Theorem subgrwlk 35314

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 29339	. . . . . 6 ⊢ (𝑆 SubGraph 𝐺 → (𝑆 ∈ V ∧ 𝐺 ∈ V))
2	1	simpld 494	. . . . 5 ⊢ (𝑆 SubGraph 𝐺 → 𝑆 ∈ V)
3		eqid 2736	. . . . . 6 ⊢ (Vtx‘𝑆) = (Vtx‘𝑆)
4		eqid 2736	. . . . . 6 ⊢ (iEdg‘𝑆) = (iEdg‘𝑆)
5	3, 4	iswlkg 29682	. . . . 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 17	. . . 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 1149	. . . . . 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 2736	. . . . . . . . . . 11 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
9		eqid 2736	. . . . . . . . . . 11 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺)
10		eqid 2736	. . . . . . . . . . 11 ⊢ (Edg‘𝑆) = (Edg‘𝑆)
11	3, 8, 4, 9, 10	subgrprop2 29343	. . . . . . . . . 10 ⊢ (𝑆 SubGraph 𝐺 → ((Vtx‘𝑆) ⊆ (Vtx‘𝐺) ∧ (iEdg‘𝑆) ⊆ (iEdg‘𝐺) ∧ (Edg‘𝑆) ⊆ 𝒫 (Vtx‘𝑆)))
12	11	simp2d 1144	. . . . . . . . 9 ⊢ (𝑆 SubGraph 𝐺 → (iEdg‘𝑆) ⊆ (iEdg‘𝐺))
13		dmss 5857	. . . . . . . . 9 ⊢ ((iEdg‘𝑆) ⊆ (iEdg‘𝐺) → dom (iEdg‘𝑆) ⊆ dom (iEdg‘𝐺))
14		sswrd 14484	. . . . . . . . 9 ⊢ (dom (iEdg‘𝑆) ⊆ dom (iEdg‘𝐺) → Word dom (iEdg‘𝑆) ⊆ Word dom (iEdg‘𝐺))
15	12, 13, 14	3syl 18	. . . . . . . 8 ⊢ (𝑆 SubGraph 𝐺 → Word dom (iEdg‘𝑆) ⊆ Word dom (iEdg‘𝐺))
16	15	sseld 3920	. . . . . . 7 ⊢ (𝑆 SubGraph 𝐺 → (𝐹 ∈ Word dom (iEdg‘𝑆) → 𝐹 ∈ Word dom (iEdg‘𝐺)))
17	11	simp1d 1143	. . . . . . . 8 ⊢ (𝑆 SubGraph 𝐺 → (Vtx‘𝑆) ⊆ (Vtx‘𝐺))
18		fss 6684	. . . . . . . . 9 ⊢ ((𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝑆) ∧ (Vtx‘𝑆) ⊆ (Vtx‘𝐺)) → 𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺))
19	18	expcom 413	. . . . . . . 8 ⊢ ((Vtx‘𝑆) ⊆ (Vtx‘𝐺) → (𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝑆) → 𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺)))
20	17, 19	syl 17	. . . . . . 7 ⊢ (𝑆 SubGraph 𝐺 → (𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝑆) → 𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺)))
21	16, 20	anim12d 610	. . . . . 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 34	. . . . 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 1150	. . . . . 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 29342	. . . . . . . . . . . . . . . . 17 ⊢ (𝑆 SubGraph 𝐺 → ((Vtx‘𝑆) ⊆ (Vtx‘𝐺) ∧ (iEdg‘𝑆) = ((iEdg‘𝐺) ↾ dom (iEdg‘𝑆)) ∧ (Edg‘𝑆) ⊆ 𝒫 (Vtx‘𝑆)))
25	24	simp2d 1144	. . . . . . . . . . . . . . . 16 ⊢ (𝑆 SubGraph 𝐺 → (iEdg‘𝑆) = ((iEdg‘𝐺) ↾ dom (iEdg‘𝑆)))
26	25	fveq1d 6842	. . . . . . . . . . . . . . 15 ⊢ (𝑆 SubGraph 𝐺 → ((iEdg‘𝑆)‘(𝐹‘𝑘)) = (((iEdg‘𝐺) ↾ dom (iEdg‘𝑆))‘(𝐹‘𝑘)))
27	26	3ad2ant1 1134	. . . . . . . . . . . . . 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 14489	. . . . . . . . . . . . . . . 16 ⊢ ((𝐹 ∈ Word dom (iEdg‘𝑆) ∧ 𝑘 ∈ (0..^(♯‘𝐹))) → (𝐹‘𝑘) ∈ dom (iEdg‘𝑆))
29	28	fvresd 6860	. . . . . . . . . . . . . . 15 ⊢ ((𝐹 ∈ Word dom (iEdg‘𝑆) ∧ 𝑘 ∈ (0..^(♯‘𝐹))) → (((iEdg‘𝐺) ↾ dom (iEdg‘𝑆))‘(𝐹‘𝑘)) = ((iEdg‘𝐺)‘(𝐹‘𝑘)))
30	29	3adant1 1131	. . . . . . . . . . . . . 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 2771	. . . . . . . . . . . . 13 ⊢ ((𝑆 SubGraph 𝐺 ∧ 𝐹 ∈ Word dom (iEdg‘𝑆) ∧ 𝑘 ∈ (0..^(♯‘𝐹))) → ((iEdg‘𝑆)‘(𝐹‘𝑘)) = ((iEdg‘𝐺)‘(𝐹‘𝑘)))
32	31	eqeq1d 2738	. . . . . . . . . . . 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 3954	. . . . . . . . . . . 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 1080	. . . . . . . . . . 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 229	. . . . . . . . . 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 1122	. . . . . . . . 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 3128	. . . . . . . 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 3077	. . . . . . . 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 17	. . . . . . 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 453	. . . . . 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 34	. . . . 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 512	. . . 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 240	. . 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 1089	. . 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 252	. 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 29682	. . 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 503	. 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 259	1 ⊢ (𝑆 SubGraph 𝐺 → (𝐹(Walks‘𝑆)𝑃 → 𝐹(Walks‘𝐺)𝑃))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395  if-wif 1063   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114  ∀wral 3051  Vcvv 3429   ⊆ wss 3889  𝒫 cpw 4541  {csn 4567  {cpr 4569   class class class wbr 5085  dom cdm 5631   ↾ cres 5633  ⟶wf 6494  ‘cfv 6498  (class class class)co 7367  0cc0 11038  1c1 11039   + caddc 11041  ...cfz 13461  ..^cfzo 13608  ♯chash 14292  Word cword 14475  Vtxcvtx 29065  iEdgciedg 29066  Edgcedg 29116   SubGraph csubgr 29336  Walkscwlks 29665

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375  ax-un 7689  ax-cnex 11094  ax-resscn 11095  ax-1cn 11096  ax-icn 11097  ax-addcl 11098  ax-addrcl 11099  ax-mulcl 11100  ax-mulrcl 11101  ax-mulcom 11102  ax-addass 11103  ax-mulass 11104  ax-distr 11105  ax-i2m1 11106  ax-1ne0 11107  ax-1rid 11108  ax-rnegex 11109  ax-rrecex 11110  ax-cnre 11111  ax-pre-lttri 11112  ax-pre-lttrn 11113  ax-pre-ltadd 11114  ax-pre-mulgt0 11115

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-ifp 1064  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3062  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-pss 3909  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-int 4890  df-iun 4935  df-br 5086  df-opab 5148  df-mpt 5167  df-tr 5193  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6265  df-ord 6326  df-on 6327  df-lim 6328  df-suc 6329  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506  df-riota 7324  df-ov 7370  df-oprab 7371  df-mpo 7372  df-om 7818  df-1st 7942  df-2nd 7943  df-frecs 8231  df-wrecs 8262  df-recs 8311  df-rdg 8349  df-1o 8405  df-er 8643  df-map 8775  df-pm 8776  df-en 8894  df-dom 8895  df-sdom 8896  df-fin 8897  df-card 9863  df-pnf 11181  df-mnf 11182  df-xr 11183  df-ltxr 11184  df-le 11185  df-sub 11379  df-neg 11380  df-nn 12175  df-n0 12438  df-z 12525  df-uz 12789  df-fz 13462  df-fzo 13609  df-hash 14293  df-word 14476  df-subgr 29337  df-wlks 29668

This theorem is referenced by:  subgrtrl  35315