = {(𝑃‘𝑘)}) ∨ (¬ (𝑃‘𝑘) = (𝑃‘(𝑘 + 1)) ∧ {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ⊆ ((iEdg‘𝐺)‘(𝐹‘𝑘)))) → ((((𝐹 ∈ Word dom (iEdg‘𝐺) ∧ 𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺)) ∧ ((𝐺 ∈ V ∧ 𝐹 ∈ V ∧ 𝑃 ∈ V) ∧ 𝐺 ∈ UPGraph)) ∧ 𝑘 ∈ (0..^(♯‘𝐹))) → ((iEdg‘𝐺)‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))}))
41	40	com12 32	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝐹 ∈ Word dom (iEdg‘𝐺) ∧ 𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺)) ∧ ((𝐺 ∈ V ∧ 𝐹 ∈ V ∧ 𝑃 ∈ V) ∧ 𝐺 ∈ UPGraph)) ∧ 𝑘 ∈ (0..^(♯‘𝐹))) → ((((𝑃‘𝑘) = (𝑃‘(𝑘 + 1)) ∧ ((iEdg‘𝐺)‘(𝐹‘𝑘)) = {(𝑃‘𝑘)}) ∨ (¬ (𝑃‘𝑘) = (𝑃‘(𝑘 + 1)) ∧ {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ⊆ ((iEdg‘𝐺)‘(𝐹‘𝑘)))) → ((iEdg‘𝐺)‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))}))
42	7, 41	biimtrid 242	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐹 ∈ Word dom (iEdg‘𝐺) ∧ 𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺)) ∧ ((𝐺 ∈ V ∧ 𝐹 ∈ V ∧ 𝑃 ∈ V) ∧ 𝐺 ∈ UPGraph)) ∧ 𝑘 ∈ (0..^(♯‘𝐹))) → (if-((𝑃‘𝑘) = (𝑃‘(𝑘 + 1)), ((iEdg‘𝐺)‘(𝐹‘𝑘)) = {(𝑃‘𝑘)}, {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ⊆ ((iEdg‘𝐺)‘(𝐹‘𝑘))) → ((iEdg‘𝐺)‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))}))
43	42	ralimdva 3141	. . . . . . . . . . . . . . 15 ⊢ (((𝐹 ∈ Word dom (iEdg‘𝐺) ∧ 𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺)) ∧ ((𝐺 ∈ V ∧ 𝐹 ∈ V ∧ 𝑃 ∈ V) ∧ 𝐺 ∈ UPGraph)) → (∀𝑘 ∈ (0..^(♯‘𝐹))if-((𝑃‘𝑘) = (𝑃‘(𝑘 + 1)), ((iEdg‘𝐺)‘(𝐹‘𝑘)) = {(𝑃‘𝑘)}, {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ⊆ ((iEdg‘𝐺)‘(𝐹‘𝑘))) → ∀𝑘 ∈ (0..^(♯‘𝐹))((iEdg‘𝐺)‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))}))
44	43	ex 412	. . . . . . . . . . . . . 14 ⊢ ((𝐹 ∈ Word dom (iEdg‘𝐺) ∧ 𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺)) → (((𝐺 ∈ V ∧ 𝐹 ∈ V ∧ 𝑃 ∈ V) ∧ 𝐺 ∈ UPGraph) → (∀𝑘 ∈ (0..^(♯‘𝐹))if-((𝑃‘𝑘) = (𝑃‘(𝑘 + 1)), ((iEdg‘𝐺)‘(𝐹‘𝑘)) = {(𝑃‘𝑘)}, {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ⊆ ((iEdg‘𝐺)‘(𝐹‘𝑘))) → ∀𝑘 ∈ (0..^(♯‘𝐹))((iEdg‘𝐺)‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))})))
45	44	com23 86	. . . . . . . . . . . . 13 ⊢ ((𝐹 ∈ Word dom (iEdg‘𝐺) ∧ 𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺)) → (∀𝑘 ∈ (0..^(♯‘𝐹))if-((𝑃‘𝑘) = (𝑃‘(𝑘 + 1)), ((iEdg‘𝐺)‘(𝐹‘𝑘)) = {(𝑃‘𝑘)}, {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ⊆ ((iEdg‘𝐺)‘(𝐹‘𝑘))) → (((𝐺 ∈ V ∧ 𝐹 ∈ V ∧ 𝑃 ∈ V) ∧ 𝐺 ∈ UPGraph) → ∀𝑘 ∈ (0..^(♯‘𝐹))((iEdg‘𝐺)‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))})))
46	45	3impia 1117	. . . . . . . . . . . 12 ⊢ ((𝐹 ∈ Word dom (iEdg‘𝐺) ∧ 𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺) ∧ ∀𝑘 ∈ (0..^(♯‘𝐹))if-((𝑃‘𝑘) = (𝑃‘(𝑘 + 1)), ((iEdg‘𝐺)‘(𝐹‘𝑘)) = {(𝑃‘𝑘)}, {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ⊆ ((iEdg‘𝐺)‘(𝐹‘𝑘)))) → (((𝐺 ∈ V ∧ 𝐹 ∈ V ∧ 𝑃 ∈ V) ∧ 𝐺 ∈ UPGraph) → ∀𝑘 ∈ (0..^(♯‘𝐹))((iEdg‘𝐺)‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))}))
47	46	impcom 407	. . . . . . . . . . 11 ⊢ ((((𝐺 ∈ V ∧ 𝐹 ∈ V ∧ 𝑃 ∈ V) ∧ 𝐺 ∈ UPGraph) ∧ (𝐹 ∈ Word dom (iEdg‘𝐺) ∧ 𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺) ∧ ∀𝑘 ∈ (0..^(♯‘𝐹))if-((𝑃‘𝑘) = (𝑃‘(𝑘 + 1)), ((iEdg‘𝐺)‘(𝐹‘𝑘)) = {(𝑃‘𝑘)}, {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ⊆ ((iEdg‘𝐺)‘(𝐹‘𝑘))))) → ∀𝑘 ∈ (0..^(♯‘𝐹))((iEdg‘𝐺)‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))})
48	5, 6, 47	3jca 1128	. . . . . . . . . 10 ⊢ ((((𝐺 ∈ V ∧ 𝐹 ∈ V ∧ 𝑃 ∈ V) ∧ 𝐺 ∈ UPGraph) ∧ (𝐹 ∈ Word dom (iEdg‘𝐺) ∧ 𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺) ∧ ∀𝑘 ∈ (0..^(♯‘𝐹))if-((𝑃‘𝑘) = (𝑃‘(𝑘 + 1)), ((iEdg‘𝐺)‘(𝐹‘𝑘)) = {(𝑃‘𝑘)}, {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ⊆ ((iEdg‘𝐺)‘(𝐹‘𝑘))))) → (𝐹 ∈ Word dom (iEdg‘𝐺) ∧ 𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺) ∧ ∀𝑘 ∈ (0..^(♯‘𝐹))((iEdg‘𝐺)‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))}))
49	48	exp31 419	. . . . . . . . 9 ⊢ ((𝐺 ∈ V ∧ 𝐹 ∈ V ∧ 𝑃 ∈ V) → (𝐺 ∈ UPGraph → ((𝐹 ∈ Word dom (iEdg‘𝐺) ∧ 𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺) ∧ ∀𝑘 ∈ (0..^(♯‘𝐹))if-((𝑃‘𝑘) = (𝑃‘(𝑘 + 1)), ((iEdg‘𝐺)‘(𝐹‘𝑘)) = {(𝑃‘𝑘)}, {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ⊆ ((iEdg‘𝐺)‘(𝐹‘𝑘)))) → (𝐹 ∈ Word dom (iEdg‘𝐺) ∧ 𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺) ∧ ∀𝑘 ∈ (0..^(♯‘𝐹))((iEdg‘𝐺)‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))}))))
50	49	com23 86	. . . . . . . 8 ⊢ ((𝐺 ∈ V ∧ 𝐹 ∈ V ∧ 𝑃 ∈ V) → ((𝐹 ∈ Word dom (iEdg‘𝐺) ∧ 𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺) ∧ ∀𝑘 ∈ (0..^(♯‘𝐹))if-((𝑃‘𝑘) = (𝑃‘(𝑘 + 1)), ((iEdg‘𝐺)‘(𝐹‘𝑘)) = {(𝑃‘𝑘)}, {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ⊆ ((iEdg‘𝐺)‘(𝐹‘𝑘)))) → (𝐺 ∈ UPGraph → (𝐹 ∈ Word dom (iEdg‘𝐺) ∧ 𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺) ∧ ∀𝑘 ∈ (0..^(♯‘𝐹))((iEdg‘𝐺)‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))}))))
51	4, 50	sylbid 240	. . . . . . 7 ⊢ ((𝐺 ∈ V ∧ 𝐹 ∈ V ∧ 𝑃 ∈ V) → (𝐹(Walks‘𝐺)𝑃 → (𝐺 ∈ UPGraph → (𝐹 ∈ Word dom (iEdg‘𝐺) ∧ 𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺) ∧ ∀𝑘 ∈ (0..^(♯‘𝐹))((iEdg‘𝐺)‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))}))))
52	51	impd 410	. . . . . 6 ⊢ ((𝐺 ∈ V ∧ 𝐹 ∈ V ∧ 𝑃 ∈ V) → ((𝐹(Walks‘𝐺)𝑃 ∧ 𝐺 ∈ UPGraph) → (𝐹 ∈ Word dom (iEdg‘𝐺) ∧ 𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺) ∧ ∀𝑘 ∈ (0..^(♯‘𝐹))((iEdg‘𝐺)‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))})))
53	52	impcom 407	. . . . 5 ⊢ (((𝐹(Walks‘𝐺)𝑃 ∧ 𝐺 ∈ UPGraph) ∧ (𝐺 ∈ V ∧ 𝐹 ∈ V ∧ 𝑃 ∈ V)) → (𝐹 ∈ Word dom (iEdg‘𝐺) ∧ 𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺) ∧ ∀𝑘 ∈ (0..^(♯‘𝐹))((iEdg‘𝐺)‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))}))
54	2, 3	isupwlk 48121	. . . . . 6 ⊢ ((𝐺 ∈ V ∧ 𝐹 ∈ V ∧ 𝑃 ∈ V) → (𝐹(UPWalks‘𝐺)𝑃 ↔ (𝐹 ∈ Word dom (iEdg‘𝐺) ∧ 𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺) ∧ ∀𝑘 ∈ (0..^(♯‘𝐹))((iEdg‘𝐺)‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))})))
55	54	adantl 481	. . . . 5 ⊢ (((𝐹(Walks‘𝐺)𝑃 ∧ 𝐺 ∈ UPGraph) ∧ (𝐺 ∈ V ∧ 𝐹 ∈ V ∧ 𝑃 ∈ V)) → (𝐹(UPWalks‘𝐺)𝑃 ↔ (𝐹 ∈ Word dom (iEdg‘𝐺) ∧ 𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺) ∧ ∀𝑘 ∈ (0..^(♯‘𝐹))((iEdg‘𝐺)‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))})))
56	53, 55	mpbird 257	. . . 4 ⊢ (((𝐹(Walks‘𝐺)𝑃 ∧ 𝐺 ∈ UPGraph) ∧ (𝐺 ∈ V ∧ 𝐹 ∈ V ∧ 𝑃 ∈ V)) → 𝐹(UPWalks‘𝐺)𝑃)
57	56	exp31 419	. . 3 ⊢ (𝐹(Walks‘𝐺)𝑃 → (𝐺 ∈ UPGraph → ((𝐺 ∈ V ∧ 𝐹 ∈ V ∧ 𝑃 ∈ V) → 𝐹(UPWalks‘𝐺)𝑃)))
58	1, 57	mpid 44	. 2 ⊢ (𝐹(Walks‘𝐺)𝑃 → (𝐺 ∈ UPGraph → 𝐹(UPWalks‘𝐺)𝑃))
59	58	impcom 407	1 ⊢ ((𝐺 ∈ UPGraph ∧ 𝐹(Walks‘𝐺)𝑃) → 𝐹(UPWalks‘𝐺)𝑃)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 847  if-wif 1062   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109   ≠ wne 2925  ∀wral 3044  Vcvv 3438   ⊆ wss 3905  {csn 4579  {cpr 4581   class class class wbr 5095  dom cdm 5623  ⟶wf 6482  ‘cfv 6486  (class class class)co 7353  0cc0 11028  1c1 11029   + caddc 11031  ...cfz 13428  ..^cfzo 13575  ♯chash 14255  Word cword 14438  Vtxcvtx 28959  iEdgciedg 28960  Edgcedg 29010  UPGraphcupgr 29043  Walkscwlks 29560  UPWalkscupwlks 48118

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5221  ax-sep 5238  ax-nul 5248  ax-pow 5307  ax-pr 5374  ax-un 7675  ax-cnex 11084  ax-resscn 11085  ax-1cn 11086  ax-icn 11087  ax-addcl 11088  ax-addrcl 11089  ax-mulcl 11090  ax-mulrcl 11091  ax-mulcom 11092  ax-addass 11093  ax-mulass 11094  ax-distr 11095  ax-i2m1 11096  ax-1ne0 11097  ax-1rid 11098  ax-rnegex 11099  ax-rrecex 11100  ax-cnre 11101  ax-pre-lttri 11102  ax-pre-lttrn 11103  ax-pre-ltadd 11104  ax-pre-mulgt0 11105

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 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-reu 3346  df-rab 3397  df-v 3440  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-pss 3925  df-nul 4287  df-if 4479  df-pw 4555  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4862  df-int 4900  df-iun 4946  df-br 5096  df-opab 5158  df-mpt 5177  df-tr 5203  df-id 5518  df-eprel 5523  df-po 5531  df-so 5532  df-fr 5576  df-we 5578  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-res 5635  df-ima 5636  df-pred 6253  df-ord 6314  df-on 6315  df-lim 6316  df-suc 6317  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494  df-riota 7310  df-ov 7356  df-oprab 7357  df-mpo 7358  df-om 7807  df-1st 7931  df-2nd 7932  df-frecs 8221  df-wrecs 8252  df-recs 8301  df-rdg 8339  df-1o 8395  df-2o 8396  df-oadd 8399  df-er 8632  df-map 8762  df-en 8880  df-dom 8881  df-sdom 8882  df-fin 8883  df-dju 9816  df-card 9854  df-pnf 11170  df-mnf 11171  df-xr 11172  df-ltxr 11173  df-le 11174  df-sub 11367  df-neg 11368  df-nn 12147  df-2 12209  df-n0 12403  df-xnn0 12476  df-z 12490  df-uz 12754  df-fz 13429  df-fzo 13576  df-hash 14256  df-word 14439  df-edg 29011  df-uhgr 29021  df-upgr 29045  df-wlks 29563  df-upwlks 48119

This theorem is referenced by:  upgrwlkupwlkb  48126