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Theorem usg2wotspth 26411

Description: A walk of length 2 between two different vertices as ordered triple corresponds to a simple path of length 2 in an undirected simple graph. (Contributed by Alexander van der Vekens, 16-Feb-2018.)

**Assertion**
Ref	Expression
usg2wotspth	⊢ ((𝑉 USGrph 𝐸 ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) ∧ 𝐴 ≠ 𝐶) → (⟨𝐴, 𝐵, 𝐶⟩ ∈ (𝐴(𝑉 2WalksOnOt 𝐸)𝐶) ↔ ∃𝑓∃𝑝(𝑓(𝑉 SPaths 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)))))

Distinct variable groups: 𝐴,𝑓,𝑝 𝐵,𝑓,𝑝 𝐶,𝑓,𝑝 𝑓,𝐸,𝑝 𝑓,𝑉,𝑝

**Proof of Theorem usg2wotspth**
Step	Hyp	Ref	Expression
1		usgrav 25867	. . . 4 ⊢ (𝑉 USGrph 𝐸 → (𝑉 ∈ V ∧ 𝐸 ∈ V))
2		el2wlkonotot 26400	. . . . 5 ⊢ (((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	. . . 4 ⊢ (((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	. . 3 ⊢ ((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	4	3adant3 1074	. 2 ⊢ ((𝑉 USGrph 𝐸 ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) ∧ 𝐴 ≠ 𝐶) → (⟨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)))))
6		simpr1 1060	. . . . . . 7 ⊢ (((𝑉 USGrph 𝐸 ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) ∧ 𝐴 ≠ 𝐶) ∧ (𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)))) → 𝑓(𝑉 Walks 𝐸)𝑝)
7		vex 3176	. . . . . . . . . . . . . . . . 17 ⊢ 𝑓 ∈ V
8		vex 3176	. . . . . . . . . . . . . . . . 17 ⊢ 𝑝 ∈ V
9	7, 8	pm3.2i 470	. . . . . . . . . . . . . . . 16 ⊢ (𝑓 ∈ V ∧ 𝑝 ∈ V)
10		is2wlk 26095	. . . . . . . . . . . . . . . 16 ⊢ (((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)}))))
11	1, 9, 10	sylancl 693	. . . . . . . . . . . . . . 15 ⊢ (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)}))))
12		usgrafun 25878	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑉 USGrph 𝐸 → Fun 𝐸)
13		c0ex 9913	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ 0 ∈ V
14	13	prid1 4241	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ 0 ∈ {0, 1}
15		fzo0to2pr 12420	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (0..^2) = {0, 1}
16	14, 15	eleqtrri 2687	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ 0 ∈ (0..^2)
17		ffvelrn 6265	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑓:(0..^2)⟶dom 𝐸 ∧ 0 ∈ (0..^2)) → (𝑓‘0) ∈ dom 𝐸)
18	16, 17	mpan2 703	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑓:(0..^2)⟶dom 𝐸 → (𝑓‘0) ∈ dom 𝐸)
19		fvelrn 6260	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((Fun 𝐸 ∧ (𝑓‘0) ∈ dom 𝐸) → (𝐸‘(𝑓‘0)) ∈ ran 𝐸)
20	18, 19	sylan2 490	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((Fun 𝐸 ∧ 𝑓:(0..^2)⟶dom 𝐸) → (𝐸‘(𝑓‘0)) ∈ ran 𝐸)
21		eleq1 2676	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝐸‘(𝑓‘0)) = {(𝑝‘0), (𝑝‘1)} → ((𝐸‘(𝑓‘0)) ∈ ran 𝐸 ↔ {(𝑝‘0), (𝑝‘1)} ∈ ran 𝐸))
22	20, 21	syl5ibcom 234	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((Fun 𝐸 ∧ 𝑓:(0..^2)⟶dom 𝐸) → ((𝐸‘(𝑓‘0)) = {(𝑝‘0), (𝑝‘1)} → {(𝑝‘0), (𝑝‘1)} ∈ ran 𝐸))
23		1ex 9914	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ 1 ∈ V
24	23	prid2 4242	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ 1 ∈ {0, 1}
25	24, 15	eleqtrri 2687	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ 1 ∈ (0..^2)
26		ffvelrn 6265	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑓:(0..^2)⟶dom 𝐸 ∧ 1 ∈ (0..^2)) → (𝑓‘1) ∈ dom 𝐸)
27	25, 26	mpan2 703	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑓:(0..^2)⟶dom 𝐸 → (𝑓‘1) ∈ dom 𝐸)
28		fvelrn 6260	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((Fun 𝐸 ∧ (𝑓‘1) ∈ dom 𝐸) → (𝐸‘(𝑓‘1)) ∈ ran 𝐸)
29	27, 28	sylan2 490	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((Fun 𝐸 ∧ 𝑓:(0..^2)⟶dom 𝐸) → (𝐸‘(𝑓‘1)) ∈ ran 𝐸)
30		eleq1 2676	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝐸‘(𝑓‘1)) = {(𝑝‘1), (𝑝‘2)} → ((𝐸‘(𝑓‘1)) ∈ ran 𝐸 ↔ {(𝑝‘1), (𝑝‘2)} ∈ ran 𝐸))
31	29, 30	syl5ibcom 234	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((Fun 𝐸 ∧ 𝑓:(0..^2)⟶dom 𝐸) → ((𝐸‘(𝑓‘1)) = {(𝑝‘1), (𝑝‘2)} → {(𝑝‘1), (𝑝‘2)} ∈ ran 𝐸))
32	22, 31	anim12d 584	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((Fun 𝐸 ∧ 𝑓:(0..^2)⟶dom 𝐸) → (((𝐸‘(𝑓‘0)) = {(𝑝‘0), (𝑝‘1)} ∧ (𝐸‘(𝑓‘1)) = {(𝑝‘1), (𝑝‘2)}) → ({(𝑝‘0), (𝑝‘1)} ∈ ran 𝐸 ∧ {(𝑝‘1), (𝑝‘2)} ∈ ran 𝐸)))
33	12, 32	sylan 487	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑉 USGrph 𝐸 ∧ 𝑓:(0..^2)⟶dom 𝐸) → (((𝐸‘(𝑓‘0)) = {(𝑝‘0), (𝑝‘1)} ∧ (𝐸‘(𝑓‘1)) = {(𝑝‘1), (𝑝‘2)}) → ({(𝑝‘0), (𝑝‘1)} ∈ ran 𝐸 ∧ {(𝑝‘1), (𝑝‘2)} ∈ ran 𝐸)))
34		simpllr 795	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((((𝑉 USGrph 𝐸 ∧ ({(𝑝‘0), (𝑝‘1)} ∈ ran 𝐸 ∧ {(𝑝‘1), (𝑝‘2)} ∈ ran 𝐸)) ∧ 𝑝:(0...2)⟶𝑉) ∧ 𝐴 ≠ 𝐶) ∧ (𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2))) → 𝑝:(0...2)⟶𝑉)
35		usgraedgrn 25910	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((𝑉 USGrph 𝐸 ∧ {(𝑝‘0), (𝑝‘1)} ∈ ran 𝐸) → (𝑝‘0) ≠ (𝑝‘1))
36	35	ex 449	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑉 USGrph 𝐸 → ({(𝑝‘0), (𝑝‘1)} ∈ ran 𝐸 → (𝑝‘0) ≠ (𝑝‘1)))
37		usgraedgrn 25910	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((𝑉 USGrph 𝐸 ∧ {(𝑝‘1), (𝑝‘2)} ∈ ran 𝐸) → (𝑝‘1) ≠ (𝑝‘2))
38	37	ex 449	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑉 USGrph 𝐸 → ({(𝑝‘1), (𝑝‘2)} ∈ ran 𝐸 → (𝑝‘1) ≠ (𝑝‘2)))
39		simplll 794	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (((((𝑝‘0) ≠ (𝑝‘1) ∧ (𝑝‘1) ≠ (𝑝‘2)) ∧ (𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2))) ∧ 𝐴 ≠ 𝐶) → (𝑝‘0) ≠ (𝑝‘1))
40		simpl 472	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ⊢ ((𝐴 = (𝑝‘0) ∧ 𝐶 = (𝑝‘2)) → 𝐴 = (𝑝‘0))
41		simpr 476	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ⊢ ((𝐴 = (𝑝‘0) ∧ 𝐶 = (𝑝‘2)) → 𝐶 = (𝑝‘2))
42	40, 41	neeq12d 2843	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ⊢ ((𝐴 = (𝑝‘0) ∧ 𝐶 = (𝑝‘2)) → (𝐴 ≠ 𝐶 ↔ (𝑝‘0) ≠ (𝑝‘2)))
43	42	biimpd 218	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ⊢ ((𝐴 = (𝑝‘0) ∧ 𝐶 = (𝑝‘2)) → (𝐴 ≠ 𝐶 → (𝑝‘0) ≠ (𝑝‘2)))
44	43	3adant2 1073	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ ((𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)) → (𝐴 ≠ 𝐶 → (𝑝‘0) ≠ (𝑝‘2)))
45	44	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ ((((𝑝‘0) ≠ (𝑝‘1) ∧ (𝑝‘1) ≠ (𝑝‘2)) ∧ (𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2))) → (𝐴 ≠ 𝐶 → (𝑝‘0) ≠ (𝑝‘2)))
46	45	imp 444	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (((((𝑝‘0) ≠ (𝑝‘1) ∧ (𝑝‘1) ≠ (𝑝‘2)) ∧ (𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2))) ∧ 𝐴 ≠ 𝐶) → (𝑝‘0) ≠ (𝑝‘2))
47		simpllr 795	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (((((𝑝‘0) ≠ (𝑝‘1) ∧ (𝑝‘1) ≠ (𝑝‘2)) ∧ (𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2))) ∧ 𝐴 ≠ 𝐶) → (𝑝‘1) ≠ (𝑝‘2))
48	39, 46, 47	3jca 1235	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (((((𝑝‘0) ≠ (𝑝‘1) ∧ (𝑝‘1) ≠ (𝑝‘2)) ∧ (𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2))) ∧ 𝐴 ≠ 𝐶) → ((𝑝‘0) ≠ (𝑝‘1) ∧ (𝑝‘0) ≠ (𝑝‘2) ∧ (𝑝‘1) ≠ (𝑝‘2)))
49	48	exp31 628	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (((𝑝‘0) ≠ (𝑝‘1) ∧ (𝑝‘1) ≠ (𝑝‘2)) → ((𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)) → (𝐴 ≠ 𝐶 → ((𝑝‘0) ≠ (𝑝‘1) ∧ (𝑝‘0) ≠ (𝑝‘2) ∧ (𝑝‘1) ≠ (𝑝‘2)))))
50	49	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑉 USGrph 𝐸 → (((𝑝‘0) ≠ (𝑝‘1) ∧ (𝑝‘1) ≠ (𝑝‘2)) → ((𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)) → (𝐴 ≠ 𝐶 → ((𝑝‘0) ≠ (𝑝‘1) ∧ (𝑝‘0) ≠ (𝑝‘2) ∧ (𝑝‘1) ≠ (𝑝‘2))))))
51	36, 38, 50	syl2and 499	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑉 USGrph 𝐸 → (({(𝑝‘0), (𝑝‘1)} ∈ ran 𝐸 ∧ {(𝑝‘1), (𝑝‘2)} ∈ ran 𝐸) → ((𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)) → (𝐴 ≠ 𝐶 → ((𝑝‘0) ≠ (𝑝‘1) ∧ (𝑝‘0) ≠ (𝑝‘2) ∧ (𝑝‘1) ≠ (𝑝‘2))))))
52	51	imp 444	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((𝑉 USGrph 𝐸 ∧ ({(𝑝‘0), (𝑝‘1)} ∈ ran 𝐸 ∧ {(𝑝‘1), (𝑝‘2)} ∈ ran 𝐸)) → ((𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)) → (𝐴 ≠ 𝐶 → ((𝑝‘0) ≠ (𝑝‘1) ∧ (𝑝‘0) ≠ (𝑝‘2) ∧ (𝑝‘1) ≠ (𝑝‘2)))))
53	52	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((𝑉 USGrph 𝐸 ∧ ({(𝑝‘0), (𝑝‘1)} ∈ ran 𝐸 ∧ {(𝑝‘1), (𝑝‘2)} ∈ ran 𝐸)) ∧ 𝑝:(0...2)⟶𝑉) → ((𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)) → (𝐴 ≠ 𝐶 → ((𝑝‘0) ≠ (𝑝‘1) ∧ (𝑝‘0) ≠ (𝑝‘2) ∧ (𝑝‘1) ≠ (𝑝‘2)))))
54	53	com23 84	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((𝑉 USGrph 𝐸 ∧ ({(𝑝‘0), (𝑝‘1)} ∈ ran 𝐸 ∧ {(𝑝‘1), (𝑝‘2)} ∈ ran 𝐸)) ∧ 𝑝:(0...2)⟶𝑉) → (𝐴 ≠ 𝐶 → ((𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)) → ((𝑝‘0) ≠ (𝑝‘1) ∧ (𝑝‘0) ≠ (𝑝‘2) ∧ (𝑝‘1) ≠ (𝑝‘2)))))
55	54	imp 444	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((((𝑉 USGrph 𝐸 ∧ ({(𝑝‘0), (𝑝‘1)} ∈ ran 𝐸 ∧ {(𝑝‘1), (𝑝‘2)} ∈ ran 𝐸)) ∧ 𝑝:(0...2)⟶𝑉) ∧ 𝐴 ≠ 𝐶) → ((𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)) → ((𝑝‘0) ≠ (𝑝‘1) ∧ (𝑝‘0) ≠ (𝑝‘2) ∧ (𝑝‘1) ≠ (𝑝‘2))))
56	55	imp 444	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((((𝑉 USGrph 𝐸 ∧ ({(𝑝‘0), (𝑝‘1)} ∈ ran 𝐸 ∧ {(𝑝‘1), (𝑝‘2)} ∈ ran 𝐸)) ∧ 𝑝:(0...2)⟶𝑉) ∧ 𝐴 ≠ 𝐶) ∧ (𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2))) → ((𝑝‘0) ≠ (𝑝‘1) ∧ (𝑝‘0) ≠ (𝑝‘2) ∧ (𝑝‘1) ≠ (𝑝‘2)))
57		eqid 2610	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (0...2) = (0...2)
58	57	f13idfv 12662	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑝:(0...2)–1-1→𝑉 ↔ (𝑝:(0...2)⟶𝑉 ∧ ((𝑝‘0) ≠ (𝑝‘1) ∧ (𝑝‘0) ≠ (𝑝‘2) ∧ (𝑝‘1) ≠ (𝑝‘2))))
59	34, 56, 58	sylanbrc 695	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((((𝑉 USGrph 𝐸 ∧ ({(𝑝‘0), (𝑝‘1)} ∈ ran 𝐸 ∧ {(𝑝‘1), (𝑝‘2)} ∈ ran 𝐸)) ∧ 𝑝:(0...2)⟶𝑉) ∧ 𝐴 ≠ 𝐶) ∧ (𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2))) → 𝑝:(0...2)–1-1→𝑉)
60		df-f1 5809	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑝:(0...2)–1-1→𝑉 ↔ (𝑝:(0...2)⟶𝑉 ∧ Fun ^◡𝑝))
61	59, 60	sylib 207	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((((𝑉 USGrph 𝐸 ∧ ({(𝑝‘0), (𝑝‘1)} ∈ ran 𝐸 ∧ {(𝑝‘1), (𝑝‘2)} ∈ ran 𝐸)) ∧ 𝑝:(0...2)⟶𝑉) ∧ 𝐴 ≠ 𝐶) ∧ (𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2))) → (𝑝:(0...2)⟶𝑉 ∧ Fun ^◡𝑝))
62	61	simprd 478	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((((𝑉 USGrph 𝐸 ∧ ({(𝑝‘0), (𝑝‘1)} ∈ ran 𝐸 ∧ {(𝑝‘1), (𝑝‘2)} ∈ ran 𝐸)) ∧ 𝑝:(0...2)⟶𝑉) ∧ 𝐴 ≠ 𝐶) ∧ (𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2))) → Fun ^◡𝑝)
63	62	exp31 628	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑉 USGrph 𝐸 ∧ ({(𝑝‘0), (𝑝‘1)} ∈ ran 𝐸 ∧ {(𝑝‘1), (𝑝‘2)} ∈ ran 𝐸)) ∧ 𝑝:(0...2)⟶𝑉) → (𝐴 ≠ 𝐶 → ((𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)) → Fun ^◡𝑝)))
64	63	exp31 628	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑉 USGrph 𝐸 → (({(𝑝‘0), (𝑝‘1)} ∈ ran 𝐸 ∧ {(𝑝‘1), (𝑝‘2)} ∈ ran 𝐸) → (𝑝:(0...2)⟶𝑉 → (𝐴 ≠ 𝐶 → ((𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)) → Fun ^◡𝑝)))))
65	64	adantr 480	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑉 USGrph 𝐸 ∧ 𝑓:(0..^2)⟶dom 𝐸) → (({(𝑝‘0), (𝑝‘1)} ∈ ran 𝐸 ∧ {(𝑝‘1), (𝑝‘2)} ∈ ran 𝐸) → (𝑝:(0...2)⟶𝑉 → (𝐴 ≠ 𝐶 → ((𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)) → Fun ^◡𝑝)))))
66	33, 65	syld 46	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑉 USGrph 𝐸 ∧ 𝑓:(0..^2)⟶dom 𝐸) → (((𝐸‘(𝑓‘0)) = {(𝑝‘0), (𝑝‘1)} ∧ (𝐸‘(𝑓‘1)) = {(𝑝‘1), (𝑝‘2)}) → (𝑝:(0...2)⟶𝑉 → (𝐴 ≠ 𝐶 → ((𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)) → Fun ^◡𝑝)))))
67	66	expcom 450	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑓:(0..^2)⟶dom 𝐸 → (𝑉 USGrph 𝐸 → (((𝐸‘(𝑓‘0)) = {(𝑝‘0), (𝑝‘1)} ∧ (𝐸‘(𝑓‘1)) = {(𝑝‘1), (𝑝‘2)}) → (𝑝:(0...2)⟶𝑉 → (𝐴 ≠ 𝐶 → ((𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)) → Fun ^◡𝑝))))))
68	67	com24 93	. . . . . . . . . . . . . . . . 17 ⊢ (𝑓:(0..^2)⟶dom 𝐸 → (𝑝:(0...2)⟶𝑉 → (((𝐸‘(𝑓‘0)) = {(𝑝‘0), (𝑝‘1)} ∧ (𝐸‘(𝑓‘1)) = {(𝑝‘1), (𝑝‘2)}) → (𝑉 USGrph 𝐸 → (𝐴 ≠ 𝐶 → ((𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)) → Fun ^◡𝑝))))))
69	68	3imp 1249	. . . . . . . . . . . . . . . 16 ⊢ ((𝑓:(0..^2)⟶dom 𝐸 ∧ 𝑝:(0...2)⟶𝑉 ∧ ((𝐸‘(𝑓‘0)) = {(𝑝‘0), (𝑝‘1)} ∧ (𝐸‘(𝑓‘1)) = {(𝑝‘1), (𝑝‘2)})) → (𝑉 USGrph 𝐸 → (𝐴 ≠ 𝐶 → ((𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)) → Fun ^◡𝑝))))
70	69	com12 32	. . . . . . . . . . . . . . 15 ⊢ (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 ≠ 𝐶 → ((𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)) → Fun ^◡𝑝))))
71	11, 70	sylbid 229	. . . . . . . . . . . . . 14 ⊢ (𝑉 USGrph 𝐸 → ((𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2) → (𝐴 ≠ 𝐶 → ((𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)) → Fun ^◡𝑝))))
72	71	com14 94	. . . . . . . . . . . . 13 ⊢ ((𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)) → ((𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2) → (𝐴 ≠ 𝐶 → (𝑉 USGrph 𝐸 → Fun ^◡𝑝))))
73	72	com12 32	. . . . . . . . . . . 12 ⊢ ((𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2) → ((𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)) → (𝐴 ≠ 𝐶 → (𝑉 USGrph 𝐸 → Fun ^◡𝑝))))
74	73	3impia 1253	. . . . . . . . . . 11 ⊢ ((𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2))) → (𝐴 ≠ 𝐶 → (𝑉 USGrph 𝐸 → Fun ^◡𝑝)))
75	74	com13 86	. . . . . . . . . 10 ⊢ (𝑉 USGrph 𝐸 → (𝐴 ≠ 𝐶 → ((𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2))) → Fun ^◡𝑝)))
76	75	a1d 25	. . . . . . . . 9 ⊢ (𝑉 USGrph 𝐸 → ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) → (𝐴 ≠ 𝐶 → ((𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2))) → Fun ^◡𝑝))))
77	76	3imp 1249	. . . . . . . 8 ⊢ ((𝑉 USGrph 𝐸 ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) ∧ 𝐴 ≠ 𝐶) → ((𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2))) → Fun ^◡𝑝))
78	77	imp 444	. . . . . . 7 ⊢ (((𝑉 USGrph 𝐸 ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) ∧ 𝐴 ≠ 𝐶) ∧ (𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)))) → Fun ^◡𝑝)
79		wlkdvspth 26138	. . . . . . 7 ⊢ ((𝑓(𝑉 Walks 𝐸)𝑝 ∧ Fun ^◡𝑝) → 𝑓(𝑉 SPaths 𝐸)𝑝)
80	6, 78, 79	syl2anc 691	. . . . . 6 ⊢ (((𝑉 USGrph 𝐸 ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) ∧ 𝐴 ≠ 𝐶) ∧ (𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)))) → 𝑓(𝑉 SPaths 𝐸)𝑝)
81		simpr2 1061	. . . . . 6 ⊢ (((𝑉 USGrph 𝐸 ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) ∧ 𝐴 ≠ 𝐶) ∧ (𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)))) → (#‘𝑓) = 2)
82		simpr3 1062	. . . . . 6 ⊢ (((𝑉 USGrph 𝐸 ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) ∧ 𝐴 ≠ 𝐶) ∧ (𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)))) → (𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)))
83	80, 81, 82	3jca 1235	. . . . 5 ⊢ (((𝑉 USGrph 𝐸 ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) ∧ 𝐴 ≠ 𝐶) ∧ (𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)))) → (𝑓(𝑉 SPaths 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2))))
84	83	ex 449	. . . 4 ⊢ ((𝑉 USGrph 𝐸 ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) ∧ 𝐴 ≠ 𝐶) → ((𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2))) → (𝑓(𝑉 SPaths 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)))))
85		spthispth 26103	. . . . . 6 ⊢ (𝑓(𝑉 SPaths 𝐸)𝑝 → 𝑓(𝑉 Paths 𝐸)𝑝)
86		pthistrl 26102	. . . . . 6 ⊢ (𝑓(𝑉 Paths 𝐸)𝑝 → 𝑓(𝑉 Trails 𝐸)𝑝)
87		trliswlk 26069	. . . . . 6 ⊢ (𝑓(𝑉 Trails 𝐸)𝑝 → 𝑓(𝑉 Walks 𝐸)𝑝)
88	85, 86, 87	3syl 18	. . . . 5 ⊢ (𝑓(𝑉 SPaths 𝐸)𝑝 → 𝑓(𝑉 Walks 𝐸)𝑝)
89	88	3anim1i 1241	. . . 4 ⊢ ((𝑓(𝑉 SPaths 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2))) → (𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2))))
90	84, 89	impbid1 214	. . 3 ⊢ ((𝑉 USGrph 𝐸 ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) ∧ 𝐴 ≠ 𝐶) → ((𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2))) ↔ (𝑓(𝑉 SPaths 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)))))
91	90	2exbidv 1839	. 2 ⊢ ((𝑉 USGrph 𝐸 ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) ∧ 𝐴 ≠ 𝐶) → (∃𝑓∃𝑝(𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2))) ↔ ∃𝑓∃𝑝(𝑓(𝑉 SPaths 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)))))
92	5, 91	bitrd 267	1 ⊢ ((𝑉 USGrph 𝐸 ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) ∧ 𝐴 ≠ 𝐶) → (⟨𝐴, 𝐵, 𝐶⟩ ∈ (𝐴(𝑉 2WalksOnOt 𝐸)𝐶) ↔ ∃𝑓∃𝑝(𝑓(𝑉 SPaths 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 ∧ w3a 1031 = wceq 1475 ∃wex 1695 ∈ wcel 1977 ≠ wne 2780 Vcvv 3173 {cpr 4127 ⟨cotp 4133 class class class wbr 4583 ^◡ccnv 5037 dom cdm 5038 ran crn 5039 Fun wfun 5798 ⟶wf 5800 –1-1→wf1 5801 ‘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 Trails ctrail 26027 Paths cpath 26028 SPaths cspath 26029 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-trail 26037 df-pth 26038 df-spth 26039 df-wlkon 26042 df-2wlkonot 26385

This theorem is referenced by: (None)

W3C validator