Theorem subgrwlk 35104

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 29233	. . . . . 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 29577	. . . . 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 29237	. . . . . . . . . 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 5849	. . . . . . . . 9 ⊢ ((iEdg‘𝑆) ⊆ (iEdg‘𝐺) → dom (iEdg‘𝑆) ⊆ dom (iEdg‘𝐺))
14		sswrd 14447	. . . . . . . . 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 3936	. . . . . . 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 6672	. . . . . . . . 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 29236	. . . . . . . . . . . . . . . . 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 6828	. . . . . . . . . . . . . . 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 14452	. . . . . . . . . . . . . . . 16 ⊢ ((𝐹 ∈ Word dom (iEdg‘𝑆) ∧ 𝑘 ∈ (0..^(♯‘𝐹))) → (𝐹‘𝑘) ∈ dom (iEdg‘𝑆))
29	28	fvresd 6846	. . . . . . . . . . . . . . 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 3970	. . . . . . . . . . . 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 3120	. . . . . . . 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 29577	. . 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 3438   ⊆ wss 3905  𝒫 cpw 4553  {csn 4579  {cpr 4581   class class class wbr 5095  dom cdm 5623   ↾ cres 5625  ⟶wf 6482  ‘cfv 6486  (class class class)co 7353  0cc0 11028  1c1 11029   + caddc 11031  ...cfz 13428  ..^cfzo 13575  ♯chash 14255  Word cword 14438  Vtxcvtx 28959  iEdgciedg 28960  Edgcedg 29010   SubGraph csubgr 29230  Walkscwlks 29560

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 5221  ax-sep 5238  ax-nul 5248  ax-pow 5307  ax-pr 5374  ax-un 7675  ax-cnex 11084  ax-resscn 11085  ax-1cn 11086  ax-icn 11087  ax-addcl 11088  ax-addrcl 11089  ax-mulcl 11090  ax-mulrcl 11091  ax-mulcom 11092  ax-addass 11093  ax-mulass 11094  ax-distr 11095  ax-i2m1 11096  ax-1ne0 11097  ax-1rid 11098  ax-rnegex 11099  ax-rrecex 11100  ax-cnre 11101  ax-pre-lttri 11102  ax-pre-lttrn 11103  ax-pre-ltadd 11104  ax-pre-mulgt0 11105

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 3346  df-rab 3397  df-v 3440  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-pss 3925  df-nul 4287  df-if 4479  df-pw 4555  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4862  df-int 4900  df-iun 4946  df-br 5096  df-opab 5158  df-mpt 5177  df-tr 5203  df-id 5518  df-eprel 5523  df-po 5531  df-so 5532  df-fr 5576  df-we 5578  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-res 5635  df-ima 5636  df-pred 6253  df-ord 6314  df-on 6315  df-lim 6316  df-suc 6317  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494  df-riota 7310  df-ov 7356  df-oprab 7357  df-mpo 7358  df-om 7807  df-1st 7931  df-2nd 7932  df-frecs 8221  df-wrecs 8252  df-recs 8301  df-rdg 8339  df-1o 8395  df-er 8632  df-map 8762  df-pm 8763  df-en 8880  df-dom 8881  df-sdom 8882  df-fin 8883  df-card 9854  df-pnf 11170  df-mnf 11171  df-xr 11172  df-ltxr 11173  df-le 11174  df-sub 11367  df-neg 11368  df-nn 12147  df-n0 12403  df-z 12490  df-uz 12754  df-fz 13429  df-fzo 13576  df-hash 14256  df-word 14439  df-subgr 29231  df-wlks 29563

This theorem is referenced by:  subgrtrl  35105