Theorem upgrwlkupwlk 48726

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 29759	. . 3 ⊢ (𝐹(Walks‘𝐺)𝑃 → (𝐺 ∈ V ∧ 𝐹 ∈ V ∧ 𝑃 ∈ V))
2		eqid 2761	. . . . . . . . 9 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
3		eqid 2761	. . . . . . . . 9 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺)
4	2, 3	iswlk 29757	. . . . . . . 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 1207	. . . . . . . . . . 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 1208	. . . . . . . . . . 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 1074	. . . . . . . . . . . . . . . . 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 4594	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ {(𝑃‘𝑘)} = {(𝑃‘𝑘), (𝑃‘𝑘)}
9		preq2 4692	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑃‘𝑘) = (𝑃‘(𝑘 + 1)) → {(𝑃‘𝑘), (𝑃‘𝑘)} = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))})
10	8, 9	eqtrid 2808	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑃‘𝑘) = (𝑃‘(𝑘 + 1)) → {(𝑃‘𝑘)} = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))})
11	10	eqeq2d 2772	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑃‘𝑘) = (𝑃‘(𝑘 + 1)) → (((iEdg‘𝐺)‘(𝐹‘𝑘)) = {(𝑃‘𝑘)} ↔ ((iEdg‘𝐺)‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))}))
12	11	biimpa 480	. . . . . . . . . . . . . . . . . . . 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 488	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝐺 ∈ V ∧ 𝐹 ∈ V ∧ 𝑃 ∈ V) ∧ 𝐺 ∈ UPGraph) → 𝐺 ∈ UPGraph)
15		simpl 486	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝐹 ∈ Word dom (iEdg‘𝐺) ∧ 𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺)) → 𝐹 ∈ Word dom (iEdg‘𝐺))
16		eqid 2761	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (Edg‘𝐺) = (Edg‘𝐺)
17	3, 16	upgredginwlk 29782	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝐺 ∈ UPGraph ∧ 𝐹 ∈ Word dom (iEdg‘𝐺)) → (𝑘 ∈ (0..^(♯‘𝐹)) → ((iEdg‘𝐺)‘(𝐹‘𝑘)) ∈ (Edg‘𝐺)))
18	14, 15, 17	syl2anr 606	. . . . . . . . . . . . . . . . . . . . . 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 410	. . . . . . . . . . . . . . . . . . . . 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 782	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((𝐹 ∈ Word dom (iEdg‘𝐺) ∧ 𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺)) ∧ ((𝐺 ∈ V ∧ 𝐹 ∈ V ∧ 𝑃 ∈ V) ∧ 𝐺 ∈ UPGraph)) → 𝐺 ∈ UPGraph)
21	20	adantr 484	. . . . . . . . . . . . . . . . . . . . . . . . . 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 484	. . . . . . . . . . . . . . . . . . . . . . . . 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 484	. . . . . . . . . . . . . . . . . . . . . . . 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 778	. . . . . . . . . . . . . . . . . . . . . . . 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 782	. . . . . . . . . . . . . . . . . . . . . . . 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 2957	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝑃‘𝑘) ≠ (𝑃‘(𝑘 + 1)) ↔ ¬ (𝑃‘𝑘) = (𝑃‘(𝑘 + 1)))
27		fvexd 6878	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝑃‘𝑘) ≠ (𝑃‘(𝑘 + 1)) → (𝑃‘𝑘) ∈ V)
28		fvexd 6878	. . . . . . . . . . . . . . . . . . . . . . . . . . . 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 1140	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝑃‘𝑘) ≠ (𝑃‘(𝑘 + 1)) → ((𝑃‘𝑘) ∈ V ∧ (𝑃‘(𝑘 + 1)) ∈ V ∧ (𝑃‘𝑘) ≠ (𝑃‘(𝑘 + 1))))
31	26, 30	sylbir 237	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (¬ (𝑃‘𝑘) = (𝑃‘(𝑘 + 1)) → ((𝑃‘𝑘) ∈ V ∧ (𝑃‘(𝑘 + 1)) ∈ V ∧ (𝑃‘𝑘) ≠ (𝑃‘(𝑘 + 1))))
32	31	adantr 484	. . . . . . . . . . . . . . . . . . . . . . . . 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 485	. . . . . . . . . . . . . . . . . . . . . . . 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 29289	. . . . . . . . . . . . . . . . . . . . . . . 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 1391	. . . . . . . . . . . . . . . . . . . . . . 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 2767	. . . . . . . . . . . . . . . . . . . . . 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 423	. . . . . . . . . . . . . . . . . . . . 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 868	. . . . . . . . . . . . . . . . . 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 244	. . . . . . . . . . . . . . . 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 3173	. . . . . . . . . . . . . . 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 416	. . . . . . . . . . . . . 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 1129	. . . . . . . . . . . 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 411	. . . . . . . . . . 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 1140	. . . . . . . . . 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 423	. . . . . . . . 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 242	. . . . . . 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 414	. . . . . 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 411	. . . . 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 48722	. . . . . 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 485	. . . . 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 259	. . . 4 ⊢ (((𝐹(Walks‘𝐺)𝑃 ∧ 𝐺 ∈ UPGraph) ∧ (𝐺 ∈ V ∧ 𝐹 ∈ V ∧ 𝑃 ∈ V)) → 𝐹(UPWalks‘𝐺)𝑃)
57	56	exp31 423	. . 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 411	1 ⊢ ((𝐺 ∈ UPGraph ∧ 𝐹(Walks‘𝐺)𝑃) → 𝐹(UPWalks‘𝐺)𝑃)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   ∧ wa 399   ∨ wo 858  if-wif 1073   ∧ w3a 1097   = wceq 1559   ∈ wcel 2141   ≠ wne 2956  ∀wral 3075  Vcvv 3453   ⊆ wss 3904  {csn 4581  {cpr 4583   class class class wbr 5099  dom cdm 5645  ⟶wf 6513  ‘cfv 6517  (class class class)co 7392  0cc0 11070  1c1 11071   + caddc 11073  ...cfz 13509  ..^cfzo 13656  ♯chash 14340  Word cword 14523  Vtxcvtx 29143  iEdgciedg 29144  Edgcedg 29194  UPGraphcupgr 29227  Walkscwlks 29743  UPWalkscupwlks 48719

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-rep 5226  ax-sep 5245  ax-nul 5255  ax-pow 5321  ax-pr 5389  ax-un 7714  ax-cnex 11126  ax-resscn 11127  ax-1cn 11128  ax-icn 11129  ax-addcl 11130  ax-addrcl 11131  ax-mulcl 11132  ax-mulrcl 11133  ax-mulcom 11134  ax-addass 11135  ax-mulass 11136  ax-distr 11137  ax-i2m1 11138  ax-1ne0 11139  ax-1rid 11140  ax-rnegex 11141  ax-rrecex 11142  ax-cnre 11143  ax-pre-lttri 11144  ax-pre-lttrn 11145  ax-pre-ltadd 11146  ax-pre-mulgt0 11147

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-ifp 1074  df-3or 1098  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-nel 3061  df-ral 3076  df-rex 3086  df-reu 3367  df-rab 3414  df-v 3455  df-sbc 3745  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-int 4905  df-iun 4950  df-br 5100  df-opab 5162  df-mpt 5181  df-tr 5207  df-id 5540  df-eprel 5545  df-po 5553  df-so 5554  df-fr 5598  df-we 5600  df-xp 5651  df-rel 5652  df-cnv 5653  df-co 5654  df-dm 5655  df-rn 5656  df-res 5657  df-ima 5658  df-pred 6284  df-ord 6345  df-on 6346  df-lim 6347  df-suc 6348  df-iota 6473  df-fun 6519  df-fn 6520  df-f 6521  df-f1 6522  df-fo 6523  df-f1o 6524  df-fv 6525  df-riota 7349  df-ov 7395  df-oprab 7396  df-mpo 7397  df-om 7843  df-1st 7966  df-2nd 7967  df-frecs 8257  df-wrecs 8288  df-recs 8337  df-rdg 8376  df-1o 8432  df-2o 8433  df-oadd 8436  df-er 8673  df-map 8805  df-en 8924  df-dom 8925  df-sdom 8926  df-fin 8927  df-dju 9856  df-card 9894  df-pnf 11215  df-mnf 11216  df-xr 11217  df-ltxr 11218  df-le 11219  df-sub 11413  df-neg 11414  df-nn 12208  df-2 12277  df-n0 12479  df-xnn0 12552  df-z 12566  df-uz 12837  df-fz 13510  df-fzo 13657  df-hash 14341  df-word 14524  df-edg 29195  df-uhgr 29205  df-upgr 29229  df-wlks 29746  df-upwlks 48720

This theorem is referenced by:  upgrwlkupwlkb  48727