Theorem subgrwlk 35119

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 29197	. . . . . 6 ⊢ (𝑆 SubGraph 𝐺 → (𝑆 ∈ V ∧ 𝐺 ∈ V))
2	1	simpld 494	. . . . 5 ⊢ (𝑆 SubGraph 𝐺 → 𝑆 ∈ V)
3		eqid 2729	. . . . . 6 ⊢ (Vtx‘𝑆) = (Vtx‘𝑆)
4		eqid 2729	. . . . . 6 ⊢ (iEdg‘𝑆) = (iEdg‘𝑆)
5	3, 4	iswlkg 29541	. . . . 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 1148	. . . . . 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 2729	. . . . . . . . . . 11 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
9		eqid 2729	. . . . . . . . . . 11 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺)
10		eqid 2729	. . . . . . . . . . 11 ⊢ (Edg‘𝑆) = (Edg‘𝑆)
11	3, 8, 4, 9, 10	subgrprop2 29201	. . . . . . . . . 10 ⊢ (𝑆 SubGraph 𝐺 → ((Vtx‘𝑆) ⊆ (Vtx‘𝐺) ∧ (iEdg‘𝑆) ⊆ (iEdg‘𝐺) ∧ (Edg‘𝑆) ⊆ 𝒫 (Vtx‘𝑆)))
12	11	simp2d 1143	. . . . . . . . 9 ⊢ (𝑆 SubGraph 𝐺 → (iEdg‘𝑆) ⊆ (iEdg‘𝐺))
13		dmss 5866	. . . . . . . . 9 ⊢ ((iEdg‘𝑆) ⊆ (iEdg‘𝐺) → dom (iEdg‘𝑆) ⊆ dom (iEdg‘𝐺))
14		sswrd 14487	. . . . . . . . 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 3945	. . . . . . 7 ⊢ (𝑆 SubGraph 𝐺 → (𝐹 ∈ Word dom (iEdg‘𝑆) → 𝐹 ∈ Word dom (iEdg‘𝐺)))
17	11	simp1d 1142	. . . . . . . 8 ⊢ (𝑆 SubGraph 𝐺 → (Vtx‘𝑆) ⊆ (Vtx‘𝐺))
18		fss 6704	. . . . . . . . 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 609	. . . . . 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 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) ∧ ∀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 29200	. . . . . . . . . . . . . . . . 17 ⊢ (𝑆 SubGraph 𝐺 → ((Vtx‘𝑆) ⊆ (Vtx‘𝐺) ∧ (iEdg‘𝑆) = ((iEdg‘𝐺) ↾ dom (iEdg‘𝑆)) ∧ (Edg‘𝑆) ⊆ 𝒫 (Vtx‘𝑆)))
25	24	simp2d 1143	. . . . . . . . . . . . . . . 16 ⊢ (𝑆 SubGraph 𝐺 → (iEdg‘𝑆) = ((iEdg‘𝐺) ↾ dom (iEdg‘𝑆)))
26	25	fveq1d 6860	. . . . . . . . . . . . . . 15 ⊢ (𝑆 SubGraph 𝐺 → ((iEdg‘𝑆)‘(𝐹‘𝑘)) = (((iEdg‘𝐺) ↾ dom (iEdg‘𝑆))‘(𝐹‘𝑘)))
27	26	3ad2ant1 1133	. . . . . . . . . . . . . 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 14492	. . . . . . . . . . . . . . . 16 ⊢ ((𝐹 ∈ Word dom (iEdg‘𝑆) ∧ 𝑘 ∈ (0..^(♯‘𝐹))) → (𝐹‘𝑘) ∈ dom (iEdg‘𝑆))
29	28	fvresd 6878	. . . . . . . . . . . . . . 15 ⊢ ((𝐹 ∈ Word dom (iEdg‘𝑆) ∧ 𝑘 ∈ (0..^(♯‘𝐹))) → (((iEdg‘𝐺) ↾ dom (iEdg‘𝑆))‘(𝐹‘𝑘)) = ((iEdg‘𝐺)‘(𝐹‘𝑘)))
30	29	3adant1 1130	. . . . . . . . . . . . . 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 2764	. . . . . . . . . . . . 13 ⊢ ((𝑆 SubGraph 𝐺 ∧ 𝐹 ∈ Word dom (iEdg‘𝑆) ∧ 𝑘 ∈ (0..^(♯‘𝐹))) → ((iEdg‘𝑆)‘(𝐹‘𝑘)) = ((iEdg‘𝐺)‘(𝐹‘𝑘)))
32	31	eqeq1d 2731	. . . . . . . . . . . 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 3979	. . . . . . . . . . . 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 1079	. . . . . . . . . . 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 1121	. . . . . . . . 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 3124	. . . . . . . 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 3069	. . . . . . . 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 1088	. . 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 29541	. . 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 1062   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109  ∀wral 3044  Vcvv 3447   ⊆ wss 3914  𝒫 cpw 4563  {csn 4589  {cpr 4591   class class class wbr 5107  dom cdm 5638   ↾ cres 5640  ⟶wf 6507  ‘cfv 6511  (class class class)co 7387  0cc0 11068  1c1 11069   + caddc 11071  ...cfz 13468  ..^cfzo 13615  ♯chash 14295  Word cword 14478  Vtxcvtx 28923  iEdgciedg 28924  Edgcedg 28974   SubGraph csubgr 29194  Walkscwlks 29524

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 5234  ax-sep 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387  ax-un 7711  ax-cnex 11124  ax-resscn 11125  ax-1cn 11126  ax-icn 11127  ax-addcl 11128  ax-addrcl 11129  ax-mulcl 11130  ax-mulrcl 11131  ax-mulcom 11132  ax-addass 11133  ax-mulass 11134  ax-distr 11135  ax-i2m1 11136  ax-1ne0 11137  ax-1rid 11138  ax-rnegex 11139  ax-rrecex 11140  ax-cnre 11141  ax-pre-lttri 11142  ax-pre-lttrn 11143  ax-pre-ltadd 11144  ax-pre-mulgt0 11145

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 3355  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3934  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-int 4911  df-iun 4957  df-br 5108  df-opab 5170  df-mpt 5189  df-tr 5215  df-id 5533  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5591  df-we 5593  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6274  df-ord 6335  df-on 6336  df-lim 6337  df-suc 6338  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518  df-fv 6519  df-riota 7344  df-ov 7390  df-oprab 7391  df-mpo 7392  df-om 7843  df-1st 7968  df-2nd 7969  df-frecs 8260  df-wrecs 8291  df-recs 8340  df-rdg 8378  df-1o 8434  df-er 8671  df-map 8801  df-pm 8802  df-en 8919  df-dom 8920  df-sdom 8921  df-fin 8922  df-card 9892  df-pnf 11210  df-mnf 11211  df-xr 11212  df-ltxr 11213  df-le 11214  df-sub 11407  df-neg 11408  df-nn 12187  df-n0 12443  df-z 12530  df-uz 12794  df-fz 13469  df-fzo 13616  df-hash 14296  df-word 14479  df-subgr 29195  df-wlks 29527

This theorem is referenced by:  subgrtrl  35120