usgr2wlkspth - Mathbox for Alexander van der Vekens

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Theorem usgr2wlkspth 40965

Description: In a simple graph, any walk of length 2 between two different vertices is a simple path. (Contributed by Alexander van der Vekens, 2-Mar-2018.) (Revised by AV, 27-Jan-2021.)

**Assertion**
Ref	Expression
usgr2wlkspth	⊢ ((𝐺 ∈ USGraph ∧ (#‘𝐹) = 2 ∧ 𝐴 ≠ 𝐵) → (𝐹(𝐴(WalksOn‘𝐺)𝐵)𝑃 ↔ 𝐹(𝐴(SPathsOn‘𝐺)𝐵)𝑃))

**Proof of Theorem usgr2wlkspth**
Step	Hyp	Ref	Expression
1		eqid 2610	. . . . . 6 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
2	1	wlkOnprop 40866	. . . . 5 ⊢ (𝐹(𝐴(WalksOn‘𝐺)𝐵)𝑃 → ((𝐺 ∈ V ∧ 𝐴 ∈ (Vtx‘𝐺) ∧ 𝐵 ∈ (Vtx‘𝐺)) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V) ∧ (𝐹(1Walks‘𝐺)𝑃 ∧ (𝑃‘0) = 𝐴 ∧ (𝑃‘(#‘𝐹)) = 𝐵)))
3		id 22	. . . . . . 7 ⊢ ((𝐴 ∈ (Vtx‘𝐺) ∧ 𝐵 ∈ (Vtx‘𝐺)) → (𝐴 ∈ (Vtx‘𝐺) ∧ 𝐵 ∈ (Vtx‘𝐺)))
4	3	3adant1 1072	. . . . . 6 ⊢ ((𝐺 ∈ V ∧ 𝐴 ∈ (Vtx‘𝐺) ∧ 𝐵 ∈ (Vtx‘𝐺)) → (𝐴 ∈ (Vtx‘𝐺) ∧ 𝐵 ∈ (Vtx‘𝐺)))
5	4	3anim1i 1241	. . . . 5 ⊢ (((𝐺 ∈ V ∧ 𝐴 ∈ (Vtx‘𝐺) ∧ 𝐵 ∈ (Vtx‘𝐺)) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V) ∧ (𝐹(1Walks‘𝐺)𝑃 ∧ (𝑃‘0) = 𝐴 ∧ (𝑃‘(#‘𝐹)) = 𝐵)) → ((𝐴 ∈ (Vtx‘𝐺) ∧ 𝐵 ∈ (Vtx‘𝐺)) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V) ∧ (𝐹(1Walks‘𝐺)𝑃 ∧ (𝑃‘0) = 𝐴 ∧ (𝑃‘(#‘𝐹)) = 𝐵)))
6	2, 5	syl 17	. . . 4 ⊢ (𝐹(𝐴(WalksOn‘𝐺)𝐵)𝑃 → ((𝐴 ∈ (Vtx‘𝐺) ∧ 𝐵 ∈ (Vtx‘𝐺)) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V) ∧ (𝐹(1Walks‘𝐺)𝑃 ∧ (𝑃‘0) = 𝐴 ∧ (𝑃‘(#‘𝐹)) = 𝐵)))
7		simpl31 1135	. . . . . . . . 9 ⊢ ((((𝐴 ∈ (Vtx‘𝐺) ∧ 𝐵 ∈ (Vtx‘𝐺)) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V) ∧ (𝐹(1Walks‘𝐺)𝑃 ∧ (𝑃‘0) = 𝐴 ∧ (𝑃‘(#‘𝐹)) = 𝐵)) ∧ (𝐺 ∈ USGraph ∧ (#‘𝐹) = 2 ∧ 𝐴 ≠ 𝐵)) → 𝐹(1Walks‘𝐺)𝑃)
8		simp2 1055	. . . . . . . . . . . . . . 15 ⊢ ((𝐹(1Walks‘𝐺)𝑃 ∧ (𝑃‘0) = 𝐴 ∧ (𝑃‘(#‘𝐹)) = 𝐵) → (𝑃‘0) = 𝐴)
9		simp3 1056	. . . . . . . . . . . . . . 15 ⊢ ((𝐹(1Walks‘𝐺)𝑃 ∧ (𝑃‘0) = 𝐴 ∧ (𝑃‘(#‘𝐹)) = 𝐵) → (𝑃‘(#‘𝐹)) = 𝐵)
10	8, 9	neeq12d 2843	. . . . . . . . . . . . . 14 ⊢ ((𝐹(1Walks‘𝐺)𝑃 ∧ (𝑃‘0) = 𝐴 ∧ (𝑃‘(#‘𝐹)) = 𝐵) → ((𝑃‘0) ≠ (𝑃‘(#‘𝐹)) ↔ 𝐴 ≠ 𝐵))
11	10	bicomd 212	. . . . . . . . . . . . 13 ⊢ ((𝐹(1Walks‘𝐺)𝑃 ∧ (𝑃‘0) = 𝐴 ∧ (𝑃‘(#‘𝐹)) = 𝐵) → (𝐴 ≠ 𝐵 ↔ (𝑃‘0) ≠ (𝑃‘(#‘𝐹))))
12	11	3anbi3d 1397	. . . . . . . . . . . 12 ⊢ ((𝐹(1Walks‘𝐺)𝑃 ∧ (𝑃‘0) = 𝐴 ∧ (𝑃‘(#‘𝐹)) = 𝐵) → ((𝐺 ∈ USGraph ∧ (#‘𝐹) = 2 ∧ 𝐴 ≠ 𝐵) ↔ (𝐺 ∈ USGraph ∧ (#‘𝐹) = 2 ∧ (𝑃‘0) ≠ (𝑃‘(#‘𝐹)))))
13		usgr2wlkspthlem1 40963	. . . . . . . . . . . . . 14 ⊢ ((𝐹(1Walks‘𝐺)𝑃 ∧ (𝐺 ∈ USGraph ∧ (#‘𝐹) = 2 ∧ (𝑃‘0) ≠ (𝑃‘(#‘𝐹)))) → Fun ^◡𝐹)
14	13	ex 449	. . . . . . . . . . . . 13 ⊢ (𝐹(1Walks‘𝐺)𝑃 → ((𝐺 ∈ USGraph ∧ (#‘𝐹) = 2 ∧ (𝑃‘0) ≠ (𝑃‘(#‘𝐹))) → Fun ^◡𝐹))
15	14	3ad2ant1 1075	. . . . . . . . . . . 12 ⊢ ((𝐹(1Walks‘𝐺)𝑃 ∧ (𝑃‘0) = 𝐴 ∧ (𝑃‘(#‘𝐹)) = 𝐵) → ((𝐺 ∈ USGraph ∧ (#‘𝐹) = 2 ∧ (𝑃‘0) ≠ (𝑃‘(#‘𝐹))) → Fun ^◡𝐹))
16	12, 15	sylbid 229	. . . . . . . . . . 11 ⊢ ((𝐹(1Walks‘𝐺)𝑃 ∧ (𝑃‘0) = 𝐴 ∧ (𝑃‘(#‘𝐹)) = 𝐵) → ((𝐺 ∈ USGraph ∧ (#‘𝐹) = 2 ∧ 𝐴 ≠ 𝐵) → Fun ^◡𝐹))
17	16	3ad2ant3 1077	. . . . . . . . . 10 ⊢ (((𝐴 ∈ (Vtx‘𝐺) ∧ 𝐵 ∈ (Vtx‘𝐺)) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V) ∧ (𝐹(1Walks‘𝐺)𝑃 ∧ (𝑃‘0) = 𝐴 ∧ (𝑃‘(#‘𝐹)) = 𝐵)) → ((𝐺 ∈ USGraph ∧ (#‘𝐹) = 2 ∧ 𝐴 ≠ 𝐵) → Fun ^◡𝐹))
18	17	imp 444	. . . . . . . . 9 ⊢ ((((𝐴 ∈ (Vtx‘𝐺) ∧ 𝐵 ∈ (Vtx‘𝐺)) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V) ∧ (𝐹(1Walks‘𝐺)𝑃 ∧ (𝑃‘0) = 𝐴 ∧ (𝑃‘(#‘𝐹)) = 𝐵)) ∧ (𝐺 ∈ USGraph ∧ (#‘𝐹) = 2 ∧ 𝐴 ≠ 𝐵)) → Fun ^◡𝐹)
19		elfvex 6131	. . . . . . . . . . . . . . 15 ⊢ (𝐴 ∈ (Vtx‘𝐺) → 𝐺 ∈ V)
20	19	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ (Vtx‘𝐺) ∧ 𝐵 ∈ (Vtx‘𝐺)) → 𝐺 ∈ V)
21	20	anim1i 590	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ (Vtx‘𝐺) ∧ 𝐵 ∈ (Vtx‘𝐺)) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V)) → (𝐺 ∈ V ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V)))
22		3anass 1035	. . . . . . . . . . . . 13 ⊢ ((𝐺 ∈ V ∧ 𝐹 ∈ V ∧ 𝑃 ∈ V) ↔ (𝐺 ∈ V ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V)))
23	21, 22	sylibr 223	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ (Vtx‘𝐺) ∧ 𝐵 ∈ (Vtx‘𝐺)) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V)) → (𝐺 ∈ V ∧ 𝐹 ∈ V ∧ 𝑃 ∈ V))
24		isTrl 40904	. . . . . . . . . . . 12 ⊢ ((𝐺 ∈ V ∧ 𝐹 ∈ V ∧ 𝑃 ∈ V) → (𝐹(TrailS‘𝐺)𝑃 ↔ (𝐹(1Walks‘𝐺)𝑃 ∧ Fun ^◡𝐹)))
25	23, 24	syl 17	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ (Vtx‘𝐺) ∧ 𝐵 ∈ (Vtx‘𝐺)) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V)) → (𝐹(TrailS‘𝐺)𝑃 ↔ (𝐹(1Walks‘𝐺)𝑃 ∧ Fun ^◡𝐹)))
26	25	3adant3 1074	. . . . . . . . . 10 ⊢ (((𝐴 ∈ (Vtx‘𝐺) ∧ 𝐵 ∈ (Vtx‘𝐺)) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V) ∧ (𝐹(1Walks‘𝐺)𝑃 ∧ (𝑃‘0) = 𝐴 ∧ (𝑃‘(#‘𝐹)) = 𝐵)) → (𝐹(TrailS‘𝐺)𝑃 ↔ (𝐹(1Walks‘𝐺)𝑃 ∧ Fun ^◡𝐹)))
27	26	adantr 480	. . . . . . . . 9 ⊢ ((((𝐴 ∈ (Vtx‘𝐺) ∧ 𝐵 ∈ (Vtx‘𝐺)) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V) ∧ (𝐹(1Walks‘𝐺)𝑃 ∧ (𝑃‘0) = 𝐴 ∧ (𝑃‘(#‘𝐹)) = 𝐵)) ∧ (𝐺 ∈ USGraph ∧ (#‘𝐹) = 2 ∧ 𝐴 ≠ 𝐵)) → (𝐹(TrailS‘𝐺)𝑃 ↔ (𝐹(1Walks‘𝐺)𝑃 ∧ Fun ^◡𝐹)))
28	7, 18, 27	mpbir2and 959	. . . . . . . 8 ⊢ ((((𝐴 ∈ (Vtx‘𝐺) ∧ 𝐵 ∈ (Vtx‘𝐺)) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V) ∧ (𝐹(1Walks‘𝐺)𝑃 ∧ (𝑃‘0) = 𝐴 ∧ (𝑃‘(#‘𝐹)) = 𝐵)) ∧ (𝐺 ∈ USGraph ∧ (#‘𝐹) = 2 ∧ 𝐴 ≠ 𝐵)) → 𝐹(TrailS‘𝐺)𝑃)
29		usgr2wlkspthlem2 40964	. . . . . . . . . . . . 13 ⊢ ((𝐹(1Walks‘𝐺)𝑃 ∧ (𝐺 ∈ USGraph ∧ (#‘𝐹) = 2 ∧ (𝑃‘0) ≠ (𝑃‘(#‘𝐹)))) → Fun ^◡𝑃)
30	29	ex 449	. . . . . . . . . . . 12 ⊢ (𝐹(1Walks‘𝐺)𝑃 → ((𝐺 ∈ USGraph ∧ (#‘𝐹) = 2 ∧ (𝑃‘0) ≠ (𝑃‘(#‘𝐹))) → Fun ^◡𝑃))
31	30	3ad2ant1 1075	. . . . . . . . . . 11 ⊢ ((𝐹(1Walks‘𝐺)𝑃 ∧ (𝑃‘0) = 𝐴 ∧ (𝑃‘(#‘𝐹)) = 𝐵) → ((𝐺 ∈ USGraph ∧ (#‘𝐹) = 2 ∧ (𝑃‘0) ≠ (𝑃‘(#‘𝐹))) → Fun ^◡𝑃))
32	12, 31	sylbid 229	. . . . . . . . . 10 ⊢ ((𝐹(1Walks‘𝐺)𝑃 ∧ (𝑃‘0) = 𝐴 ∧ (𝑃‘(#‘𝐹)) = 𝐵) → ((𝐺 ∈ USGraph ∧ (#‘𝐹) = 2 ∧ 𝐴 ≠ 𝐵) → Fun ^◡𝑃))
33	32	3ad2ant3 1077	. . . . . . . . 9 ⊢ (((𝐴 ∈ (Vtx‘𝐺) ∧ 𝐵 ∈ (Vtx‘𝐺)) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V) ∧ (𝐹(1Walks‘𝐺)𝑃 ∧ (𝑃‘0) = 𝐴 ∧ (𝑃‘(#‘𝐹)) = 𝐵)) → ((𝐺 ∈ USGraph ∧ (#‘𝐹) = 2 ∧ 𝐴 ≠ 𝐵) → Fun ^◡𝑃))
34	33	imp 444	. . . . . . . 8 ⊢ ((((𝐴 ∈ (Vtx‘𝐺) ∧ 𝐵 ∈ (Vtx‘𝐺)) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V) ∧ (𝐹(1Walks‘𝐺)𝑃 ∧ (𝑃‘0) = 𝐴 ∧ (𝑃‘(#‘𝐹)) = 𝐵)) ∧ (𝐺 ∈ USGraph ∧ (#‘𝐹) = 2 ∧ 𝐴 ≠ 𝐵)) → Fun ^◡𝑃)
35	21	3adant3 1074	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ (Vtx‘𝐺) ∧ 𝐵 ∈ (Vtx‘𝐺)) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V) ∧ (𝐹(1Walks‘𝐺)𝑃 ∧ (𝑃‘0) = 𝐴 ∧ (𝑃‘(#‘𝐹)) = 𝐵)) → (𝐺 ∈ V ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V)))
36	35, 22	sylibr 223	. . . . . . . . . 10 ⊢ (((𝐴 ∈ (Vtx‘𝐺) ∧ 𝐵 ∈ (Vtx‘𝐺)) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V) ∧ (𝐹(1Walks‘𝐺)𝑃 ∧ (𝑃‘0) = 𝐴 ∧ (𝑃‘(#‘𝐹)) = 𝐵)) → (𝐺 ∈ V ∧ 𝐹 ∈ V ∧ 𝑃 ∈ V))
37		issPth 40930	. . . . . . . . . 10 ⊢ ((𝐺 ∈ V ∧ 𝐹 ∈ V ∧ 𝑃 ∈ V) → (𝐹(SPathS‘𝐺)𝑃 ↔ (𝐹(TrailS‘𝐺)𝑃 ∧ Fun ^◡𝑃)))
38	36, 37	syl 17	. . . . . . . . 9 ⊢ (((𝐴 ∈ (Vtx‘𝐺) ∧ 𝐵 ∈ (Vtx‘𝐺)) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V) ∧ (𝐹(1Walks‘𝐺)𝑃 ∧ (𝑃‘0) = 𝐴 ∧ (𝑃‘(#‘𝐹)) = 𝐵)) → (𝐹(SPathS‘𝐺)𝑃 ↔ (𝐹(TrailS‘𝐺)𝑃 ∧ Fun ^◡𝑃)))
39	38	adantr 480	. . . . . . . 8 ⊢ ((((𝐴 ∈ (Vtx‘𝐺) ∧ 𝐵 ∈ (Vtx‘𝐺)) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V) ∧ (𝐹(1Walks‘𝐺)𝑃 ∧ (𝑃‘0) = 𝐴 ∧ (𝑃‘(#‘𝐹)) = 𝐵)) ∧ (𝐺 ∈ USGraph ∧ (#‘𝐹) = 2 ∧ 𝐴 ≠ 𝐵)) → (𝐹(SPathS‘𝐺)𝑃 ↔ (𝐹(TrailS‘𝐺)𝑃 ∧ Fun ^◡𝑃)))
40	28, 34, 39	mpbir2and 959	. . . . . . 7 ⊢ ((((𝐴 ∈ (Vtx‘𝐺) ∧ 𝐵 ∈ (Vtx‘𝐺)) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V) ∧ (𝐹(1Walks‘𝐺)𝑃 ∧ (𝑃‘0) = 𝐴 ∧ (𝑃‘(#‘𝐹)) = 𝐵)) ∧ (𝐺 ∈ USGraph ∧ (#‘𝐹) = 2 ∧ 𝐴 ≠ 𝐵)) → 𝐹(SPathS‘𝐺)𝑃)
41		3simpc 1053	. . . . . . . . 9 ⊢ ((𝐹(1Walks‘𝐺)𝑃 ∧ (𝑃‘0) = 𝐴 ∧ (𝑃‘(#‘𝐹)) = 𝐵) → ((𝑃‘0) = 𝐴 ∧ (𝑃‘(#‘𝐹)) = 𝐵))
42	41	3ad2ant3 1077	. . . . . . . 8 ⊢ (((𝐴 ∈ (Vtx‘𝐺) ∧ 𝐵 ∈ (Vtx‘𝐺)) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V) ∧ (𝐹(1Walks‘𝐺)𝑃 ∧ (𝑃‘0) = 𝐴 ∧ (𝑃‘(#‘𝐹)) = 𝐵)) → ((𝑃‘0) = 𝐴 ∧ (𝑃‘(#‘𝐹)) = 𝐵))
43	42	adantr 480	. . . . . . 7 ⊢ ((((𝐴 ∈ (Vtx‘𝐺) ∧ 𝐵 ∈ (Vtx‘𝐺)) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V) ∧ (𝐹(1Walks‘𝐺)𝑃 ∧ (𝑃‘0) = 𝐴 ∧ (𝑃‘(#‘𝐹)) = 𝐵)) ∧ (𝐺 ∈ USGraph ∧ (#‘𝐹) = 2 ∧ 𝐴 ≠ 𝐵)) → ((𝑃‘0) = 𝐴 ∧ (𝑃‘(#‘𝐹)) = 𝐵))
44		3anass 1035	. . . . . . 7 ⊢ ((𝐹(SPathS‘𝐺)𝑃 ∧ (𝑃‘0) = 𝐴 ∧ (𝑃‘(#‘𝐹)) = 𝐵) ↔ (𝐹(SPathS‘𝐺)𝑃 ∧ ((𝑃‘0) = 𝐴 ∧ (𝑃‘(#‘𝐹)) = 𝐵)))
45	40, 43, 44	sylanbrc 695	. . . . . 6 ⊢ ((((𝐴 ∈ (Vtx‘𝐺) ∧ 𝐵 ∈ (Vtx‘𝐺)) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V) ∧ (𝐹(1Walks‘𝐺)𝑃 ∧ (𝑃‘0) = 𝐴 ∧ (𝑃‘(#‘𝐹)) = 𝐵)) ∧ (𝐺 ∈ USGraph ∧ (#‘𝐹) = 2 ∧ 𝐴 ≠ 𝐵)) → (𝐹(SPathS‘𝐺)𝑃 ∧ (𝑃‘0) = 𝐴 ∧ (𝑃‘(#‘𝐹)) = 𝐵))
46		3simpa 1051	. . . . . . . 8 ⊢ (((𝐴 ∈ (Vtx‘𝐺) ∧ 𝐵 ∈ (Vtx‘𝐺)) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V) ∧ (𝐹(1Walks‘𝐺)𝑃 ∧ (𝑃‘0) = 𝐴 ∧ (𝑃‘(#‘𝐹)) = 𝐵)) → ((𝐴 ∈ (Vtx‘𝐺) ∧ 𝐵 ∈ (Vtx‘𝐺)) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V)))
47	46	adantr 480	. . . . . . 7 ⊢ ((((𝐴 ∈ (Vtx‘𝐺) ∧ 𝐵 ∈ (Vtx‘𝐺)) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V) ∧ (𝐹(1Walks‘𝐺)𝑃 ∧ (𝑃‘0) = 𝐴 ∧ (𝑃‘(#‘𝐹)) = 𝐵)) ∧ (𝐺 ∈ USGraph ∧ (#‘𝐹) = 2 ∧ 𝐴 ≠ 𝐵)) → ((𝐴 ∈ (Vtx‘𝐺) ∧ 𝐵 ∈ (Vtx‘𝐺)) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V)))
48	1	isspthonpth-av 40955	. . . . . . 7 ⊢ (((𝐴 ∈ (Vtx‘𝐺) ∧ 𝐵 ∈ (Vtx‘𝐺)) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V)) → (𝐹(𝐴(SPathsOn‘𝐺)𝐵)𝑃 ↔ (𝐹(SPathS‘𝐺)𝑃 ∧ (𝑃‘0) = 𝐴 ∧ (𝑃‘(#‘𝐹)) = 𝐵)))
49	47, 48	syl 17	. . . . . 6 ⊢ ((((𝐴 ∈ (Vtx‘𝐺) ∧ 𝐵 ∈ (Vtx‘𝐺)) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V) ∧ (𝐹(1Walks‘𝐺)𝑃 ∧ (𝑃‘0) = 𝐴 ∧ (𝑃‘(#‘𝐹)) = 𝐵)) ∧ (𝐺 ∈ USGraph ∧ (#‘𝐹) = 2 ∧ 𝐴 ≠ 𝐵)) → (𝐹(𝐴(SPathsOn‘𝐺)𝐵)𝑃 ↔ (𝐹(SPathS‘𝐺)𝑃 ∧ (𝑃‘0) = 𝐴 ∧ (𝑃‘(#‘𝐹)) = 𝐵)))
50	45, 49	mpbird 246	. . . . 5 ⊢ ((((𝐴 ∈ (Vtx‘𝐺) ∧ 𝐵 ∈ (Vtx‘𝐺)) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V) ∧ (𝐹(1Walks‘𝐺)𝑃 ∧ (𝑃‘0) = 𝐴 ∧ (𝑃‘(#‘𝐹)) = 𝐵)) ∧ (𝐺 ∈ USGraph ∧ (#‘𝐹) = 2 ∧ 𝐴 ≠ 𝐵)) → 𝐹(𝐴(SPathsOn‘𝐺)𝐵)𝑃)
51	50	ex 449	. . . 4 ⊢ (((𝐴 ∈ (Vtx‘𝐺) ∧ 𝐵 ∈ (Vtx‘𝐺)) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V) ∧ (𝐹(1Walks‘𝐺)𝑃 ∧ (𝑃‘0) = 𝐴 ∧ (𝑃‘(#‘𝐹)) = 𝐵)) → ((𝐺 ∈ USGraph ∧ (#‘𝐹) = 2 ∧ 𝐴 ≠ 𝐵) → 𝐹(𝐴(SPathsOn‘𝐺)𝐵)𝑃))
52	6, 51	syl 17	. . 3 ⊢ (𝐹(𝐴(WalksOn‘𝐺)𝐵)𝑃 → ((𝐺 ∈ USGraph ∧ (#‘𝐹) = 2 ∧ 𝐴 ≠ 𝐵) → 𝐹(𝐴(SPathsOn‘𝐺)𝐵)𝑃))
53	52	com12 32	. 2 ⊢ ((𝐺 ∈ USGraph ∧ (#‘𝐹) = 2 ∧ 𝐴 ≠ 𝐵) → (𝐹(𝐴(WalksOn‘𝐺)𝐵)𝑃 → 𝐹(𝐴(SPathsOn‘𝐺)𝐵)𝑃))
54		spthonpthon 40957	. . 3 ⊢ (𝐹(𝐴(SPathsOn‘𝐺)𝐵)𝑃 → 𝐹(𝐴(PathsOn‘𝐺)𝐵)𝑃)
55		pthontrlon 40953	. . 3 ⊢ (𝐹(𝐴(PathsOn‘𝐺)𝐵)𝑃 → 𝐹(𝐴(TrailsOn‘𝐺)𝐵)𝑃)
56		trlsonwlkon 40917	. . 3 ⊢ (𝐹(𝐴(TrailsOn‘𝐺)𝐵)𝑃 → 𝐹(𝐴(WalksOn‘𝐺)𝐵)𝑃)
57	54, 55, 56	3syl 18	. 2 ⊢ (𝐹(𝐴(SPathsOn‘𝐺)𝐵)𝑃 → 𝐹(𝐴(WalksOn‘𝐺)𝐵)𝑃)
58	53, 57	impbid1 214	1 ⊢ ((𝐺 ∈ USGraph ∧ (#‘𝐹) = 2 ∧ 𝐴 ≠ 𝐵) → (𝐹(𝐴(WalksOn‘𝐺)𝐵)𝑃 ↔ 𝐹(𝐴(SPathsOn‘𝐺)𝐵)𝑃))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 Vcvv 3173 class class class wbr 4583 ^◡ccnv 5037 Fun wfun 5798 ‘cfv 5804 (class class class)co 6549 0cc0 9815 2c2 10947 #chash 12979 Vtxcvtx 25673 USGraph cusgr 40379 1Walksc1wlks 40796 WalksOncwlkson 40798 TrailSctrls 40899 TrailsOnctrlson 40900 SPathScspths 40920 PathsOncpthson 40921 SPathsOncspthson 40922

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-ifp 1007 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-2o 7448 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-3 10957 df-n0 11170 df-z 11255 df-uz 11564 df-fz 12198 df-fzo 12335 df-hash 12980 df-word 13154 df-concat 13156 df-s1 13157 df-s2 13444 df-s3 13445 df-uhgr 25724 df-upgr 25749 df-umgr 25750 df-edga 25793 df-uspgr 40380 df-usgr 40381 df-1wlks 40800 df-wlks 40801 df-wlkson 40802 df-trls 40901 df-trlson 40902 df-pths 40923 df-spths 40924 df-pthson 40925 df-spthson 40926

This theorem is referenced by: usgr2trlspth 40967 wpthswwlks2on 41164

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