Theorem subgrwlk 35490

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 29525	. . . . . 6 ⊢ (𝑆 SubGraph 𝐺 → (𝑆 ∈ V ∧ 𝐺 ∈ V))
2	1	simpld 499	. . . . 5 ⊢ (𝑆 SubGraph 𝐺 → 𝑆 ∈ V)
3		eqid 2765	. . . . . 6 ⊢ (Vtx‘𝑆) = (Vtx‘𝑆)
4		eqid 2765	. . . . . 6 ⊢ (iEdg‘𝑆) = (iEdg‘𝑆)
5	3, 4	iswlkg 29868	. . . . 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 1164	. . . . . 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 2765	. . . . . . . . . . 11 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
9		eqid 2765	. . . . . . . . . . 11 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺)
10		eqid 2765	. . . . . . . . . . 11 ⊢ (Edg‘𝑆) = (Edg‘𝑆)
11	3, 8, 4, 9, 10	subgrprop2 29529	. . . . . . . . . 10 ⊢ (𝑆 SubGraph 𝐺 → ((Vtx‘𝑆) ⊆ (Vtx‘𝐺) ∧ (iEdg‘𝑆) ⊆ (iEdg‘𝐺) ∧ (Edg‘𝑆) ⊆ 𝒫 (Vtx‘𝑆)))
12	11	simp2d 1159	. . . . . . . . 9 ⊢ (𝑆 SubGraph 𝐺 → (iEdg‘𝑆) ⊆ (iEdg‘𝐺))
13		dmss 5882	. . . . . . . . 9 ⊢ ((iEdg‘𝑆) ⊆ (iEdg‘𝐺) → dom (iEdg‘𝑆) ⊆ dom (iEdg‘𝐺))
14		sswrd 14547	. . . . . . . . 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 3938	. . . . . . 7 ⊢ (𝑆 SubGraph 𝐺 → (𝐹 ∈ Word dom (iEdg‘𝑆) → 𝐹 ∈ Word dom (iEdg‘𝐺)))
17	11	simp1d 1158	. . . . . . . 8 ⊢ (𝑆 SubGraph 𝐺 → (Vtx‘𝑆) ⊆ (Vtx‘𝐺))
18		fss 6712	. . . . . . . . 9 ⊢ ((𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝑆) ∧ (Vtx‘𝑆) ⊆ (Vtx‘𝐺)) → 𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺))
19	18	expcom 418	. . . . . . . 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 620	. . . . . 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 1165	. . . . . 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 29528	. . . . . . . . . . . . . . . . 17 ⊢ (𝑆 SubGraph 𝐺 → ((Vtx‘𝑆) ⊆ (Vtx‘𝐺) ∧ (iEdg‘𝑆) = ((iEdg‘𝐺) ↾ dom (iEdg‘𝑆)) ∧ (Edg‘𝑆) ⊆ 𝒫 (Vtx‘𝑆)))
25	24	simp2d 1159	. . . . . . . . . . . . . . . 16 ⊢ (𝑆 SubGraph 𝐺 → (iEdg‘𝑆) = ((iEdg‘𝐺) ↾ dom (iEdg‘𝑆)))
26	25	fveq1d 6873	. . . . . . . . . . . . . . 15 ⊢ (𝑆 SubGraph 𝐺 → ((iEdg‘𝑆)‘(𝐹‘𝑘)) = (((iEdg‘𝐺) ↾ dom (iEdg‘𝑆))‘(𝐹‘𝑘)))
27	26	3ad2ant1 1149	. . . . . . . . . . . . . 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 14552	. . . . . . . . . . . . . . . 16 ⊢ ((𝐹 ∈ Word dom (iEdg‘𝑆) ∧ 𝑘 ∈ (0..^(♯‘𝐹))) → (𝐹‘𝑘) ∈ dom (iEdg‘𝑆))
29	28	fvresd 6891	. . . . . . . . . . . . . . 15 ⊢ ((𝐹 ∈ Word dom (iEdg‘𝑆) ∧ 𝑘 ∈ (0..^(♯‘𝐹))) → (((iEdg‘𝐺) ↾ dom (iEdg‘𝑆))‘(𝐹‘𝑘)) = ((iEdg‘𝐺)‘(𝐹‘𝑘)))
30	29	3adant1 1146	. . . . . . . . . . . . . 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 2800	. . . . . . . . . . . . 13 ⊢ ((𝑆 SubGraph 𝐺 ∧ 𝐹 ∈ Word dom (iEdg‘𝑆) ∧ 𝑘 ∈ (0..^(♯‘𝐹))) → ((iEdg‘𝑆)‘(𝐹‘𝑘)) = ((iEdg‘𝐺)‘(𝐹‘𝑘)))
32	31	eqeq1d 2767	. . . . . . . . . . . 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 3971	. . . . . . . . . . . 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 1094	. . . . . . . . . . 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 1137	. . . . . . . . 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 3156	. . . . . . . 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 3105	. . . . . . . 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 458	. . . . . 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 521	. . . 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 1103	. . 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 29868	. . 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 512	. 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))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400  if-wif 1076   ∧ w3a 1101   = wceq 1563   ∈ wcel 2145  ∀wral 3079  Vcvv 3457   ⊆ wss 3907  𝒫 cpw 4558  {csn 4585  {cpr 4587   class class class wbr 5104  dom cdm 5651   ↾ cres 5653  ⟶wf 6521  ‘cfv 6525  (class class class)co 7400  0cc0 11088  1c1 11089   + caddc 11091  ...cfz 13523  ..^cfzo 13670  ♯chash 14354  Word cword 14538  Vtxcvtx 29251  iEdgciedg 29252  Edgcedg 29302   SubGraph csubgr 29522  Walkscwlks 29851

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737  ax-rep 5231  ax-sep 5250  ax-nul 5260  ax-pow 5326  ax-pr 5394  ax-un 7722  ax-cnex 11144  ax-resscn 11145  ax-1cn 11146  ax-icn 11147  ax-addcl 11148  ax-addrcl 11149  ax-mulcl 11150  ax-mulrcl 11151  ax-mulcom 11152  ax-addass 11153  ax-mulass 11154  ax-distr 11155  ax-i2m1 11156  ax-1ne0 11157  ax-1rid 11158  ax-rnegex 11159  ax-rrecex 11160  ax-cnre 11161  ax-pre-lttri 11162  ax-pre-lttrn 11163  ax-pre-ltadd 11164  ax-pre-mulgt0 11165

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-ifp 1077  df-3or 1102  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-nel 3065  df-ral 3080  df-rex 3090  df-reu 3371  df-rab 3418  df-v 3459  df-sbc 3748  df-csb 3856  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-pss 3927  df-nul 4289  df-if 4484  df-pw 4560  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-int 4908  df-iun 4953  df-br 5105  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-pred 6291  df-ord 6352  df-on 6353  df-lim 6354  df-suc 6355  df-iota 6481  df-fun 6527  df-fn 6528  df-f 6529  df-f1 6530  df-fo 6531  df-f1o 6532  df-fv 6533  df-riota 7357  df-ov 7403  df-oprab 7404  df-mpo 7405  df-om 7851  df-1st 7974  df-2nd 7975  df-frecs 8266  df-wrecs 8297  df-recs 8346  df-rdg 8385  df-1o 8441  df-er 8682  df-map 8814  df-pm 8815  df-en 8932  df-dom 8933  df-sdom 8934  df-fin 8935  df-card 9913  df-pnf 11233  df-mnf 11234  df-xr 11235  df-ltxr 11236  df-le 11237  df-sub 11431  df-neg 11432  df-nn 12222  df-n0 12493  df-z 12580  df-uz 12851  df-fz 13524  df-fzo 13671  df-hash 14355  df-word 14539  df-subgr 29523  df-wlks 29854

This theorem is referenced by:  subgrtrl  35491