Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  usg2wotspth Structured version   Visualization version   GIF version

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 ∧ (𝐴𝑉𝐵𝑉𝐶𝑉) ∧ 𝐴𝐶) → (⟨𝐴, 𝐵, 𝐶⟩ ∈ (𝐴(𝑉 2WalksOnOt 𝐸)𝐶) ↔ ∃𝑓𝑝(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
StepHypRef Expression
1 usgrav 25867 . . . 4 (𝑉 USGrph 𝐸 → (𝑉 ∈ V ∧ 𝐸 ∈ V))
2 el2wlkonotot 26400 . . . . 5 (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐴𝑉𝐶𝑉)) → (⟨𝐴, 𝐵, 𝐶⟩ ∈ (𝐴(𝑉 2WalksOnOt 𝐸)𝐶) ↔ ∃𝑓𝑝(𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)))))
323adantr2 1214 . . . 4 (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐴𝑉𝐵𝑉𝐶𝑉)) → (⟨𝐴, 𝐵, 𝐶⟩ ∈ (𝐴(𝑉 2WalksOnOt 𝐸)𝐶) ↔ ∃𝑓𝑝(𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)))))
41, 3sylan 487 . . 3 ((𝑉 USGrph 𝐸 ∧ (𝐴𝑉𝐵𝑉𝐶𝑉)) → (⟨𝐴, 𝐵, 𝐶⟩ ∈ (𝐴(𝑉 2WalksOnOt 𝐸)𝐶) ↔ ∃𝑓𝑝(𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)))))
543adant3 1074 . 2 ((𝑉 USGrph 𝐸 ∧ (𝐴𝑉𝐵𝑉𝐶𝑉) ∧ 𝐴𝐶) → (⟨𝐴, 𝐵, 𝐶⟩ ∈ (𝐴(𝑉 2WalksOnOt 𝐸)𝐶) ↔ ∃𝑓𝑝(𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)))))
6 simpr1 1060 . . . . . . 7 (((𝑉 USGrph 𝐸 ∧ (𝐴𝑉𝐵𝑉𝐶𝑉) ∧ 𝐴𝐶) ∧ (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
97, 8pm3.2i 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 𝐸𝑝:(0...2)⟶𝑉 ∧ ((𝐸‘(𝑓‘0)) = {(𝑝‘0), (𝑝‘1)} ∧ (𝐸‘(𝑓‘1)) = {(𝑝‘1), (𝑝‘2)}))))
111, 9, 10sylancl 693 . . . . . . . . . . . . . . 15 (𝑉 USGrph 𝐸 → ((𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2) ↔ (𝑓:(0..^2)⟶dom 𝐸𝑝:(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
1413prid1 4241 . . . . . . . . . . . . . . . . . . . . . . . . . 26 0 ∈ {0, 1}
15 fzo0to2pr 12420 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (0..^2) = {0, 1}
1614, 15eleqtrri 2687 . . . . . . . . . . . . . . . . . . . . . . . . 25 0 ∈ (0..^2)
17 ffvelrn 6265 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑓:(0..^2)⟶dom 𝐸 ∧ 0 ∈ (0..^2)) → (𝑓‘0) ∈ dom 𝐸)
1816, 17mpan2 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)
2018, 19sylan2 490 . . . . . . . . . . . . . . . . . . . . . . 23 ((Fun 𝐸𝑓:(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))
2220, 21syl5ibcom 234 . . . . . . . . . . . . . . . . . . . . . 22 ((Fun 𝐸𝑓:(0..^2)⟶dom 𝐸) → ((𝐸‘(𝑓‘0)) = {(𝑝‘0), (𝑝‘1)} → {(𝑝‘0), (𝑝‘1)} ∈ ran 𝐸))
23 1ex 9914 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 1 ∈ V
2423prid2 4242 . . . . . . . . . . . . . . . . . . . . . . . . . 26 1 ∈ {0, 1}
2524, 15eleqtrri 2687 . . . . . . . . . . . . . . . . . . . . . . . . 25 1 ∈ (0..^2)
26 ffvelrn 6265 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑓:(0..^2)⟶dom 𝐸 ∧ 1 ∈ (0..^2)) → (𝑓‘1) ∈ dom 𝐸)
2725, 26mpan2 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)
2927, 28sylan2 490 . . . . . . . . . . . . . . . . . . . . . . 23 ((Fun 𝐸𝑓:(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))
3129, 30syl5ibcom 234 . . . . . . . . . . . . . . . . . . . . . 22 ((Fun 𝐸𝑓:(0..^2)⟶dom 𝐸) → ((𝐸‘(𝑓‘1)) = {(𝑝‘1), (𝑝‘2)} → {(𝑝‘1), (𝑝‘2)} ∈ ran 𝐸))
3222, 31anim12d 584 . . . . . . . . . . . . . . . . . . . . 21 ((Fun 𝐸𝑓:(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)))
3312, 32sylan 487 . . . . . . . . . . . . . . . . . . . 20 ((𝑉 USGrph 𝐸𝑓:(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) ∧ 𝐴𝐶) ∧ (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))
3635ex 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))
3837ex 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))) ∧ 𝐴𝐶) → (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))
4240, 41neeq12d 2843 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ((𝐴 = (𝑝‘0) ∧ 𝐶 = (𝑝‘2)) → (𝐴𝐶 ↔ (𝑝‘0) ≠ (𝑝‘2)))
4342biimpd 218 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ((𝐴 = (𝑝‘0) ∧ 𝐶 = (𝑝‘2)) → (𝐴𝐶 → (𝑝‘0) ≠ (𝑝‘2)))
44433adant2 1073 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ((𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)) → (𝐴𝐶 → (𝑝‘0) ≠ (𝑝‘2)))
4544adantl 481 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ((((𝑝‘0) ≠ (𝑝‘1) ∧ (𝑝‘1) ≠ (𝑝‘2)) ∧ (𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2))) → (𝐴𝐶 → (𝑝‘0) ≠ (𝑝‘2)))
4645imp 444 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (((((𝑝‘0) ≠ (𝑝‘1) ∧ (𝑝‘1) ≠ (𝑝‘2)) ∧ (𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2))) ∧ 𝐴𝐶) → (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))) ∧ 𝐴𝐶) → (𝑝‘1) ≠ (𝑝‘2))
4839, 46, 473jca 1235 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (((((𝑝‘0) ≠ (𝑝‘1) ∧ (𝑝‘1) ≠ (𝑝‘2)) ∧ (𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2))) ∧ 𝐴𝐶) → ((𝑝‘0) ≠ (𝑝‘1) ∧ (𝑝‘0) ≠ (𝑝‘2) ∧ (𝑝‘1) ≠ (𝑝‘2)))
4948exp31 628 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (((𝑝‘0) ≠ (𝑝‘1) ∧ (𝑝‘1) ≠ (𝑝‘2)) → ((𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)) → (𝐴𝐶 → ((𝑝‘0) ≠ (𝑝‘1) ∧ (𝑝‘0) ≠ (𝑝‘2) ∧ (𝑝‘1) ≠ (𝑝‘2)))))
5049a1i 11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝑉 USGrph 𝐸 → (((𝑝‘0) ≠ (𝑝‘1) ∧ (𝑝‘1) ≠ (𝑝‘2)) → ((𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)) → (𝐴𝐶 → ((𝑝‘0) ≠ (𝑝‘1) ∧ (𝑝‘0) ≠ (𝑝‘2) ∧ (𝑝‘1) ≠ (𝑝‘2))))))
5136, 38, 50syl2and 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)) → (𝐴𝐶 → ((𝑝‘0) ≠ (𝑝‘1) ∧ (𝑝‘0) ≠ (𝑝‘2) ∧ (𝑝‘1) ≠ (𝑝‘2))))))
5251imp 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)) → (𝐴𝐶 → ((𝑝‘0) ≠ (𝑝‘1) ∧ (𝑝‘0) ≠ (𝑝‘2) ∧ (𝑝‘1) ≠ (𝑝‘2)))))
5352adantr 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)) → (𝐴𝐶 → ((𝑝‘0) ≠ (𝑝‘1) ∧ (𝑝‘0) ≠ (𝑝‘2) ∧ (𝑝‘1) ≠ (𝑝‘2)))))
5453com23 84 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((𝑉 USGrph 𝐸 ∧ ({(𝑝‘0), (𝑝‘1)} ∈ ran 𝐸 ∧ {(𝑝‘1), (𝑝‘2)} ∈ ran 𝐸)) ∧ 𝑝:(0...2)⟶𝑉) → (𝐴𝐶 → ((𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)) → ((𝑝‘0) ≠ (𝑝‘1) ∧ (𝑝‘0) ≠ (𝑝‘2) ∧ (𝑝‘1) ≠ (𝑝‘2)))))
5554imp 444 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((((𝑉 USGrph 𝐸 ∧ ({(𝑝‘0), (𝑝‘1)} ∈ ran 𝐸 ∧ {(𝑝‘1), (𝑝‘2)} ∈ ran 𝐸)) ∧ 𝑝:(0...2)⟶𝑉) ∧ 𝐴𝐶) → ((𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)) → ((𝑝‘0) ≠ (𝑝‘1) ∧ (𝑝‘0) ≠ (𝑝‘2) ∧ (𝑝‘1) ≠ (𝑝‘2))))
5655imp 444 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((((𝑉 USGrph 𝐸 ∧ ({(𝑝‘0), (𝑝‘1)} ∈ ran 𝐸 ∧ {(𝑝‘1), (𝑝‘2)} ∈ ran 𝐸)) ∧ 𝑝:(0...2)⟶𝑉) ∧ 𝐴𝐶) ∧ (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)
5857f13idfv 12662 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑝:(0...2)–1-1𝑉 ↔ (𝑝:(0...2)⟶𝑉 ∧ ((𝑝‘0) ≠ (𝑝‘1) ∧ (𝑝‘0) ≠ (𝑝‘2) ∧ (𝑝‘1) ≠ (𝑝‘2))))
5934, 56, 58sylanbrc 695 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((((𝑉 USGrph 𝐸 ∧ ({(𝑝‘0), (𝑝‘1)} ∈ ran 𝐸 ∧ {(𝑝‘1), (𝑝‘2)} ∈ ran 𝐸)) ∧ 𝑝:(0...2)⟶𝑉) ∧ 𝐴𝐶) ∧ (𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2))) → 𝑝:(0...2)–1-1𝑉)
60 df-f1 5809 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑝:(0...2)–1-1𝑉 ↔ (𝑝:(0...2)⟶𝑉 ∧ Fun 𝑝))
6159, 60sylib 207 . . . . . . . . . . . . . . . . . . . . . . . 24 (((((𝑉 USGrph 𝐸 ∧ ({(𝑝‘0), (𝑝‘1)} ∈ ran 𝐸 ∧ {(𝑝‘1), (𝑝‘2)} ∈ ran 𝐸)) ∧ 𝑝:(0...2)⟶𝑉) ∧ 𝐴𝐶) ∧ (𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2))) → (𝑝:(0...2)⟶𝑉 ∧ Fun 𝑝))
6261simprd 478 . . . . . . . . . . . . . . . . . . . . . . 23 (((((𝑉 USGrph 𝐸 ∧ ({(𝑝‘0), (𝑝‘1)} ∈ ran 𝐸 ∧ {(𝑝‘1), (𝑝‘2)} ∈ ran 𝐸)) ∧ 𝑝:(0...2)⟶𝑉) ∧ 𝐴𝐶) ∧ (𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2))) → Fun 𝑝)
6362exp31 628 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑉 USGrph 𝐸 ∧ ({(𝑝‘0), (𝑝‘1)} ∈ ran 𝐸 ∧ {(𝑝‘1), (𝑝‘2)} ∈ ran 𝐸)) ∧ 𝑝:(0...2)⟶𝑉) → (𝐴𝐶 → ((𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)) → Fun 𝑝)))
6463exp31 628 . . . . . . . . . . . . . . . . . . . . 21 (𝑉 USGrph 𝐸 → (({(𝑝‘0), (𝑝‘1)} ∈ ran 𝐸 ∧ {(𝑝‘1), (𝑝‘2)} ∈ ran 𝐸) → (𝑝:(0...2)⟶𝑉 → (𝐴𝐶 → ((𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)) → Fun 𝑝)))))
6564adantr 480 . . . . . . . . . . . . . . . . . . . 20 ((𝑉 USGrph 𝐸𝑓:(0..^2)⟶dom 𝐸) → (({(𝑝‘0), (𝑝‘1)} ∈ ran 𝐸 ∧ {(𝑝‘1), (𝑝‘2)} ∈ ran 𝐸) → (𝑝:(0...2)⟶𝑉 → (𝐴𝐶 → ((𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)) → Fun 𝑝)))))
6633, 65syld 46 . . . . . . . . . . . . . . . . . . 19 ((𝑉 USGrph 𝐸𝑓:(0..^2)⟶dom 𝐸) → (((𝐸‘(𝑓‘0)) = {(𝑝‘0), (𝑝‘1)} ∧ (𝐸‘(𝑓‘1)) = {(𝑝‘1), (𝑝‘2)}) → (𝑝:(0...2)⟶𝑉 → (𝐴𝐶 → ((𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)) → Fun 𝑝)))))
6766expcom 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 → (𝐴𝐶 → ((𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)) → Fun 𝑝))))))
6867com24 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 → (𝐴𝐶 → ((𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)) → Fun 𝑝))))))
69683imp 1249 . . . . . . . . . . . . . . . 16 ((𝑓:(0..^2)⟶dom 𝐸𝑝:(0...2)⟶𝑉 ∧ ((𝐸‘(𝑓‘0)) = {(𝑝‘0), (𝑝‘1)} ∧ (𝐸‘(𝑓‘1)) = {(𝑝‘1), (𝑝‘2)})) → (𝑉 USGrph 𝐸 → (𝐴𝐶 → ((𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)) → Fun 𝑝))))
7069com12 32 . . . . . . . . . . . . . . 15 (𝑉 USGrph 𝐸 → ((𝑓:(0..^2)⟶dom 𝐸𝑝:(0...2)⟶𝑉 ∧ ((𝐸‘(𝑓‘0)) = {(𝑝‘0), (𝑝‘1)} ∧ (𝐸‘(𝑓‘1)) = {(𝑝‘1), (𝑝‘2)})) → (𝐴𝐶 → ((𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)) → Fun 𝑝))))
7111, 70sylbid 229 . . . . . . . . . . . . . 14 (𝑉 USGrph 𝐸 → ((𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2) → (𝐴𝐶 → ((𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)) → Fun 𝑝))))
7271com14 94 . . . . . . . . . . . . 13 ((𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)) → ((𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2) → (𝐴𝐶 → (𝑉 USGrph 𝐸 → Fun 𝑝))))
7372com12 32 . . . . . . . . . . . 12 ((𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2) → ((𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)) → (𝐴𝐶 → (𝑉 USGrph 𝐸 → Fun 𝑝))))
74733impia 1253 . . . . . . . . . . 11 ((𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2))) → (𝐴𝐶 → (𝑉 USGrph 𝐸 → Fun 𝑝)))
7574com13 86 . . . . . . . . . 10 (𝑉 USGrph 𝐸 → (𝐴𝐶 → ((𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2))) → Fun 𝑝)))
7675a1d 25 . . . . . . . . 9 (𝑉 USGrph 𝐸 → ((𝐴𝑉𝐵𝑉𝐶𝑉) → (𝐴𝐶 → ((𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2))) → Fun 𝑝))))
77763imp 1249 . . . . . . . 8 ((𝑉 USGrph 𝐸 ∧ (𝐴𝑉𝐵𝑉𝐶𝑉) ∧ 𝐴𝐶) → ((𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2))) → Fun 𝑝))
7877imp 444 . . . . . . 7 (((𝑉 USGrph 𝐸 ∧ (𝐴𝑉𝐵𝑉𝐶𝑉) ∧ 𝐴𝐶) ∧ (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)
806, 78, 79syl2anc 691 . . . . . 6 (((𝑉 USGrph 𝐸 ∧ (𝐴𝑉𝐵𝑉𝐶𝑉) ∧ 𝐴𝐶) ∧ (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 ∧ (𝐴𝑉𝐵𝑉𝐶𝑉) ∧ 𝐴𝐶) ∧ (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 ∧ (𝐴𝑉𝐵𝑉𝐶𝑉) ∧ 𝐴𝐶) ∧ (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)))
8380, 81, 823jca 1235 . . . . 5 (((𝑉 USGrph 𝐸 ∧ (𝐴𝑉𝐵𝑉𝐶𝑉) ∧ 𝐴𝐶) ∧ (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))))
8483ex 449 . . . 4 ((𝑉 USGrph 𝐸 ∧ (𝐴𝑉𝐵𝑉𝐶𝑉) ∧ 𝐴𝐶) → ((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)𝑝𝑓(𝑉 Paths 𝐸)𝑝)
86 pthistrl 26102 . . . . . 6 (𝑓(𝑉 Paths 𝐸)𝑝𝑓(𝑉 Trails 𝐸)𝑝)
87 trliswlk 26069 . . . . . 6 (𝑓(𝑉 Trails 𝐸)𝑝𝑓(𝑉 Walks 𝐸)𝑝)
8885, 86, 873syl 18 . . . . 5 (𝑓(𝑉 SPaths 𝐸)𝑝𝑓(𝑉 Walks 𝐸)𝑝)
89883anim1i 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))))
9084, 89impbid1 214 . . 3 ((𝑉 USGrph 𝐸 ∧ (𝐴𝑉𝐵𝑉𝐶𝑉) ∧ 𝐴𝐶) → ((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)))))
91902exbidv 1839 . 2 ((𝑉 USGrph 𝐸 ∧ (𝐴𝑉𝐵𝑉𝐶𝑉) ∧ 𝐴𝐶) → (∃𝑓𝑝(𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2))) ↔ ∃𝑓𝑝(𝑓(𝑉 SPaths 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)))))
925, 91bitrd 267 1 ((𝑉 USGrph 𝐸 ∧ (𝐴𝑉𝐵𝑉𝐶𝑉) ∧ 𝐴𝐶) → (⟨𝐴, 𝐵, 𝐶⟩ ∈ (𝐴(𝑉 2WalksOnOt 𝐸)𝐶) ↔ ∃𝑓𝑝(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)
 Copyright terms: Public domain W3C validator