Theorem usg2wlkonot 26410

Description: A walk of length 2 between two vertices as ordered triple in an undirected simple graph. This theorem would also hold for undirected multigraphs, but to prove this the cases 𝐴 = 𝐵 and/or 𝐵 = 𝐶 must be considered separately. (Contributed by Alexander van der Vekens, 18-Feb-2018.)

Assertion

Ref	Expression
usg2wlkonot	⊢ ((𝑉 USGrph 𝐸 ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) → (⟨𝐴, 𝐵, 𝐶⟩ ∈ (𝐴(𝑉 2WalksOnOt 𝐸)𝐶) ↔ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸)))

Proof of Theorem usg2wlkonot

Dummy variables 𝑓 𝑝 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		usgrav 25867	. . 3 ⊢ (𝑉 USGrph 𝐸 → (𝑉 ∈ V ∧ 𝐸 ∈ V))
2		el2wlkonotot 26400	. . . 4 ⊢ (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) → (⟨𝐴, 𝐵, 𝐶⟩ ∈ (𝐴(𝑉 2WalksOnOt 𝐸)𝐶) ↔ ∃𝑓∃𝑝(𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)))))
3	2	3adantr2 1214	. . 3 ⊢ (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) → (⟨𝐴, 𝐵, 𝐶⟩ ∈ (𝐴(𝑉 2WalksOnOt 𝐸)𝐶) ↔ ∃𝑓∃𝑝(𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)))))
4	1, 3	sylan 487	. 2 ⊢ ((𝑉 USGrph 𝐸 ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) → (⟨𝐴, 𝐵, 𝐶⟩ ∈ (𝐴(𝑉 2WalksOnOt 𝐸)𝐶) ↔ ∃𝑓∃𝑝(𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)))))
5		vex 3176	. . . . . . . . . . . 12 ⊢ 𝑓 ∈ V
6		vex 3176	. . . . . . . . . . . 12 ⊢ 𝑝 ∈ V
7	5, 6	pm3.2i 470	. . . . . . . . . . 11 ⊢ (𝑓 ∈ V ∧ 𝑝 ∈ V)
8		is2wlk 26095	. . . . . . . . . . 11 ⊢ (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝑓 ∈ V ∧ 𝑝 ∈ V)) → ((𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2) ↔ (𝑓:(0..^2)⟶dom 𝐸 ∧ 𝑝:(0...2)⟶𝑉 ∧ ((𝐸‘(𝑓‘0)) = {(𝑝‘0), (𝑝‘1)} ∧ (𝐸‘(𝑓‘1)) = {(𝑝‘1), (𝑝‘2)}))))
9	1, 7, 8	sylancl 693	. . . . . . . . . 10 ⊢ (𝑉 USGrph 𝐸 → ((𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2) ↔ (𝑓:(0..^2)⟶dom 𝐸 ∧ 𝑝:(0...2)⟶𝑉 ∧ ((𝐸‘(𝑓‘0)) = {(𝑝‘0), (𝑝‘1)} ∧ (𝐸‘(𝑓‘1)) = {(𝑝‘1), (𝑝‘2)}))))
10		preq12 4214	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1)) → {𝐴, 𝐵} = {(𝑝‘0), (𝑝‘1)})
11	10	3adant3 1074	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)) → {𝐴, 𝐵} = {(𝑝‘0), (𝑝‘1)})
12	11	eqeq2d 2620	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)) → ((𝐸‘(𝑓‘0)) = {𝐴, 𝐵} ↔ (𝐸‘(𝑓‘0)) = {(𝑝‘0), (𝑝‘1)}))
13		preq12 4214	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)) → {𝐵, 𝐶} = {(𝑝‘1), (𝑝‘2)})
14	13	3adant1 1072	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)) → {𝐵, 𝐶} = {(𝑝‘1), (𝑝‘2)})
15	14	eqeq2d 2620	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)) → ((𝐸‘(𝑓‘1)) = {𝐵, 𝐶} ↔ (𝐸‘(𝑓‘1)) = {(𝑝‘1), (𝑝‘2)}))
16	12, 15	anbi12d 743	. . . . . . . . . . . . . 14 ⊢ ((𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)) → (((𝐸‘(𝑓‘0)) = {𝐴, 𝐵} ∧ (𝐸‘(𝑓‘1)) = {𝐵, 𝐶}) ↔ ((𝐸‘(𝑓‘0)) = {(𝑝‘0), (𝑝‘1)} ∧ (𝐸‘(𝑓‘1)) = {(𝑝‘1), (𝑝‘2)})))
17	16	bicomd 212	. . . . . . . . . . . . 13 ⊢ ((𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)) → (((𝐸‘(𝑓‘0)) = {(𝑝‘0), (𝑝‘1)} ∧ (𝐸‘(𝑓‘1)) = {(𝑝‘1), (𝑝‘2)}) ↔ ((𝐸‘(𝑓‘0)) = {𝐴, 𝐵} ∧ (𝐸‘(𝑓‘1)) = {𝐵, 𝐶})))
18	17	3anbi3d 1397	. . . . . . . . . . . 12 ⊢ ((𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)) → ((𝑓:(0..^2)⟶dom 𝐸 ∧ 𝑝:(0...2)⟶𝑉 ∧ ((𝐸‘(𝑓‘0)) = {(𝑝‘0), (𝑝‘1)} ∧ (𝐸‘(𝑓‘1)) = {(𝑝‘1), (𝑝‘2)})) ↔ (𝑓:(0..^2)⟶dom 𝐸 ∧ 𝑝:(0...2)⟶𝑉 ∧ ((𝐸‘(𝑓‘0)) = {𝐴, 𝐵} ∧ (𝐸‘(𝑓‘1)) = {𝐵, 𝐶}))))
19		usgrafun 25878	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑉 USGrph 𝐸 → Fun 𝐸)
20		c0ex 9913	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ 0 ∈ V
21	20	prid1 4241	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ 0 ∈ {0, 1}
22		fzo0to2pr 12420	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (0..^2) = {0, 1}
23	21, 22	eleqtrri 2687	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ 0 ∈ (0..^2)
24		ffvelrn 6265	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑓:(0..^2)⟶dom 𝐸 ∧ 0 ∈ (0..^2)) → (𝑓‘0) ∈ dom 𝐸)
25	23, 24	mpan2 703	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑓:(0..^2)⟶dom 𝐸 → (𝑓‘0) ∈ dom 𝐸)
26		fvelrn 6260	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((Fun 𝐸 ∧ (𝑓‘0) ∈ dom 𝐸) → (𝐸‘(𝑓‘0)) ∈ ran 𝐸)
27	25, 26	sylan2 490	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((Fun 𝐸 ∧ 𝑓:(0..^2)⟶dom 𝐸) → (𝐸‘(𝑓‘0)) ∈ ran 𝐸)
28		eleq1 2676	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐸‘(𝑓‘0)) = {𝐴, 𝐵} → ((𝐸‘(𝑓‘0)) ∈ ran 𝐸 ↔ {𝐴, 𝐵} ∈ ran 𝐸))
29	27, 28	syl5ibcom 234	. . . . . . . . . . . . . . . . . . 19 ⊢ ((Fun 𝐸 ∧ 𝑓:(0..^2)⟶dom 𝐸) → ((𝐸‘(𝑓‘0)) = {𝐴, 𝐵} → {𝐴, 𝐵} ∈ ran 𝐸))
30		1ex 9914	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ 1 ∈ V
31	30	prid2 4242	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ 1 ∈ {0, 1}
32	31, 22	eleqtrri 2687	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ 1 ∈ (0..^2)
33		ffvelrn 6265	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑓:(0..^2)⟶dom 𝐸 ∧ 1 ∈ (0..^2)) → (𝑓‘1) ∈ dom 𝐸)
34	32, 33	mpan2 703	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑓:(0..^2)⟶dom 𝐸 → (𝑓‘1) ∈ dom 𝐸)
35		fvelrn 6260	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((Fun 𝐸 ∧ (𝑓‘1) ∈ dom 𝐸) → (𝐸‘(𝑓‘1)) ∈ ran 𝐸)
36	34, 35	sylan2 490	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((Fun 𝐸 ∧ 𝑓:(0..^2)⟶dom 𝐸) → (𝐸‘(𝑓‘1)) ∈ ran 𝐸)
37		eleq1 2676	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐸‘(𝑓‘1)) = {𝐵, 𝐶} → ((𝐸‘(𝑓‘1)) ∈ ran 𝐸 ↔ {𝐵, 𝐶} ∈ ran 𝐸))
38	36, 37	syl5ibcom 234	. . . . . . . . . . . . . . . . . . 19 ⊢ ((Fun 𝐸 ∧ 𝑓:(0..^2)⟶dom 𝐸) → ((𝐸‘(𝑓‘1)) = {𝐵, 𝐶} → {𝐵, 𝐶} ∈ ran 𝐸))
39	29, 38	anim12d 584	. . . . . . . . . . . . . . . . . 18 ⊢ ((Fun 𝐸 ∧ 𝑓:(0..^2)⟶dom 𝐸) → (((𝐸‘(𝑓‘0)) = {𝐴, 𝐵} ∧ (𝐸‘(𝑓‘1)) = {𝐵, 𝐶}) → ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸)))
40	19, 39	sylan 487	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑉 USGrph 𝐸 ∧ 𝑓:(0..^2)⟶dom 𝐸) → (((𝐸‘(𝑓‘0)) = {𝐴, 𝐵} ∧ (𝐸‘(𝑓‘1)) = {𝐵, 𝐶}) → ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸)))
41	40	a1d 25	. . . . . . . . . . . . . . . 16 ⊢ ((𝑉 USGrph 𝐸 ∧ 𝑓:(0..^2)⟶dom 𝐸) → ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) → (((𝐸‘(𝑓‘0)) = {𝐴, 𝐵} ∧ (𝐸‘(𝑓‘1)) = {𝐵, 𝐶}) → ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸))))
42	41	expcom 450	. . . . . . . . . . . . . . 15 ⊢ (𝑓:(0..^2)⟶dom 𝐸 → (𝑉 USGrph 𝐸 → ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) → (((𝐸‘(𝑓‘0)) = {𝐴, 𝐵} ∧ (𝐸‘(𝑓‘1)) = {𝐵, 𝐶}) → ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸)))))
43	42	com24 93	. . . . . . . . . . . . . 14 ⊢ (𝑓:(0..^2)⟶dom 𝐸 → (((𝐸‘(𝑓‘0)) = {𝐴, 𝐵} ∧ (𝐸‘(𝑓‘1)) = {𝐵, 𝐶}) → ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) → (𝑉 USGrph 𝐸 → ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸)))))
44	43	a1d 25	. . . . . . . . . . . . 13 ⊢ (𝑓:(0..^2)⟶dom 𝐸 → (𝑝:(0...2)⟶𝑉 → (((𝐸‘(𝑓‘0)) = {𝐴, 𝐵} ∧ (𝐸‘(𝑓‘1)) = {𝐵, 𝐶}) → ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) → (𝑉 USGrph 𝐸 → ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸))))))
45	44	3imp 1249	. . . . . . . . . . . 12 ⊢ ((𝑓:(0..^2)⟶dom 𝐸 ∧ 𝑝:(0...2)⟶𝑉 ∧ ((𝐸‘(𝑓‘0)) = {𝐴, 𝐵} ∧ (𝐸‘(𝑓‘1)) = {𝐵, 𝐶})) → ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) → (𝑉 USGrph 𝐸 → ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸))))
46	18, 45	syl6bi 242	. . . . . . . . . . 11 ⊢ ((𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)) → ((𝑓:(0..^2)⟶dom 𝐸 ∧ 𝑝:(0...2)⟶𝑉 ∧ ((𝐸‘(𝑓‘0)) = {(𝑝‘0), (𝑝‘1)} ∧ (𝐸‘(𝑓‘1)) = {(𝑝‘1), (𝑝‘2)})) → ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) → (𝑉 USGrph 𝐸 → ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸)))))
47	46	com14 94	. . . . . . . . . 10 ⊢ (𝑉 USGrph 𝐸 → ((𝑓:(0..^2)⟶dom 𝐸 ∧ 𝑝:(0...2)⟶𝑉 ∧ ((𝐸‘(𝑓‘0)) = {(𝑝‘0), (𝑝‘1)} ∧ (𝐸‘(𝑓‘1)) = {(𝑝‘1), (𝑝‘2)})) → ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) → ((𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)) → ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸)))))
48	9, 47	sylbid 229	. . . . . . . . 9 ⊢ (𝑉 USGrph 𝐸 → ((𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2) → ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) → ((𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)) → ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸)))))
49	48	com14 94	. . . . . . . 8 ⊢ ((𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)) → ((𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2) → ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) → (𝑉 USGrph 𝐸 → ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸)))))
50	49	expdcom 454	. . . . . . 7 ⊢ (𝑓(𝑉 Walks 𝐸)𝑝 → ((#‘𝑓) = 2 → ((𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)) → ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) → (𝑉 USGrph 𝐸 → ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸))))))
51	50	3imp 1249	. . . . . 6 ⊢ ((𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2))) → ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) → (𝑉 USGrph 𝐸 → ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸))))
52	51	com13 86	. . . . 5 ⊢ (𝑉 USGrph 𝐸 → ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) → ((𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2))) → ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸))))
53	52	imp 444	. . . 4 ⊢ ((𝑉 USGrph 𝐸 ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) → ((𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2))) → ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸)))
54	53	exlimdvv 1849	. . 3 ⊢ ((𝑉 USGrph 𝐸 ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) → (∃𝑓∃𝑝(𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2))) → ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸)))
55		usg2wlk 26145	. . . . 5 ⊢ ((𝑉 USGrph 𝐸 ∧ {𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸) → ∃𝑓∃𝑝(𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2))))
56	55	3expib 1260	. . . 4 ⊢ (𝑉 USGrph 𝐸 → (({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸) → ∃𝑓∃𝑝(𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)))))
57	56	adantr 480	. . 3 ⊢ ((𝑉 USGrph 𝐸 ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) → (({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸) → ∃𝑓∃𝑝(𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)))))
58	54, 57	impbid 201	. 2 ⊢ ((𝑉 USGrph 𝐸 ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) → (∃𝑓∃𝑝(𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2))) ↔ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸)))
59	4, 58	bitrd 267	1 ⊢ ((𝑉 USGrph 𝐸 ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) → (⟨𝐴, 𝐵, 𝐶⟩ ∈ (𝐴(𝑉 2WalksOnOt 𝐸)𝐶) ↔ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸)))

Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475  ∃wex 1695   ∈ wcel 1977  Vcvv 3173  {cpr 4127  ⟨cotp 4133   class class class wbr 4583  dom cdm 5038  ran crn 5039  Fun wfun 5798  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549  0cc0 9815  1c1 9816  2c2 10947  ...cfz 12197  ..^cfzo 12334  #chash 12979   USGrph cusg 25859   Walks cwalk 26026   2WalksOnOt c2wlkonot 26382

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-ot 4134  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  df-2wlkonot 26385

This theorem is referenced by:  usg2spthonot  26415  usg2spthonot0  26416  frg2woteu  26582  frg2woteqm  26586