Theorem usgra2adedgwlkonALT 26144

Description: Alternate proof for usgra2adedgwlkon 26143, using usgra2adedgwlk 26142, but with a longer proof! In an undirected simple graph, two adjacent edges form a walk between two (different) vertices. (Contributed by Alexander van der Vekens, 18-Feb-2018.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypotheses

Ref	Expression
usgra2adedgspth.f	⊢ 𝐹 = {⟨0, (◡𝐸‘{𝐴, 𝐵})⟩, ⟨1, (◡𝐸‘{𝐵, 𝐶})⟩}
usgra2adedgspth.p	⊢ 𝑃 = {⟨0, 𝐴⟩, ⟨1, 𝐵⟩, ⟨2, 𝐶⟩}

Assertion

Ref	Expression
usgra2adedgwlkonALT	⊢ (𝑉 USGrph 𝐸 → (({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸) → 𝐹(𝐴(𝑉 WalkOn 𝐸)𝐶)𝑃))

Proof of Theorem usgra2adedgwlkonALT

Step	Hyp	Ref	Expression
1		usgra2adedgspth.f	. . 3 ⊢ 𝐹 = {⟨0, (◡𝐸‘{𝐴, 𝐵})⟩, ⟨1, (◡𝐸‘{𝐵, 𝐶})⟩}
2		usgra2adedgspth.p	. . 3 ⊢ 𝑃 = {⟨0, 𝐴⟩, ⟨1, 𝐵⟩, ⟨2, 𝐶⟩}
3	1, 2	usgra2adedgwlk 26142	. 2 ⊢ (𝑉 USGrph 𝐸 → (({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸) → (𝐹(𝑉 Walks 𝐸)𝑃 ∧ (#‘𝐹) = 2 ∧ (𝐴 = (𝑃‘0) ∧ 𝐵 = (𝑃‘1) ∧ 𝐶 = (𝑃‘2)))))
4		simp1 1054	. . . 4 ⊢ ((𝐹(𝑉 Walks 𝐸)𝑃 ∧ (#‘𝐹) = 2 ∧ (𝐴 = (𝑃‘0) ∧ 𝐵 = (𝑃‘1) ∧ 𝐶 = (𝑃‘2))) → 𝐹(𝑉 Walks 𝐸)𝑃)
5		eqcom 2617	. . . . . . 7 ⊢ (𝐴 = (𝑃‘0) ↔ (𝑃‘0) = 𝐴)
6	5	biimpi 205	. . . . . 6 ⊢ (𝐴 = (𝑃‘0) → (𝑃‘0) = 𝐴)
7	6	3ad2ant1 1075	. . . . 5 ⊢ ((𝐴 = (𝑃‘0) ∧ 𝐵 = (𝑃‘1) ∧ 𝐶 = (𝑃‘2)) → (𝑃‘0) = 𝐴)
8	7	3ad2ant3 1077	. . . 4 ⊢ ((𝐹(𝑉 Walks 𝐸)𝑃 ∧ (#‘𝐹) = 2 ∧ (𝐴 = (𝑃‘0) ∧ 𝐵 = (𝑃‘1) ∧ 𝐶 = (𝑃‘2))) → (𝑃‘0) = 𝐴)
9		fveq2 6103	. . . . . 6 ⊢ ((#‘𝐹) = 2 → (𝑃‘(#‘𝐹)) = (𝑃‘2))
10		eqcom 2617	. . . . . . . 8 ⊢ (𝐶 = (𝑃‘2) ↔ (𝑃‘2) = 𝐶)
11	10	biimpi 205	. . . . . . 7 ⊢ (𝐶 = (𝑃‘2) → (𝑃‘2) = 𝐶)
12	11	3ad2ant3 1077	. . . . . 6 ⊢ ((𝐴 = (𝑃‘0) ∧ 𝐵 = (𝑃‘1) ∧ 𝐶 = (𝑃‘2)) → (𝑃‘2) = 𝐶)
13	9, 12	sylan9eq 2664	. . . . 5 ⊢ (((#‘𝐹) = 2 ∧ (𝐴 = (𝑃‘0) ∧ 𝐵 = (𝑃‘1) ∧ 𝐶 = (𝑃‘2))) → (𝑃‘(#‘𝐹)) = 𝐶)
14	13	3adant1 1072	. . . 4 ⊢ ((𝐹(𝑉 Walks 𝐸)𝑃 ∧ (#‘𝐹) = 2 ∧ (𝐴 = (𝑃‘0) ∧ 𝐵 = (𝑃‘1) ∧ 𝐶 = (𝑃‘2))) → (𝑃‘(#‘𝐹)) = 𝐶)
15	4, 8, 14	3jca 1235	. . 3 ⊢ ((𝐹(𝑉 Walks 𝐸)𝑃 ∧ (#‘𝐹) = 2 ∧ (𝐴 = (𝑃‘0) ∧ 𝐵 = (𝑃‘1) ∧ 𝐶 = (𝑃‘2))) → (𝐹(𝑉 Walks 𝐸)𝑃 ∧ (𝑃‘0) = 𝐴 ∧ (𝑃‘(#‘𝐹)) = 𝐶))
16		wlkbprop 26051	. . . . . 6 ⊢ (𝐹(𝑉 Walks 𝐸)𝑃 → ((#‘𝐹) ∈ ℕ0 ∧ (𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V)))
17		2mwlk 26049	. . . . . 6 ⊢ (𝐹(𝑉 Walks 𝐸)𝑃 → (𝐹 ∈ Word dom 𝐸 ∧ 𝑃:(0...(#‘𝐹))⟶𝑉))
18		simp-4l 802	. . . . . . . . 9 ⊢ ((((((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V)) ∧ (𝐹 ∈ Word dom 𝐸 ∧ 𝑃:(0...(#‘𝐹))⟶𝑉)) ∧ (#‘𝐹) = 2) ∧ (𝐴 = (𝑃‘0) ∧ 𝐵 = (𝑃‘1) ∧ 𝐶 = (𝑃‘2))) → (𝑉 ∈ V ∧ 𝐸 ∈ V))
19		simp-4r 803	. . . . . . . . 9 ⊢ ((((((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V)) ∧ (𝐹 ∈ Word dom 𝐸 ∧ 𝑃:(0...(#‘𝐹))⟶𝑉)) ∧ (#‘𝐹) = 2) ∧ (𝐴 = (𝑃‘0) ∧ 𝐵 = (𝑃‘1) ∧ 𝐶 = (𝑃‘2))) → (𝐹 ∈ V ∧ 𝑃 ∈ V))
20		oveq2 6557	. . . . . . . . . . . . . . . 16 ⊢ ((#‘𝐹) = 2 → (0...(#‘𝐹)) = (0...2))
21	20	feq2d 5944	. . . . . . . . . . . . . . 15 ⊢ ((#‘𝐹) = 2 → (𝑃:(0...(#‘𝐹))⟶𝑉 ↔ 𝑃:(0...2)⟶𝑉))
22	21	anbi2d 736	. . . . . . . . . . . . . 14 ⊢ ((#‘𝐹) = 2 → ((𝐹 ∈ Word dom 𝐸 ∧ 𝑃:(0...(#‘𝐹))⟶𝑉) ↔ (𝐹 ∈ Word dom 𝐸 ∧ 𝑃:(0...2)⟶𝑉)))
23	22	anbi2d 736	. . . . . . . . . . . . 13 ⊢ ((#‘𝐹) = 2 → ((((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V)) ∧ (𝐹 ∈ Word dom 𝐸 ∧ 𝑃:(0...(#‘𝐹))⟶𝑉)) ↔ (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V)) ∧ (𝐹 ∈ Word dom 𝐸 ∧ 𝑃:(0...2)⟶𝑉))))
24		c0ex 9913	. . . . . . . . . . . . . . . . . . 19 ⊢ 0 ∈ V
25	24	tpid1 4246	. . . . . . . . . . . . . . . . . 18 ⊢ 0 ∈ {0, 1, 2}
26		fz0tp 12309	. . . . . . . . . . . . . . . . . 18 ⊢ (0...2) = {0, 1, 2}
27	25, 26	eleqtrri 2687	. . . . . . . . . . . . . . . . 17 ⊢ 0 ∈ (0...2)
28		ffvelrn 6265	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑃:(0...2)⟶𝑉 ∧ 0 ∈ (0...2)) → (𝑃‘0) ∈ 𝑉)
29	27, 28	mpan2 703	. . . . . . . . . . . . . . . 16 ⊢ (𝑃:(0...2)⟶𝑉 → (𝑃‘0) ∈ 𝑉)
30		2z 11286	. . . . . . . . . . . . . . . . . . . 20 ⊢ 2 ∈ ℤ
31	30	elexi 3186	. . . . . . . . . . . . . . . . . . 19 ⊢ 2 ∈ V
32	31	tpid3 4250	. . . . . . . . . . . . . . . . . 18 ⊢ 2 ∈ {0, 1, 2}
33	32, 26	eleqtrri 2687	. . . . . . . . . . . . . . . . 17 ⊢ 2 ∈ (0...2)
34		ffvelrn 6265	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑃:(0...2)⟶𝑉 ∧ 2 ∈ (0...2)) → (𝑃‘2) ∈ 𝑉)
35	33, 34	mpan2 703	. . . . . . . . . . . . . . . 16 ⊢ (𝑃:(0...2)⟶𝑉 → (𝑃‘2) ∈ 𝑉)
36	29, 35	jca 553	. . . . . . . . . . . . . . 15 ⊢ (𝑃:(0...2)⟶𝑉 → ((𝑃‘0) ∈ 𝑉 ∧ (𝑃‘2) ∈ 𝑉))
37	36	adantl 481	. . . . . . . . . . . . . 14 ⊢ ((𝐹 ∈ Word dom 𝐸 ∧ 𝑃:(0...2)⟶𝑉) → ((𝑃‘0) ∈ 𝑉 ∧ (𝑃‘2) ∈ 𝑉))
38	37	adantl 481	. . . . . . . . . . . . 13 ⊢ ((((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V)) ∧ (𝐹 ∈ Word dom 𝐸 ∧ 𝑃:(0...2)⟶𝑉)) → ((𝑃‘0) ∈ 𝑉 ∧ (𝑃‘2) ∈ 𝑉))
39	23, 38	syl6bi 242	. . . . . . . . . . . 12 ⊢ ((#‘𝐹) = 2 → ((((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V)) ∧ (𝐹 ∈ Word dom 𝐸 ∧ 𝑃:(0...(#‘𝐹))⟶𝑉)) → ((𝑃‘0) ∈ 𝑉 ∧ (𝑃‘2) ∈ 𝑉)))
40	39	impcom 445	. . . . . . . . . . 11 ⊢ (((((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V)) ∧ (𝐹 ∈ Word dom 𝐸 ∧ 𝑃:(0...(#‘𝐹))⟶𝑉)) ∧ (#‘𝐹) = 2) → ((𝑃‘0) ∈ 𝑉 ∧ (𝑃‘2) ∈ 𝑉))
41	40	adantr 480	. . . . . . . . . 10 ⊢ ((((((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V)) ∧ (𝐹 ∈ Word dom 𝐸 ∧ 𝑃:(0...(#‘𝐹))⟶𝑉)) ∧ (#‘𝐹) = 2) ∧ (𝐴 = (𝑃‘0) ∧ 𝐵 = (𝑃‘1) ∧ 𝐶 = (𝑃‘2))) → ((𝑃‘0) ∈ 𝑉 ∧ (𝑃‘2) ∈ 𝑉))
42		eleq1 2676	. . . . . . . . . . . . 13 ⊢ (𝐴 = (𝑃‘0) → (𝐴 ∈ 𝑉 ↔ (𝑃‘0) ∈ 𝑉))
43		eleq1 2676	. . . . . . . . . . . . 13 ⊢ (𝐶 = (𝑃‘2) → (𝐶 ∈ 𝑉 ↔ (𝑃‘2) ∈ 𝑉))
44	42, 43	bi2anan9 913	. . . . . . . . . . . 12 ⊢ ((𝐴 = (𝑃‘0) ∧ 𝐶 = (𝑃‘2)) → ((𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) ↔ ((𝑃‘0) ∈ 𝑉 ∧ (𝑃‘2) ∈ 𝑉)))
45	44	3adant2 1073	. . . . . . . . . . 11 ⊢ ((𝐴 = (𝑃‘0) ∧ 𝐵 = (𝑃‘1) ∧ 𝐶 = (𝑃‘2)) → ((𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) ↔ ((𝑃‘0) ∈ 𝑉 ∧ (𝑃‘2) ∈ 𝑉)))
46	45	adantl 481	. . . . . . . . . 10 ⊢ ((((((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V)) ∧ (𝐹 ∈ Word dom 𝐸 ∧ 𝑃:(0...(#‘𝐹))⟶𝑉)) ∧ (#‘𝐹) = 2) ∧ (𝐴 = (𝑃‘0) ∧ 𝐵 = (𝑃‘1) ∧ 𝐶 = (𝑃‘2))) → ((𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) ↔ ((𝑃‘0) ∈ 𝑉 ∧ (𝑃‘2) ∈ 𝑉)))
47	41, 46	mpbird 246	. . . . . . . . 9 ⊢ ((((((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V)) ∧ (𝐹 ∈ Word dom 𝐸 ∧ 𝑃:(0...(#‘𝐹))⟶𝑉)) ∧ (#‘𝐹) = 2) ∧ (𝐴 = (𝑃‘0) ∧ 𝐵 = (𝑃‘1) ∧ 𝐶 = (𝑃‘2))) → (𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉))
48	18, 19, 47	3jca 1235	. . . . . . . 8 ⊢ ((((((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V)) ∧ (𝐹 ∈ Word dom 𝐸 ∧ 𝑃:(0...(#‘𝐹))⟶𝑉)) ∧ (#‘𝐹) = 2) ∧ (𝐴 = (𝑃‘0) ∧ 𝐵 = (𝑃‘1) ∧ 𝐶 = (𝑃‘2))) → ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V) ∧ (𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)))
49	48	exp41 636	. . . . . . 7 ⊢ (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V)) → ((𝐹 ∈ Word dom 𝐸 ∧ 𝑃:(0...(#‘𝐹))⟶𝑉) → ((#‘𝐹) = 2 → ((𝐴 = (𝑃‘0) ∧ 𝐵 = (𝑃‘1) ∧ 𝐶 = (𝑃‘2)) → ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V) ∧ (𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉))))))
50	49	3adant1 1072	. . . . . 6 ⊢ (((#‘𝐹) ∈ ℕ0 ∧ (𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V)) → ((𝐹 ∈ Word dom 𝐸 ∧ 𝑃:(0...(#‘𝐹))⟶𝑉) → ((#‘𝐹) = 2 → ((𝐴 = (𝑃‘0) ∧ 𝐵 = (𝑃‘1) ∧ 𝐶 = (𝑃‘2)) → ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V) ∧ (𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉))))))
51	16, 17, 50	sylc 63	. . . . 5 ⊢ (𝐹(𝑉 Walks 𝐸)𝑃 → ((#‘𝐹) = 2 → ((𝐴 = (𝑃‘0) ∧ 𝐵 = (𝑃‘1) ∧ 𝐶 = (𝑃‘2)) → ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V) ∧ (𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)))))
52	51	3imp 1249	. . . 4 ⊢ ((𝐹(𝑉 Walks 𝐸)𝑃 ∧ (#‘𝐹) = 2 ∧ (𝐴 = (𝑃‘0) ∧ 𝐵 = (𝑃‘1) ∧ 𝐶 = (𝑃‘2))) → ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V) ∧ (𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)))
53		iswlkon 26062	. . . 4 ⊢ (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V) ∧ (𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) → (𝐹(𝐴(𝑉 WalkOn 𝐸)𝐶)𝑃 ↔ (𝐹(𝑉 Walks 𝐸)𝑃 ∧ (𝑃‘0) = 𝐴 ∧ (𝑃‘(#‘𝐹)) = 𝐶)))
54	52, 53	syl 17	. . 3 ⊢ ((𝐹(𝑉 Walks 𝐸)𝑃 ∧ (#‘𝐹) = 2 ∧ (𝐴 = (𝑃‘0) ∧ 𝐵 = (𝑃‘1) ∧ 𝐶 = (𝑃‘2))) → (𝐹(𝐴(𝑉 WalkOn 𝐸)𝐶)𝑃 ↔ (𝐹(𝑉 Walks 𝐸)𝑃 ∧ (𝑃‘0) = 𝐴 ∧ (𝑃‘(#‘𝐹)) = 𝐶)))
55	15, 54	mpbird 246	. 2 ⊢ ((𝐹(𝑉 Walks 𝐸)𝑃 ∧ (#‘𝐹) = 2 ∧ (𝐴 = (𝑃‘0) ∧ 𝐵 = (𝑃‘1) ∧ 𝐶 = (𝑃‘2))) → 𝐹(𝐴(𝑉 WalkOn 𝐸)𝐶)𝑃)
56	3, 55	syl6 34	1 ⊢ (𝑉 USGrph 𝐸 → (({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸) → 𝐹(𝐴(𝑉 WalkOn 𝐸)𝐶)𝑃))

Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  Vcvv 3173  {cpr 4127  {ctp 4129  ⟨cop 4131   class class class wbr 4583  ^◡ccnv 5037  dom cdm 5038  ran crn 5039  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549  0cc0 9815  1c1 9816  2c2 10947  ℕ₀cn0 11169  ℤcz 11254  ...cfz 12197  #chash 12979  Word cword 13146   USGrph cusg 25859   Walks cwalk 26026   WalkOn cwlkon 26030

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-er 7629  df-map 7746  df-pm 7747  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-card 8648  df-cda 8873  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-2 10956  df-n0 11170  df-z 11255  df-uz 11564  df-fz 12198  df-fzo 12335  df-hash 12980  df-word 13154  df-usgra 25862  df-wlk 26036  df-wlkon 26042

This theorem is referenced by: (None)