Theorem subgrwlk 30156

Description: If a walk exists in a subgraph of a graph 𝐺, then that walk also exists in 𝐺. (Contributed by BTernaryTau, 22-Oct-2023.)

Assertion

Ref	Expression
subgrwlk	⊢ (𝑆 SubGraph 𝐺 → (𝐹(Walks‘𝑆)𝑃 → 𝐹(Walks‘𝐺)𝑃))

Proof of Theorem subgrwlk

Dummy variable 𝑘 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		subgrv 29738	. . . . . 6 ⊢ (𝑆 SubGraph 𝐺 → (𝑆 ∈ V ∧ 𝐺 ∈ V))
2	1	simpld 500	. . . . 5 ⊢ (𝑆 SubGraph 𝐺 → 𝑆 ∈ V)
3		eqid 2762	. . . . . 6 ⊢ (Vtx‘𝑆) = (Vtx‘𝑆)
4		eqid 2762	. . . . . 6 ⊢ (iEdg‘𝑆) = (iEdg‘𝑆)
5	3, 4	iswlkg 30081	. . . . 5 ⊢ (𝑆 ∈ V → (𝐹(Walks‘𝑆)𝑃 ↔ (𝐹 ∈ Word dom (iEdg‘𝑆) ∧ 𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝑆) ∧ ∀𝑘 ∈ (0..^(♯‘𝐹))if-((𝑃‘𝑘) = (𝑃‘(𝑘 + 1)), ((iEdg‘𝑆)‘(𝐹‘𝑘)) = {(𝑃‘𝑘)}, {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ⊆ ((iEdg‘𝑆)‘(𝐹‘𝑘))))))
6	2, 5	syl 18	. . . 4 ⊢ (𝑆 SubGraph 𝐺 → (𝐹(Walks‘𝑆)𝑃 ↔ (𝐹 ∈ Word dom (iEdg‘𝑆) ∧ 𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝑆) ∧ ∀𝑘 ∈ (0..^(♯‘𝐹))if-((𝑃‘𝑘) = (𝑃‘(𝑘 + 1)), ((iEdg‘𝑆)‘(𝐹‘𝑘)) = {(𝑃‘𝑘)}, {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ⊆ ((iEdg‘𝑆)‘(𝐹‘𝑘))))))
7		3simpa 1166	. . . . . 6 ⊢ ((𝐹 ∈ Word dom (iEdg‘𝑆) ∧ 𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝑆) ∧ ∀𝑘 ∈ (0..^(♯‘𝐹))if-((𝑃‘𝑘) = (𝑃‘(𝑘 + 1)), ((iEdg‘𝑆)‘(𝐹‘𝑘)) = {(𝑃‘𝑘)}, {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ⊆ ((iEdg‘𝑆)‘(𝐹‘𝑘)))) → (𝐹 ∈ Word dom (iEdg‘𝑆) ∧ 𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝑆)))
8		eqid 2762	. . . . . . . . . . 11 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
9		eqid 2762	. . . . . . . . . . 11 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺)
10		eqid 2762	. . . . . . . . . . 11 ⊢ (Edg‘𝑆) = (Edg‘𝑆)
11	3, 8, 4, 9, 10	subgrprop2 29742	. . . . . . . . . 10 ⊢ (𝑆 SubGraph 𝐺 → ((Vtx‘𝑆) ⊆ (Vtx‘𝐺) ∧ (iEdg‘𝑆) ⊆ (iEdg‘𝐺) ∧ (Edg‘𝑆) ⊆ 𝒫 (Vtx‘𝑆)))
12	11	simp2d 1161	. . . . . . . . 9 ⊢ (𝑆 SubGraph 𝐺 → (iEdg‘𝑆) ⊆ (iEdg‘𝐺))
13		dmss 5890	. . . . . . . . 9 ⊢ ((iEdg‘𝑆) ⊆ (iEdg‘𝐺) → dom (iEdg‘𝑆) ⊆ dom (iEdg‘𝐺))
14		sswrd 14591	. . . . . . . . 9 ⊢ (dom (iEdg‘𝑆) ⊆ dom (iEdg‘𝐺) → Word dom (iEdg‘𝑆) ⊆ Word dom (iEdg‘𝐺))
15	12, 13, 14	3syl 19	. . . . . . . 8 ⊢ (𝑆 SubGraph 𝐺 → Word dom (iEdg‘𝑆) ⊆ Word dom (iEdg‘𝐺))
16	15	sseld 3933	. . . . . . 7 ⊢ (𝑆 SubGraph 𝐺 → (𝐹 ∈ Word dom (iEdg‘𝑆) → 𝐹 ∈ Word dom (iEdg‘𝐺)))
17	11	simp1d 1160	. . . . . . . 8 ⊢ (𝑆 SubGraph 𝐺 → (Vtx‘𝑆) ⊆ (Vtx‘𝐺))
18		fss 6723	. . . . . . . . 9 ⊢ ((𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝑆) ∧ (Vtx‘𝑆) ⊆ (Vtx‘𝐺)) → 𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺))
19	18	expcom 419	. . . . . . . 8 ⊢ ((Vtx‘𝑆) ⊆ (Vtx‘𝐺) → (𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝑆) → 𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺)))
20	17, 19	syl 18	. . . . . . 7 ⊢ (𝑆 SubGraph 𝐺 → (𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝑆) → 𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺)))
21	16, 20	anim12d 621	. . . . . 6 ⊢ (𝑆 SubGraph 𝐺 → ((𝐹 ∈ Word dom (iEdg‘𝑆) ∧ 𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝑆)) → (𝐹 ∈ Word dom (iEdg‘𝐺) ∧ 𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺))))
22	7, 21	syl5 35	. . . . 5 ⊢ (𝑆 SubGraph 𝐺 → ((𝐹 ∈ Word dom (iEdg‘𝑆) ∧ 𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝑆) ∧ ∀𝑘 ∈ (0..^(♯‘𝐹))if-((𝑃‘𝑘) = (𝑃‘(𝑘 + 1)), ((iEdg‘𝑆)‘(𝐹‘𝑘)) = {(𝑃‘𝑘)}, {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ⊆ ((iEdg‘𝑆)‘(𝐹‘𝑘)))) → (𝐹 ∈ Word dom (iEdg‘𝐺) ∧ 𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺))))
23		3simpb 1167	. . . . . 6 ⊢ ((𝐹 ∈ Word dom (iEdg‘𝑆) ∧ 𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝑆) ∧ ∀𝑘 ∈ (0..^(♯‘𝐹))if-((𝑃‘𝑘) = (𝑃‘(𝑘 + 1)), ((iEdg‘𝑆)‘(𝐹‘𝑘)) = {(𝑃‘𝑘)}, {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ⊆ ((iEdg‘𝑆)‘(𝐹‘𝑘)))) → (𝐹 ∈ Word dom (iEdg‘𝑆) ∧ ∀𝑘 ∈ (0..^(♯‘𝐹))if-((𝑃‘𝑘) = (𝑃‘(𝑘 + 1)), ((iEdg‘𝑆)‘(𝐹‘𝑘)) = {(𝑃‘𝑘)}, {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ⊆ ((iEdg‘𝑆)‘(𝐹‘𝑘)))))
24	3, 8, 4, 9, 10	subgrprop 29741	. . . . . . . . . . . . . . . . 17 ⊢ (𝑆 SubGraph 𝐺 → ((Vtx‘𝑆) ⊆ (Vtx‘𝐺) ∧ (iEdg‘𝑆) = ((iEdg‘𝐺) ↾ dom (iEdg‘𝑆)) ∧ (Edg‘𝑆) ⊆ 𝒫 (Vtx‘𝑆)))
25	24	simp2d 1161	. . . . . . . . . . . . . . . 16 ⊢ (𝑆 SubGraph 𝐺 → (iEdg‘𝑆) = ((iEdg‘𝐺) ↾ dom (iEdg‘𝑆)))
26	25	fveq1d 6884	. . . . . . . . . . . . . . 15 ⊢ (𝑆 SubGraph 𝐺 → ((iEdg‘𝑆)‘(𝐹‘𝑘)) = (((iEdg‘𝐺) ↾ dom (iEdg‘𝑆))‘(𝐹‘𝑘)))
27	26	3ad2ant1 1151	. . . . . . . . . . . . . 14 ⊢ ((𝑆 SubGraph 𝐺 ∧ 𝐹 ∈ Word dom (iEdg‘𝑆) ∧ 𝑘 ∈ (0..^(♯‘𝐹))) → ((iEdg‘𝑆)‘(𝐹‘𝑘)) = (((iEdg‘𝐺) ↾ dom (iEdg‘𝑆))‘(𝐹‘𝑘)))
28		wrdsymbcl 14596	. . . . . . . . . . . . . . . 16 ⊢ ((𝐹 ∈ Word dom (iEdg‘𝑆) ∧ 𝑘 ∈ (0..^(♯‘𝐹))) → (𝐹‘𝑘) ∈ dom (iEdg‘𝑆))
29	28	fvresd 6902	. . . . . . . . . . . . . . 15 ⊢ ((𝐹 ∈ Word dom (iEdg‘𝑆) ∧ 𝑘 ∈ (0..^(♯‘𝐹))) → (((iEdg‘𝐺) ↾ dom (iEdg‘𝑆))‘(𝐹‘𝑘)) = ((iEdg‘𝐺)‘(𝐹‘𝑘)))
30	29	3adant1 1148	. . . . . . . . . . . . . 14 ⊢ ((𝑆 SubGraph 𝐺 ∧ 𝐹 ∈ Word dom (iEdg‘𝑆) ∧ 𝑘 ∈ (0..^(♯‘𝐹))) → (((iEdg‘𝐺) ↾ dom (iEdg‘𝑆))‘(𝐹‘𝑘)) = ((iEdg‘𝐺)‘(𝐹‘𝑘)))
31	27, 30	eqtrd 2797	. . . . . . . . . . . . 13 ⊢ ((𝑆 SubGraph 𝐺 ∧ 𝐹 ∈ Word dom (iEdg‘𝑆) ∧ 𝑘 ∈ (0..^(♯‘𝐹))) → ((iEdg‘𝑆)‘(𝐹‘𝑘)) = ((iEdg‘𝐺)‘(𝐹‘𝑘)))
32	31	eqeq1d 2764	. . . . . . . . . . . 12 ⊢ ((𝑆 SubGraph 𝐺 ∧ 𝐹 ∈ Word dom (iEdg‘𝑆) ∧ 𝑘 ∈ (0..^(♯‘𝐹))) → (((iEdg‘𝑆)‘(𝐹‘𝑘)) = {(𝑃‘𝑘)} ↔ ((iEdg‘𝐺)‘(𝐹‘𝑘)) = {(𝑃‘𝑘)}))
33	31	sseq2d 3966	. . . . . . . . . . . 12 ⊢ ((𝑆 SubGraph 𝐺 ∧ 𝐹 ∈ Word dom (iEdg‘𝑆) ∧ 𝑘 ∈ (0..^(♯‘𝐹))) → ({(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ⊆ ((iEdg‘𝑆)‘(𝐹‘𝑘)) ↔ {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ⊆ ((iEdg‘𝐺)‘(𝐹‘𝑘))))
34	32, 33	ifpbi23d 1096	. . . . . . . . . . 11 ⊢ ((𝑆 SubGraph 𝐺 ∧ 𝐹 ∈ Word dom (iEdg‘𝑆) ∧ 𝑘 ∈ (0..^(♯‘𝐹))) → (if-((𝑃‘𝑘) = (𝑃‘(𝑘 + 1)), ((iEdg‘𝑆)‘(𝐹‘𝑘)) = {(𝑃‘𝑘)}, {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ⊆ ((iEdg‘𝑆)‘(𝐹‘𝑘))) ↔ if-((𝑃‘𝑘) = (𝑃‘(𝑘 + 1)), ((iEdg‘𝐺)‘(𝐹‘𝑘)) = {(𝑃‘𝑘)}, {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ⊆ ((iEdg‘𝐺)‘(𝐹‘𝑘)))))
35	34	biimpd 232	. . . . . . . . . 10 ⊢ ((𝑆 SubGraph 𝐺 ∧ 𝐹 ∈ Word dom (iEdg‘𝑆) ∧ 𝑘 ∈ (0..^(♯‘𝐹))) → (if-((𝑃‘𝑘) = (𝑃‘(𝑘 + 1)), ((iEdg‘𝑆)‘(𝐹‘𝑘)) = {(𝑃‘𝑘)}, {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ⊆ ((iEdg‘𝑆)‘(𝐹‘𝑘))) → if-((𝑃‘𝑘) = (𝑃‘(𝑘 + 1)), ((iEdg‘𝐺)‘(𝐹‘𝑘)) = {(𝑃‘𝑘)}, {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ⊆ ((iEdg‘𝐺)‘(𝐹‘𝑘)))))
36	35	3expia 1139	. . . . . . . . 9 ⊢ ((𝑆 SubGraph 𝐺 ∧ 𝐹 ∈ Word dom (iEdg‘𝑆)) → (𝑘 ∈ (0..^(♯‘𝐹)) → (if-((𝑃‘𝑘) = (𝑃‘(𝑘 + 1)), ((iEdg‘𝑆)‘(𝐹‘𝑘)) = {(𝑃‘𝑘)}, {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ⊆ ((iEdg‘𝑆)‘(𝐹‘𝑘))) → if-((𝑃‘𝑘) = (𝑃‘(𝑘 + 1)), ((iEdg‘𝐺)‘(𝐹‘𝑘)) = {(𝑃‘𝑘)}, {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ⊆ ((iEdg‘𝐺)‘(𝐹‘𝑘))))))
37	36	ralrimiv 3155	. . . . . . . 8 ⊢ ((𝑆 SubGraph 𝐺 ∧ 𝐹 ∈ Word dom (iEdg‘𝑆)) → ∀𝑘 ∈ (0..^(♯‘𝐹))(if-((𝑃‘𝑘) = (𝑃‘(𝑘 + 1)), ((iEdg‘𝑆)‘(𝐹‘𝑘)) = {(𝑃‘𝑘)}, {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ⊆ ((iEdg‘𝑆)‘(𝐹‘𝑘))) → if-((𝑃‘𝑘) = (𝑃‘(𝑘 + 1)), ((iEdg‘𝐺)‘(𝐹‘𝑘)) = {(𝑃‘𝑘)}, {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ⊆ ((iEdg‘𝐺)‘(𝐹‘𝑘)))))
38		ralim 3104	. . . . . . . 8 ⊢ (∀𝑘 ∈ (0..^(♯‘𝐹))(if-((𝑃‘𝑘) = (𝑃‘(𝑘 + 1)), ((iEdg‘𝑆)‘(𝐹‘𝑘)) = {(𝑃‘𝑘)}, {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ⊆ ((iEdg‘𝑆)‘(𝐹‘𝑘))) → if-((𝑃‘𝑘) = (𝑃‘(𝑘 + 1)), ((iEdg‘𝐺)‘(𝐹‘𝑘)) = {(𝑃‘𝑘)}, {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ⊆ ((iEdg‘𝐺)‘(𝐹‘𝑘)))) → (∀𝑘 ∈ (0..^(♯‘𝐹))if-((𝑃‘𝑘) = (𝑃‘(𝑘 + 1)), ((iEdg‘𝑆)‘(𝐹‘𝑘)) = {(𝑃‘𝑘)}, {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ⊆ ((iEdg‘𝑆)‘(𝐹‘𝑘))) → ∀𝑘 ∈ (0..^(♯‘𝐹))if-((𝑃‘𝑘) = (𝑃‘(𝑘 + 1)), ((iEdg‘𝐺)‘(𝐹‘𝑘)) = {(𝑃‘𝑘)}, {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ⊆ ((iEdg‘𝐺)‘(𝐹‘𝑘)))))
39	37, 38	syl 18	. . . . . . 7 ⊢ ((𝑆 SubGraph 𝐺 ∧ 𝐹 ∈ Word dom (iEdg‘𝑆)) → (∀𝑘 ∈ (0..^(♯‘𝐹))if-((𝑃‘𝑘) = (𝑃‘(𝑘 + 1)), ((iEdg‘𝑆)‘(𝐹‘𝑘)) = {(𝑃‘𝑘)}, {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ⊆ ((iEdg‘𝑆)‘(𝐹‘𝑘))) → ∀𝑘 ∈ (0..^(♯‘𝐹))if-((𝑃‘𝑘) = (𝑃‘(𝑘 + 1)), ((iEdg‘𝐺)‘(𝐹‘𝑘)) = {(𝑃‘𝑘)}, {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ⊆ ((iEdg‘𝐺)‘(𝐹‘𝑘)))))
40	39	expimpd 459	. . . . . 6 ⊢ (𝑆 SubGraph 𝐺 → ((𝐹 ∈ Word dom (iEdg‘𝑆) ∧ ∀𝑘 ∈ (0..^(♯‘𝐹))if-((𝑃‘𝑘) = (𝑃‘(𝑘 + 1)), ((iEdg‘𝑆)‘(𝐹‘𝑘)) = {(𝑃‘𝑘)}, {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ⊆ ((iEdg‘𝑆)‘(𝐹‘𝑘)))) → ∀𝑘 ∈ (0..^(♯‘𝐹))if-((𝑃‘𝑘) = (𝑃‘(𝑘 + 1)), ((iEdg‘𝐺)‘(𝐹‘𝑘)) = {(𝑃‘𝑘)}, {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ⊆ ((iEdg‘𝐺)‘(𝐹‘𝑘)))))
41	23, 40	syl5 35	. . . . 5 ⊢ (𝑆 SubGraph 𝐺 → ((𝐹 ∈ Word dom (iEdg‘𝑆) ∧ 𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝑆) ∧ ∀𝑘 ∈ (0..^(♯‘𝐹))if-((𝑃‘𝑘) = (𝑃‘(𝑘 + 1)), ((iEdg‘𝑆)‘(𝐹‘𝑘)) = {(𝑃‘𝑘)}, {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ⊆ ((iEdg‘𝑆)‘(𝐹‘𝑘)))) → ∀𝑘 ∈ (0..^(♯‘𝐹))if-((𝑃‘𝑘) = (𝑃‘(𝑘 + 1)), ((iEdg‘𝐺)‘(𝐹‘𝑘)) = {(𝑃‘𝑘)}, {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ⊆ ((iEdg‘𝐺)‘(𝐹‘𝑘)))))
42	22, 41	jcad 522	. . . 4 ⊢ (𝑆 SubGraph 𝐺 → ((𝐹 ∈ Word dom (iEdg‘𝑆) ∧ 𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝑆) ∧ ∀𝑘 ∈ (0..^(♯‘𝐹))if-((𝑃‘𝑘) = (𝑃‘(𝑘 + 1)), ((iEdg‘𝑆)‘(𝐹‘𝑘)) = {(𝑃‘𝑘)}, {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ⊆ ((iEdg‘𝑆)‘(𝐹‘𝑘)))) → ((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))))))
43	6, 42	sylbid 243	. . 3 ⊢ (𝑆 SubGraph 𝐺 → (𝐹(Walks‘𝑆)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))))))
44		df-3an 1105	. . 3 ⊢ ((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))if-((𝑃‘𝑘) = (𝑃‘(𝑘 + 1)), ((iEdg‘𝐺)‘(𝐹‘𝑘)) = {(𝑃‘𝑘)}, {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ⊆ ((iEdg‘𝐺)‘(𝐹‘𝑘)))))
45	43, 44	imbitrrdi 255	. 2 ⊢ (𝑆 SubGraph 𝐺 → (𝐹(Walks‘𝑆)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))))))
46	8, 9	iswlkg 30081	. . 3 ⊢ (𝐺 ∈ 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))))))
47	1, 46	simpl2im 513	. 2 ⊢ (𝑆 SubGraph 𝐺 → (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))))))
48	45, 47	sylibrd 262	1 ⊢ (𝑆 SubGraph 𝐺 → (𝐹(Walks‘𝑆)𝑃 → 𝐹(Walks‘𝐺)𝑃))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401  if-wif 1078   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145  ∀wral 3078  Vcvv 3453   ⊆ wss 3902  𝒫 cpw 4560  {csn 4587  {cpr 4589   class class class wbr 5107  dom cdm 5659   ↾ cres 5661  ⟶wf 6533  ‘cfv 6537  (class class class)co 7417  0cc0 11128  1c1 11129   + caddc 11131  ...cfz 13565  ..^cfzo 13713  ♯chash 14398  Word cword 14582  Vtxcvtx 29461  iEdgciedg 29462  Edgcedg 29512   SubGraph csubgr 29735  Walkscwlks 30064

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-rep 5236  ax-sep 5255  ax-nul 5267  ax-pow 5334  ax-pr 5402  ax-un 7740  ax-cnex 11184  ax-resscn 11185  ax-1cn 11186  ax-icn 11187  ax-addcl 11188  ax-addrcl 11189  ax-mulcl 11190  ax-mulrcl 11191  ax-mulcom 11192  ax-addass 11193  ax-mulass 11194  ax-distr 11195  ax-i2m1 11196  ax-1ne0 11197  ax-1rid 11198  ax-rnegex 11199  ax-rrecex 11200  ax-cnre 11201  ax-pre-lttri 11202  ax-pre-lttrn 11203  ax-pre-ltadd 11204  ax-pre-mulgt0 11205

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-ifp 1079  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-nel 3064  df-ral 3079  df-rex 3089  df-reu 3368  df-rab 3415  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-int 4911  df-iun 4956  df-br 5108  df-opab 5172  df-mpt 5191  df-tr 5217  df-id 5554  df-eprel 5559  df-po 5567  df-so 5568  df-fr 5612  df-we 5614  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-pred 6303  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7374  df-ov 7420  df-oprab 7421  df-mpo 7422  df-om 7867  df-1st 7990  df-2nd 7991  df-frecs 8284  df-wrecs 8315  df-recs 8364  df-rdg 8403  df-1o 8459  df-er 8700  df-map 8832  df-pm 8833  df-en 8957  df-dom 8958  df-sdom 8959  df-fin 8960  df-card 9948  df-pnf 11273  df-mnf 11274  df-xr 11275  df-ltxr 11276  df-le 11277  df-sub 11471  df-neg 11472  df-nn 12262  df-n0 12533  df-z 12620  df-uz 12892  df-fz 13566  df-fzo 13714  df-hash 14399  df-word 14583  df-subgr 29736  df-wlks 30067

This theorem is used by:  subgrtrl  30181