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Theorem umgrwwlks2on 30027
Description: A walk of length 2 between two vertices as word in a multigraph. This theorem would also hold for pseudographs, 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.) (Revised by AV, 12-May-2021.)
Hypotheses
Ref Expression
s3wwlks2on.v 𝑉 = (Vtx‘𝐺)
usgrwwlks2on.e 𝐸 = (Edg‘𝐺)
Assertion
Ref Expression
umgrwwlks2on ((𝐺 ∈ UMGraph ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) → (⟨“𝐴𝐵𝐶”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐶) ↔ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸)))
Proof of Theorem umgrwwlks2on
Dummy variables 𝑓 𝑝 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 umgrupgr 29172 . . . 4 (𝐺 ∈ UMGraph → 𝐺 ∈ UPGraph)
21adantr 480 . . 3 ((𝐺 ∈ UMGraph ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) → 𝐺 ∈ UPGraph)
3 simp1 1137 . . . 4 ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) → 𝐴 ∈ 𝑉)
43adantl 481 . . 3 ((𝐺 ∈ UMGraph ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) → 𝐴 ∈ 𝑉)
5 simpr3 1198 . . 3 ((𝐺 ∈ UMGraph ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) → 𝐶 ∈ 𝑉)
6 s3wwlks2on.v . . . 4 𝑉 = (Vtx‘𝐺)
76s3wwlks2on 30024 . . 3 ((𝐺 ∈ UPGraph ∧ 𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) → (⟨“𝐴𝐵𝐶”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐶) ↔ ∃𝑓(𝑓(Walks‘𝐺)⟨“𝐴𝐵𝐶”⟩ ∧ (♯‘𝑓) = 2)))
82, 4, 5, 7syl3anc 1374 . 2 ((𝐺 ∈ UMGraph ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) → (⟨“𝐴𝐵𝐶”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐶) ↔ ∃𝑓(𝑓(Walks‘𝐺)⟨“𝐴𝐵𝐶”⟩ ∧ (♯‘𝑓) = 2)))
9 eqid 2736 . . . . . . . 8 (iEdg‘𝐺) = (iEdg‘𝐺)
106, 9upgr2wlk 29735 . . . . . . 7 (𝐺 ∈ UPGraph → ((𝑓(Walks‘𝐺)⟨“𝐴𝐵𝐶”⟩ ∧ (♯‘𝑓) = 2) ↔ (𝑓:(0..^2)⟶dom (iEdg‘𝐺) ∧ ⟨“𝐴𝐵𝐶”⟩:(0...2)⟶𝑉 ∧ (((iEdg‘𝐺)‘(𝑓‘0)) = {(⟨“𝐴𝐵𝐶”⟩‘0), (⟨“𝐴𝐵𝐶”⟩‘1)} ∧ ((iEdg‘𝐺)‘(𝑓‘1)) = {(⟨“𝐴𝐵𝐶”⟩‘1), (⟨“𝐴𝐵𝐶”⟩‘2)}))))
111, 10syl 17 . . . . . 6 (𝐺 ∈ UMGraph → ((𝑓(Walks‘𝐺)⟨“𝐴𝐵𝐶”⟩ ∧ (♯‘𝑓) = 2) ↔ (𝑓:(0..^2)⟶dom (iEdg‘𝐺) ∧ ⟨“𝐴𝐵𝐶”⟩:(0...2)⟶𝑉 ∧ (((iEdg‘𝐺)‘(𝑓‘0)) = {(⟨“𝐴𝐵𝐶”⟩‘0), (⟨“𝐴𝐵𝐶”⟩‘1)} ∧ ((iEdg‘𝐺)‘(𝑓‘1)) = {(⟨“𝐴𝐵𝐶”⟩‘1), (⟨“𝐴𝐵𝐶”⟩‘2)}))))
1211adantr 480 . . . . 5 ((𝐺 ∈ UMGraph ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) → ((𝑓(Walks‘𝐺)⟨“𝐴𝐵𝐶”⟩ ∧ (♯‘𝑓) = 2) ↔ (𝑓:(0..^2)⟶dom (iEdg‘𝐺) ∧ ⟨“𝐴𝐵𝐶”⟩:(0...2)⟶𝑉 ∧ (((iEdg‘𝐺)‘(𝑓‘0)) = {(⟨“𝐴𝐵𝐶”⟩‘0), (⟨“𝐴𝐵𝐶”⟩‘1)} ∧ ((iEdg‘𝐺)‘(𝑓‘1)) = {(⟨“𝐴𝐵𝐶”⟩‘1), (⟨“𝐴𝐵𝐶”⟩‘2)}))))
13 s3fv0 14853 . . . . . . . . . . . 12 (𝐴 ∈ 𝑉 → (⟨“𝐴𝐵𝐶”⟩‘0) = 𝐴)
14133ad2ant1 1134 . . . . . . . . . . 11 ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) → (⟨“𝐴𝐵𝐶”⟩‘0) = 𝐴)
15 s3fv1 14854 . . . . . . . . . . . 12 (𝐵 ∈ 𝑉 → (⟨“𝐴𝐵𝐶”⟩‘1) = 𝐵)
16153ad2ant2 1135 . . . . . . . . . . 11 ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) → (⟨“𝐴𝐵𝐶”⟩‘1) = 𝐵)
1714, 16preq12d 4685 . . . . . . . . . 10 ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) → {(⟨“𝐴𝐵𝐶”⟩‘0), (⟨“𝐴𝐵𝐶”⟩‘1)} = {𝐴, 𝐵})
1817eqeq2d 2747 . . . . . . . . 9 ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) → (((iEdg‘𝐺)‘(𝑓‘0)) = {(⟨“𝐴𝐵𝐶”⟩‘0), (⟨“𝐴𝐵𝐶”⟩‘1)} ↔ ((iEdg‘𝐺)‘(𝑓‘0)) = {𝐴, 𝐵}))
19 s3fv2 14855 . . . . . . . . . . . 12 (𝐶 ∈ 𝑉 → (⟨“𝐴𝐵𝐶”⟩‘2) = 𝐶)
20193ad2ant3 1136 . . . . . . . . . . 11 ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) → (⟨“𝐴𝐵𝐶”⟩‘2) = 𝐶)
2116, 20preq12d 4685 . . . . . . . . . 10 ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) → {(⟨“𝐴𝐵𝐶”⟩‘1), (⟨“𝐴𝐵𝐶”⟩‘2)} = {𝐵, 𝐶})
2221eqeq2d 2747 . . . . . . . . 9 ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) → (((iEdg‘𝐺)‘(𝑓‘1)) = {(⟨“𝐴𝐵𝐶”⟩‘1), (⟨“𝐴𝐵𝐶”⟩‘2)} ↔ ((iEdg‘𝐺)‘(𝑓‘1)) = {𝐵, 𝐶}))
2318, 22anbi12d 633 . . . . . . . 8 ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) → ((((iEdg‘𝐺)‘(𝑓‘0)) = {(⟨“𝐴𝐵𝐶”⟩‘0), (⟨“𝐴𝐵𝐶”⟩‘1)} ∧ ((iEdg‘𝐺)‘(𝑓‘1)) = {(⟨“𝐴𝐵𝐶”⟩‘1), (⟨“𝐴𝐵𝐶”⟩‘2)}) ↔ (((iEdg‘𝐺)‘(𝑓‘0)) = {𝐴, 𝐵} ∧ ((iEdg‘𝐺)‘(𝑓‘1)) = {𝐵, 𝐶})))
2423adantl 481 . . . . . . 7 ((𝐺 ∈ UMGraph ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) → ((((iEdg‘𝐺)‘(𝑓‘0)) = {(⟨“𝐴𝐵𝐶”⟩‘0), (⟨“𝐴𝐵𝐶”⟩‘1)} ∧ ((iEdg‘𝐺)‘(𝑓‘1)) = {(⟨“𝐴𝐵𝐶”⟩‘1), (⟨“𝐴𝐵𝐶”⟩‘2)}) ↔ (((iEdg‘𝐺)‘(𝑓‘0)) = {𝐴, 𝐵} ∧ ((iEdg‘𝐺)‘(𝑓‘1)) = {𝐵, 𝐶})))
25243anbi3d 1445 . . . . . 6 ((𝐺 ∈ UMGraph ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) → ((𝑓:(0..^2)⟶dom (iEdg‘𝐺) ∧ ⟨“𝐴𝐵𝐶”⟩:(0...2)⟶𝑉 ∧ (((iEdg‘𝐺)‘(𝑓‘0)) = {(⟨“𝐴𝐵𝐶”⟩‘0), (⟨“𝐴𝐵𝐶”⟩‘1)} ∧ ((iEdg‘𝐺)‘(𝑓‘1)) = {(⟨“𝐴𝐵𝐶”⟩‘1), (⟨“𝐴𝐵𝐶”⟩‘2)})) ↔ (𝑓:(0..^2)⟶dom (iEdg‘𝐺) ∧ ⟨“𝐴𝐵𝐶”⟩:(0...2)⟶𝑉 ∧ (((iEdg‘𝐺)‘(𝑓‘0)) = {𝐴, 𝐵} ∧ ((iEdg‘𝐺)‘(𝑓‘1)) = {𝐵, 𝐶}))))
26 umgruhgr 29173 . . . . . . . . . . 11 (𝐺 ∈ UMGraph → 𝐺 ∈ UHGraph)
279uhgrfun 29135 . . . . . . . . . . 11 (𝐺 ∈ UHGraph → Fun (iEdg‘𝐺))
28 fdmrn 6699 . . . . . . . . . . . 12 (Fun (iEdg‘𝐺) ↔ (iEdg‘𝐺):dom (iEdg‘𝐺)⟶ran (iEdg‘𝐺))
29 simpr 484 . . . . . . . . . . . . . . . . 17 ((𝑓:(0..^2)⟶dom (iEdg‘𝐺) ∧ (iEdg‘𝐺):dom (iEdg‘𝐺)⟶ran (iEdg‘𝐺)) → (iEdg‘𝐺):dom (iEdg‘𝐺)⟶ran (iEdg‘𝐺))
30 id 22 . . . . . . . . . . . . . . . . . . 19 (𝑓:(0..^2)⟶dom (iEdg‘𝐺) → 𝑓:(0..^2)⟶dom (iEdg‘𝐺))
31 c0ex 11138 . . . . . . . . . . . . . . . . . . . . . 22 0 ∈ V
3231prid1 4706 . . . . . . . . . . . . . . . . . . . . 21 0 ∈ {0, 1}
33 fzo0to2pr 13705 . . . . . . . . . . . . . . . . . . . . 21 (0..^2) = {0, 1}
3432, 33eleqtrri 2835 . . . . . . . . . . . . . . . . . . . 20 0 ∈ (0..^2)
3534a1i 11 . . . . . . . . . . . . . . . . . . 19 (𝑓:(0..^2)⟶dom (iEdg‘𝐺) → 0 ∈ (0..^2))
3630, 35ffvelcdmd 7037 . . . . . . . . . . . . . . . . . 18 (𝑓:(0..^2)⟶dom (iEdg‘𝐺) → (𝑓‘0) ∈ dom (iEdg‘𝐺))
3736adantr 480 . . . . . . . . . . . . . . . . 17 ((𝑓:(0..^2)⟶dom (iEdg‘𝐺) ∧ (iEdg‘𝐺):dom (iEdg‘𝐺)⟶ran (iEdg‘𝐺)) → (𝑓‘0) ∈ dom (iEdg‘𝐺))
3829, 37ffvelcdmd 7037 . . . . . . . . . . . . . . . 16 ((𝑓:(0..^2)⟶dom (iEdg‘𝐺) ∧ (iEdg‘𝐺):dom (iEdg‘𝐺)⟶ran (iEdg‘𝐺)) → ((iEdg‘𝐺)‘(𝑓‘0)) ∈ ran (iEdg‘𝐺))
39 1ex 11140 . . . . . . . . . . . . . . . . . . . . . 22 1 ∈ V
4039prid2 4707 . . . . . . . . . . . . . . . . . . . . 21 1 ∈ {0, 1}
4140, 33eleqtrri 2835 . . . . . . . . . . . . . . . . . . . 20 1 ∈ (0..^2)
4241a1i 11 . . . . . . . . . . . . . . . . . . 19 (𝑓:(0..^2)⟶dom (iEdg‘𝐺) → 1 ∈ (0..^2))
4330, 42ffvelcdmd 7037 . . . . . . . . . . . . . . . . . 18 (𝑓:(0..^2)⟶dom (iEdg‘𝐺) → (𝑓‘1) ∈ dom (iEdg‘𝐺))
4443adantr 480 . . . . . . . . . . . . . . . . 17 ((𝑓:(0..^2)⟶dom (iEdg‘𝐺) ∧ (iEdg‘𝐺):dom (iEdg‘𝐺)⟶ran (iEdg‘𝐺)) → (𝑓‘1) ∈ dom (iEdg‘𝐺))
4529, 44ffvelcdmd 7037 . . . . . . . . . . . . . . . 16 ((𝑓:(0..^2)⟶dom (iEdg‘𝐺) ∧ (iEdg‘𝐺):dom (iEdg‘𝐺)⟶ran (iEdg‘𝐺)) → ((iEdg‘𝐺)‘(𝑓‘1)) ∈ ran (iEdg‘𝐺))
4638, 45jca 511 . . . . . . . . . . . . . . 15 ((𝑓:(0..^2)⟶dom (iEdg‘𝐺) ∧ (iEdg‘𝐺):dom (iEdg‘𝐺)⟶ran (iEdg‘𝐺)) → (((iEdg‘𝐺)‘(𝑓‘0)) ∈ ran (iEdg‘𝐺) ∧ ((iEdg‘𝐺)‘(𝑓‘1)) ∈ ran (iEdg‘𝐺)))
4746ex 412 . . . . . . . . . . . . . 14 (𝑓:(0..^2)⟶dom (iEdg‘𝐺) → ((iEdg‘𝐺):dom (iEdg‘𝐺)⟶ran (iEdg‘𝐺) → (((iEdg‘𝐺)‘(𝑓‘0)) ∈ ran (iEdg‘𝐺) ∧ ((iEdg‘𝐺)‘(𝑓‘1)) ∈ ran (iEdg‘𝐺))))
48473ad2ant1 1134 . . . . . . . . . . . . 13 ((𝑓:(0..^2)⟶dom (iEdg‘𝐺) ∧ ⟨“𝐴𝐵𝐶”⟩:(0...2)⟶𝑉 ∧ (((iEdg‘𝐺)‘(𝑓‘0)) = {𝐴, 𝐵} ∧ ((iEdg‘𝐺)‘(𝑓‘1)) = {𝐵, 𝐶})) → ((iEdg‘𝐺):dom (iEdg‘𝐺)⟶ran (iEdg‘𝐺) → (((iEdg‘𝐺)‘(𝑓‘0)) ∈ ran (iEdg‘𝐺) ∧ ((iEdg‘𝐺)‘(𝑓‘1)) ∈ ran (iEdg‘𝐺))))
4948com12 32 . . . . . . . . . . . 12 ((iEdg‘𝐺):dom (iEdg‘𝐺)⟶ran (iEdg‘𝐺) → ((𝑓:(0..^2)⟶dom (iEdg‘𝐺) ∧ ⟨“𝐴𝐵𝐶”⟩:(0...2)⟶𝑉 ∧ (((iEdg‘𝐺)‘(𝑓‘0)) = {𝐴, 𝐵} ∧ ((iEdg‘𝐺)‘(𝑓‘1)) = {𝐵, 𝐶})) → (((iEdg‘𝐺)‘(𝑓‘0)) ∈ ran (iEdg‘𝐺) ∧ ((iEdg‘𝐺)‘(𝑓‘1)) ∈ ran (iEdg‘𝐺))))
5028, 49sylbi 217 . . . . . . . . . . 11 (Fun (iEdg‘𝐺) → ((𝑓:(0..^2)⟶dom (iEdg‘𝐺) ∧ ⟨“𝐴𝐵𝐶”⟩:(0...2)⟶𝑉 ∧ (((iEdg‘𝐺)‘(𝑓‘0)) = {𝐴, 𝐵} ∧ ((iEdg‘𝐺)‘(𝑓‘1)) = {𝐵, 𝐶})) → (((iEdg‘𝐺)‘(𝑓‘0)) ∈ ran (iEdg‘𝐺) ∧ ((iEdg‘𝐺)‘(𝑓‘1)) ∈ ran (iEdg‘𝐺))))
5126, 27, 503syl 18 . . . . . . . . . 10 (𝐺 ∈ UMGraph → ((𝑓:(0..^2)⟶dom (iEdg‘𝐺) ∧ ⟨“𝐴𝐵𝐶”⟩:(0...2)⟶𝑉 ∧ (((iEdg‘𝐺)‘(𝑓‘0)) = {𝐴, 𝐵} ∧ ((iEdg‘𝐺)‘(𝑓‘1)) = {𝐵, 𝐶})) → (((iEdg‘𝐺)‘(𝑓‘0)) ∈ ran (iEdg‘𝐺) ∧ ((iEdg‘𝐺)‘(𝑓‘1)) ∈ ran (iEdg‘𝐺))))
5251imp 406 . . . . . . . . 9 ((𝐺 ∈ UMGraph ∧ (𝑓:(0..^2)⟶dom (iEdg‘𝐺) ∧ ⟨“𝐴𝐵𝐶”⟩:(0...2)⟶𝑉 ∧ (((iEdg‘𝐺)‘(𝑓‘0)) = {𝐴, 𝐵} ∧ ((iEdg‘𝐺)‘(𝑓‘1)) = {𝐵, 𝐶}))) → (((iEdg‘𝐺)‘(𝑓‘0)) ∈ ran (iEdg‘𝐺) ∧ ((iEdg‘𝐺)‘(𝑓‘1)) ∈ ran (iEdg‘𝐺)))
53 eqcom 2743 . . . . . . . . . . . . . . 15 (((iEdg‘𝐺)‘(𝑓‘0)) = {𝐴, 𝐵} ↔ {𝐴, 𝐵} = ((iEdg‘𝐺)‘(𝑓‘0)))
5453biimpi 216 . . . . . . . . . . . . . 14 (((iEdg‘𝐺)‘(𝑓‘0)) = {𝐴, 𝐵} → {𝐴, 𝐵} = ((iEdg‘𝐺)‘(𝑓‘0)))
5554adantr 480 . . . . . . . . . . . . 13 ((((iEdg‘𝐺)‘(𝑓‘0)) = {𝐴, 𝐵} ∧ ((iEdg‘𝐺)‘(𝑓‘1)) = {𝐵, 𝐶}) → {𝐴, 𝐵} = ((iEdg‘𝐺)‘(𝑓‘0)))
56553ad2ant3 1136 . . . . . . . . . . . 12 ((𝑓:(0..^2)⟶dom (iEdg‘𝐺) ∧ ⟨“𝐴𝐵𝐶”⟩:(0...2)⟶𝑉 ∧ (((iEdg‘𝐺)‘(𝑓‘0)) = {𝐴, 𝐵} ∧ ((iEdg‘𝐺)‘(𝑓‘1)) = {𝐵, 𝐶})) → {𝐴, 𝐵} = ((iEdg‘𝐺)‘(𝑓‘0)))
5756adantl 481 . . . . . . . . . . 11 ((𝐺 ∈ UMGraph ∧ (𝑓:(0..^2)⟶dom (iEdg‘𝐺) ∧ ⟨“𝐴𝐵𝐶”⟩:(0...2)⟶𝑉 ∧ (((iEdg‘𝐺)‘(𝑓‘0)) = {𝐴, 𝐵} ∧ ((iEdg‘𝐺)‘(𝑓‘1)) = {𝐵, 𝐶}))) → {𝐴, 𝐵} = ((iEdg‘𝐺)‘(𝑓‘0)))
58 usgrwwlks2on.e . . . . . . . . . . . . 13 𝐸 = (Edg‘𝐺)
59 edgval 29118 . . . . . . . . . . . . 13 (Edg‘𝐺) = ran (iEdg‘𝐺)
6058, 59eqtri 2759 . . . . . . . . . . . 12 𝐸 = ran (iEdg‘𝐺)
6160a1i 11 . . . . . . . . . . 11 ((𝐺 ∈ UMGraph ∧ (𝑓:(0..^2)⟶dom (iEdg‘𝐺) ∧ ⟨“𝐴𝐵𝐶”⟩:(0...2)⟶𝑉 ∧ (((iEdg‘𝐺)‘(𝑓‘0)) = {𝐴, 𝐵} ∧ ((iEdg‘𝐺)‘(𝑓‘1)) = {𝐵, 𝐶}))) → 𝐸 = ran (iEdg‘𝐺))
6257, 61eleq12d 2830 . . . . . . . . . 10 ((𝐺 ∈ UMGraph ∧ (𝑓:(0..^2)⟶dom (iEdg‘𝐺) ∧ ⟨“𝐴𝐵𝐶”⟩:(0...2)⟶𝑉 ∧ (((iEdg‘𝐺)‘(𝑓‘0)) = {𝐴, 𝐵} ∧ ((iEdg‘𝐺)‘(𝑓‘1)) = {𝐵, 𝐶}))) → ({𝐴, 𝐵} ∈ 𝐸 ↔ ((iEdg‘𝐺)‘(𝑓‘0)) ∈ ran (iEdg‘𝐺)))
63 eqcom 2743 . . . . . . . . . . . . . . 15 (((iEdg‘𝐺)‘(𝑓‘1)) = {𝐵, 𝐶} ↔ {𝐵, 𝐶} = ((iEdg‘𝐺)‘(𝑓‘1)))
6463biimpi 216 . . . . . . . . . . . . . 14 (((iEdg‘𝐺)‘(𝑓‘1)) = {𝐵, 𝐶} → {𝐵, 𝐶} = ((iEdg‘𝐺)‘(𝑓‘1)))
6564adantl 481 . . . . . . . . . . . . 13 ((((iEdg‘𝐺)‘(𝑓‘0)) = {𝐴, 𝐵} ∧ ((iEdg‘𝐺)‘(𝑓‘1)) = {𝐵, 𝐶}) → {𝐵, 𝐶} = ((iEdg‘𝐺)‘(𝑓‘1)))
66653ad2ant3 1136 . . . . . . . . . . . 12 ((𝑓:(0..^2)⟶dom (iEdg‘𝐺) ∧ ⟨“𝐴𝐵𝐶”⟩:(0...2)⟶𝑉 ∧ (((iEdg‘𝐺)‘(𝑓‘0)) = {𝐴, 𝐵} ∧ ((iEdg‘𝐺)‘(𝑓‘1)) = {𝐵, 𝐶})) → {𝐵, 𝐶} = ((iEdg‘𝐺)‘(𝑓‘1)))
6766adantl 481 . . . . . . . . . . 11 ((𝐺 ∈ UMGraph ∧ (𝑓:(0..^2)⟶dom (iEdg‘𝐺) ∧ ⟨“𝐴𝐵𝐶”⟩:(0...2)⟶𝑉 ∧ (((iEdg‘𝐺)‘(𝑓‘0)) = {𝐴, 𝐵} ∧ ((iEdg‘𝐺)‘(𝑓‘1)) = {𝐵, 𝐶}))) → {𝐵, 𝐶} = ((iEdg‘𝐺)‘(𝑓‘1)))
6867, 61eleq12d 2830 . . . . . . . . . 10 ((𝐺 ∈ UMGraph ∧ (𝑓:(0..^2)⟶dom (iEdg‘𝐺) ∧ ⟨“𝐴𝐵𝐶”⟩:(0...2)⟶𝑉 ∧ (((iEdg‘𝐺)‘(𝑓‘0)) = {𝐴, 𝐵} ∧ ((iEdg‘𝐺)‘(𝑓‘1)) = {𝐵, 𝐶}))) → ({𝐵, 𝐶} ∈ 𝐸 ↔ ((iEdg‘𝐺)‘(𝑓‘1)) ∈ ran (iEdg‘𝐺)))
6962, 68anbi12d 633 . . . . . . . . 9 ((𝐺 ∈ UMGraph ∧ (𝑓:(0..^2)⟶dom (iEdg‘𝐺) ∧ ⟨“𝐴𝐵𝐶”⟩:(0...2)⟶𝑉 ∧ (((iEdg‘𝐺)‘(𝑓‘0)) = {𝐴, 𝐵} ∧ ((iEdg‘𝐺)‘(𝑓‘1)) = {𝐵, 𝐶}))) → (({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸) ↔ (((iEdg‘𝐺)‘(𝑓‘0)) ∈ ran (iEdg‘𝐺) ∧ ((iEdg‘𝐺)‘(𝑓‘1)) ∈ ran (iEdg‘𝐺))))
7052, 69mpbird 257 . . . . . . . 8 ((𝐺 ∈ UMGraph ∧ (𝑓:(0..^2)⟶dom (iEdg‘𝐺) ∧ ⟨“𝐴𝐵𝐶”⟩:(0...2)⟶𝑉 ∧ (((iEdg‘𝐺)‘(𝑓‘0)) = {𝐴, 𝐵} ∧ ((iEdg‘𝐺)‘(𝑓‘1)) = {𝐵, 𝐶}))) → ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸))
7170ex 412 . . . . . . 7 (𝐺 ∈ UMGraph → ((𝑓:(0..^2)⟶dom (iEdg‘𝐺) ∧ ⟨“𝐴𝐵𝐶”⟩:(0...2)⟶𝑉 ∧ (((iEdg‘𝐺)‘(𝑓‘0)) = {𝐴, 𝐵} ∧ ((iEdg‘𝐺)‘(𝑓‘1)) = {𝐵, 𝐶})) → ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸)))
7271adantr 480 . . . . . 6 ((𝐺 ∈ UMGraph ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) → ((𝑓:(0..^2)⟶dom (iEdg‘𝐺) ∧ ⟨“𝐴𝐵𝐶”⟩:(0...2)⟶𝑉 ∧ (((iEdg‘𝐺)‘(𝑓‘0)) = {𝐴, 𝐵} ∧ ((iEdg‘𝐺)‘(𝑓‘1)) = {𝐵, 𝐶})) → ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸)))
7325, 72sylbid 240 . . . . 5 ((𝐺 ∈ UMGraph ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) → ((𝑓:(0..^2)⟶dom (iEdg‘𝐺) ∧ ⟨“𝐴𝐵𝐶”⟩:(0...2)⟶𝑉 ∧ (((iEdg‘𝐺)‘(𝑓‘0)) = {(⟨“𝐴𝐵𝐶”⟩‘0), (⟨“𝐴𝐵𝐶”⟩‘1)} ∧ ((iEdg‘𝐺)‘(𝑓‘1)) = {(⟨“𝐴𝐵𝐶”⟩‘1), (⟨“𝐴𝐵𝐶”⟩‘2)})) → ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸)))
7412, 73sylbid 240 . . . 4 ((𝐺 ∈ UMGraph ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) → ((𝑓(Walks‘𝐺)⟨“𝐴𝐵𝐶”⟩ ∧ (♯‘𝑓) = 2) → ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸)))
7574exlimdv 1935 . . 3 ((𝐺 ∈ UMGraph ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) → (∃𝑓(𝑓(Walks‘𝐺)⟨“𝐴𝐵𝐶”⟩ ∧ (♯‘𝑓) = 2) → ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸)))
7658umgr2wlk 30017 . . . . . . 7 ((𝐺 ∈ UMGraph ∧ {𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸) → ∃𝑓∃𝑝(𝑓(Walks‘𝐺)𝑝 ∧ (♯‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2))))
77 wlklenvp1 29687 . . . . . . . . . . . . . . . . . . . 20 (𝑓(Walks‘𝐺)𝑝 → (♯‘𝑝) = ((♯‘𝑓) + 1))
78 oveq1 7374 . . . . . . . . . . . . . . . . . . . . . 22 ((♯‘𝑓) = 2 → ((♯‘𝑓) + 1) = (2 + 1))
79 2p1e3 12318 . . . . . . . . . . . . . . . . . . . . . 22 (2 + 1) = 3
8078, 79eqtrdi 2787 . . . . . . . . . . . . . . . . . . . . 21 ((♯‘𝑓) = 2 → ((♯‘𝑓) + 1) = 3)
8180adantr 480 . . . . . . . . . . . . . . . . . . . 20 (((♯‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2))) → ((♯‘𝑓) + 1) = 3)
8277, 81sylan9eq 2791 . . . . . . . . . . . . . . . . . . 19 ((𝑓(Walks‘𝐺)𝑝 ∧ ((♯‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)))) → (♯‘𝑝) = 3)
83 eqcom 2743 . . . . . . . . . . . . . . . . . . . . . . 23 (𝐴 = (𝑝‘0) ↔ (𝑝‘0) = 𝐴)
84 eqcom 2743 . . . . . . . . . . . . . . . . . . . . . . 23 (𝐵 = (𝑝‘1) ↔ (𝑝‘1) = 𝐵)
85 eqcom 2743 . . . . . . . . . . . . . . . . . . . . . . 23 (𝐶 = (𝑝‘2) ↔ (𝑝‘2) = 𝐶)
8683, 84, 853anbi123i 1156 . . . . . . . . . . . . . . . . . . . . . 22 ((𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)) ↔ ((𝑝‘0) = 𝐴 ∧ (𝑝‘1) = 𝐵 ∧ (𝑝‘2) = 𝐶))
8786biimpi 216 . . . . . . . . . . . . . . . . . . . . 21 ((𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)) → ((𝑝‘0) = 𝐴 ∧ (𝑝‘1) = 𝐵 ∧ (𝑝‘2) = 𝐶))
8887adantl 481 . . . . . . . . . . . . . . . . . . . 20 (((♯‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2))) → ((𝑝‘0) = 𝐴 ∧ (𝑝‘1) = 𝐵 ∧ (𝑝‘2) = 𝐶))
8988adantl 481 . . . . . . . . . . . . . . . . . . 19 ((𝑓(Walks‘𝐺)𝑝 ∧ ((♯‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)))) → ((𝑝‘0) = 𝐴 ∧ (𝑝‘1) = 𝐵 ∧ (𝑝‘2) = 𝐶))
9082, 89jca 511 . . . . . . . . . . . . . . . . . 18 ((𝑓(Walks‘𝐺)𝑝 ∧ ((♯‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)))) → ((♯‘𝑝) = 3 ∧ ((𝑝‘0) = 𝐴 ∧ (𝑝‘1) = 𝐵 ∧ (𝑝‘2) = 𝐶)))
916wlkpwrd 29686 . . . . . . . . . . . . . . . . . . . . 21 (𝑓(Walks‘𝐺)𝑝 → 𝑝 ∈ Word 𝑉)
9280eqeq2d 2747 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((♯‘𝑓) = 2 → ((♯‘𝑝) = ((♯‘𝑓) + 1) ↔ (♯‘𝑝) = 3))
9392adantl 481 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑝 ∈ Word 𝑉 ∧ (♯‘𝑓) = 2) → ((♯‘𝑝) = ((♯‘𝑓) + 1) ↔ (♯‘𝑝) = 3))
94 simp1 1137 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑝 ∈ Word 𝑉 ∧ (♯‘𝑝) = 3 ∧ (𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2))) → 𝑝 ∈ Word 𝑉)
95 oveq2 7375 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((♯‘𝑝) = 3 → (0..^(♯‘𝑝)) = (0..^3))
96 fzo0to3tp 13707 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (0..^3) = {0, 1, 2}
9795, 96eqtrdi 2787 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((♯‘𝑝) = 3 → (0..^(♯‘𝑝)) = {0, 1, 2})
9831tpid1 4712 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 0 ∈ {0, 1, 2}
99 eleq2 2825 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((0..^(♯‘𝑝)) = {0, 1, 2} → (0 ∈ (0..^(♯‘𝑝)) ↔ 0 ∈ {0, 1, 2}))
10098, 99mpbiri 258 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((0..^(♯‘𝑝)) = {0, 1, 2} → 0 ∈ (0..^(♯‘𝑝)))
101 wrdsymbcl 14489 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((𝑝 ∈ Word 𝑉 ∧ 0 ∈ (0..^(♯‘𝑝))) → (𝑝‘0) ∈ 𝑉)
102100, 101sylan2 594 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((𝑝 ∈ Word 𝑉 ∧ (0..^(♯‘𝑝)) = {0, 1, 2}) → (𝑝‘0) ∈ 𝑉)
10339tpid2 4714 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 1 ∈ {0, 1, 2}
104 eleq2 2825 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((0..^(♯‘𝑝)) = {0, 1, 2} → (1 ∈ (0..^(♯‘𝑝)) ↔ 1 ∈ {0, 1, 2}))
105103, 104mpbiri 258 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((0..^(♯‘𝑝)) = {0, 1, 2} → 1 ∈ (0..^(♯‘𝑝)))
106 wrdsymbcl 14489 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((𝑝 ∈ Word 𝑉 ∧ 1 ∈ (0..^(♯‘𝑝))) → (𝑝‘1) ∈ 𝑉)
107105, 106sylan2 594 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((𝑝 ∈ Word 𝑉 ∧ (0..^(♯‘𝑝)) = {0, 1, 2}) → (𝑝‘1) ∈ 𝑉)
108 2ex 12258 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 2 ∈ V
109108tpid3 4717 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 2 ∈ {0, 1, 2}
110 eleq2 2825 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((0..^(♯‘𝑝)) = {0, 1, 2} → (2 ∈ (0..^(♯‘𝑝)) ↔ 2 ∈ {0, 1, 2}))
111109, 110mpbiri 258 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((0..^(♯‘𝑝)) = {0, 1, 2} → 2 ∈ (0..^(♯‘𝑝)))
112 wrdsymbcl 14489 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((𝑝 ∈ Word 𝑉 ∧ 2 ∈ (0..^(♯‘𝑝))) → (𝑝‘2) ∈ 𝑉)
113111, 112sylan2 594 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((𝑝 ∈ Word 𝑉 ∧ (0..^(♯‘𝑝)) = {0, 1, 2}) → (𝑝‘2) ∈ 𝑉)
114102, 107, 1133jca 1129 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((𝑝 ∈ Word 𝑉 ∧ (0..^(♯‘𝑝)) = {0, 1, 2}) → ((𝑝‘0) ∈ 𝑉 ∧ (𝑝‘1) ∈ 𝑉 ∧ (𝑝‘2) ∈ 𝑉))
11597, 114sylan2 594 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝑝 ∈ Word 𝑉 ∧ (♯‘𝑝) = 3) → ((𝑝‘0) ∈ 𝑉 ∧ (𝑝‘1) ∈ 𝑉 ∧ (𝑝‘2) ∈ 𝑉))
1161153adant3 1133 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝑝 ∈ Word 𝑉 ∧ (♯‘𝑝) = 3 ∧ (𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2))) → ((𝑝‘0) ∈ 𝑉 ∧ (𝑝‘1) ∈ 𝑉 ∧ (𝑝‘2) ∈ 𝑉))
117 eleq1 2824 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝐴 = (𝑝‘0) → (𝐴 ∈ 𝑉 ↔ (𝑝‘0) ∈ 𝑉))
1181173ad2ant1 1134 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)) → (𝐴 ∈ 𝑉 ↔ (𝑝‘0) ∈ 𝑉))
119 eleq1 2824 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝐵 = (𝑝‘1) → (𝐵 ∈ 𝑉 ↔ (𝑝‘1) ∈ 𝑉))
1201193ad2ant2 1135 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)) → (𝐵 ∈ 𝑉 ↔ (𝑝‘1) ∈ 𝑉))
121 eleq1 2824 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝐶 = (𝑝‘2) → (𝐶 ∈ 𝑉 ↔ (𝑝‘2) ∈ 𝑉))
1221213ad2ant3 1136 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)) → (𝐶 ∈ 𝑉 ↔ (𝑝‘2) ∈ 𝑉))
123118, 120, 1223anbi123d 1439 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)) → ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) ↔ ((𝑝‘0) ∈ 𝑉 ∧ (𝑝‘1) ∈ 𝑉 ∧ (𝑝‘2) ∈ 𝑉)))
1241233ad2ant3 1136 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝑝 ∈ Word 𝑉 ∧ (♯‘𝑝) = 3 ∧ (𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2))) → ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) ↔ ((𝑝‘0) ∈ 𝑉 ∧ (𝑝‘1) ∈ 𝑉 ∧ (𝑝‘2) ∈ 𝑉)))
125116, 124mpbird 257 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑝 ∈ Word 𝑉 ∧ (♯‘𝑝) = 3 ∧ (𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2))) → (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉))
12694, 125jca 511 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑝 ∈ Word 𝑉 ∧ (♯‘𝑝) = 3 ∧ (𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2))) → (𝑝 ∈ Word 𝑉 ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)))
1271263exp 1120 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑝 ∈ Word 𝑉 → ((♯‘𝑝) = 3 → ((𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)) → (𝑝 ∈ Word 𝑉 ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)))))
128127adantr 480 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑝 ∈ Word 𝑉 ∧ (♯‘𝑓) = 2) → ((♯‘𝑝) = 3 → ((𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)) → (𝑝 ∈ Word 𝑉 ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)))))
12993, 128sylbid 240 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑝 ∈ Word 𝑉 ∧ (♯‘𝑓) = 2) → ((♯‘𝑝) = ((♯‘𝑓) + 1) → ((𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)) → (𝑝 ∈ Word 𝑉 ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)))))
130129impancom 451 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑝 ∈ Word 𝑉 ∧ (♯‘𝑝) = ((♯‘𝑓) + 1)) → ((♯‘𝑓) = 2 → ((𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)) → (𝑝 ∈ Word 𝑉 ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)))))
131130impd 410 . . . . . . . . . . . . . . . . . . . . 21 ((𝑝 ∈ Word 𝑉 ∧ (♯‘𝑝) = ((♯‘𝑓) + 1)) → (((♯‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2))) → (𝑝 ∈ Word 𝑉 ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉))))
13291, 77, 131syl2anc 585 . . . . . . . . . . . . . . . . . . . 20 (𝑓(Walks‘𝐺)𝑝 → (((♯‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2))) → (𝑝 ∈ Word 𝑉 ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉))))
133132imp 406 . . . . . . . . . . . . . . . . . . 19 ((𝑓(Walks‘𝐺)𝑝 ∧ ((♯‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)))) → (𝑝 ∈ Word 𝑉 ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)))
134 eqwrds3 14923 . . . . . . . . . . . . . . . . . . 19 ((𝑝 ∈ Word 𝑉 ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) → (𝑝 = ⟨“𝐴𝐵𝐶”⟩ ↔ ((♯‘𝑝) = 3 ∧ ((𝑝‘0) = 𝐴 ∧ (𝑝‘1) = 𝐵 ∧ (𝑝‘2) = 𝐶))))
135133, 134syl 17 . . . . . . . . . . . . . . . . . 18 ((𝑓(Walks‘𝐺)𝑝 ∧ ((♯‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)))) → (𝑝 = ⟨“𝐴𝐵𝐶”⟩ ↔ ((♯‘𝑝) = 3 ∧ ((𝑝‘0) = 𝐴 ∧ (𝑝‘1) = 𝐵 ∧ (𝑝‘2) = 𝐶))))
13690, 135mpbird 257 . . . . . . . . . . . . . . . . 17 ((𝑓(Walks‘𝐺)𝑝 ∧ ((♯‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)))) → 𝑝 = ⟨“𝐴𝐵𝐶”⟩)
137136breq2d 5097 . . . . . . . . . . . . . . . 16 ((𝑓(Walks‘𝐺)𝑝 ∧ ((♯‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)))) → (𝑓(Walks‘𝐺)𝑝 ↔ 𝑓(Walks‘𝐺)⟨“𝐴𝐵𝐶”⟩))
138137biimpd 229 . . . . . . . . . . . . . . 15 ((𝑓(Walks‘𝐺)𝑝 ∧ ((♯‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)))) → (𝑓(Walks‘𝐺)𝑝 → 𝑓(Walks‘𝐺)⟨“𝐴𝐵𝐶”⟩))
139138ex 412 . . . . . . . . . . . . . 14 (𝑓(Walks‘𝐺)𝑝 → (((♯‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2))) → (𝑓(Walks‘𝐺)𝑝 → 𝑓(Walks‘𝐺)⟨“𝐴𝐵𝐶”⟩)))
140139pm2.43a 54 . . . . . . . . . . . . 13 (𝑓(Walks‘𝐺)𝑝 → (((♯‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2))) → 𝑓(Walks‘𝐺)⟨“𝐴𝐵𝐶”⟩))
1411403impib 1117 . . . . . . . . . . . 12 ((𝑓(Walks‘𝐺)𝑝 ∧ (♯‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2))) → 𝑓(Walks‘𝐺)⟨“𝐴𝐵𝐶”⟩)
142141adantl 481 . . . . . . . . . . 11 (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) ∧ (𝑓(Walks‘𝐺)𝑝 ∧ (♯‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)))) → 𝑓(Walks‘𝐺)⟨“𝐴𝐵𝐶”⟩)
143 simpr2 1197 . . . . . . . . . . 11 (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) ∧ (𝑓(Walks‘𝐺)𝑝 ∧ (♯‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)))) → (♯‘𝑓) = 2)
144142, 143jca 511 . . . . . . . . . 10 (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) ∧ (𝑓(Walks‘𝐺)𝑝 ∧ (♯‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)))) → (𝑓(Walks‘𝐺)⟨“𝐴𝐵𝐶”⟩ ∧ (♯‘𝑓) = 2))
145144ex 412 . . . . . . . . 9 ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) → ((𝑓(Walks‘𝐺)𝑝 ∧ (♯‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2))) → (𝑓(Walks‘𝐺)⟨“𝐴𝐵𝐶”⟩ ∧ (♯‘𝑓) = 2)))
146145exlimdv 1935 . . . . . . . 8 ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) → (∃𝑝(𝑓(Walks‘𝐺)𝑝 ∧ (♯‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2))) → (𝑓(Walks‘𝐺)⟨“𝐴𝐵𝐶”⟩ ∧ (♯‘𝑓) = 2)))
147146eximdv 1919 . . . . . . 7 ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) → (∃𝑓∃𝑝(𝑓(Walks‘𝐺)𝑝 ∧ (♯‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2))) → ∃𝑓(𝑓(Walks‘𝐺)⟨“𝐴𝐵𝐶”⟩ ∧ (♯‘𝑓) = 2)))
14876, 147syl5com 31 . . . . . 6 ((𝐺 ∈ UMGraph ∧ {𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸) → ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) → ∃𝑓(𝑓(Walks‘𝐺)⟨“𝐴𝐵𝐶”⟩ ∧ (♯‘𝑓) = 2)))
1491483expib 1123 . . . . 5 (𝐺 ∈ UMGraph → (({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸) → ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) → ∃𝑓(𝑓(Walks‘𝐺)⟨“𝐴𝐵𝐶”⟩ ∧ (♯‘𝑓) = 2))))
150149com23 86 . . . 4 (𝐺 ∈ UMGraph → ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) → (({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸) → ∃𝑓(𝑓(Walks‘𝐺)⟨“𝐴𝐵𝐶”⟩ ∧ (♯‘𝑓) = 2))))
151150imp 406 . . 3 ((𝐺 ∈ UMGraph ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) → (({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸) → ∃𝑓(𝑓(Walks‘𝐺)⟨“𝐴𝐵𝐶”⟩ ∧ (♯‘𝑓) = 2)))
15275, 151impbid 212 . 2 ((𝐺 ∈ UMGraph ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) → (∃𝑓(𝑓(Walks‘𝐺)⟨“𝐴𝐵𝐶”⟩ ∧ (♯‘𝑓) = 2) ↔ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸)))
1538, 152bitrd 279 1 ((𝐺 ∈ UMGraph ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) → (⟨“𝐴𝐵𝐶”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐶) ↔ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸)))
Colors of variables: wff setvar class
Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1542  ∃wex 1781   ∈ wcel 2114  {cpr 4569  {ctp 4571   class class class wbr 5085  dom cdm 5631  ran crn 5632  Fun wfun 6492  ⟶wf 6494  ‘cfv 6498  (class class class)co 7367  0cc0 11038  1c1 11039   + caddc 11041  2c2 12236  3c3 12237  ...cfz 13461  ..^cfzo 13608  ♯chash 14292  Word cword 14475  ⟨“cs3 14804  Vtxcvtx 29065  iEdgciedg 29066  Edgcedg 29116  UHGraphcuhgr 29125  UPGraphcupgr 29149  UMGraphcumgr 29150  Walkscwlks 29665   WWalksNOn cwwlksnon 29895
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375  ax-un 7689  ax-ac2 10385  ax-cnex 11094  ax-resscn 11095  ax-1cn 11096  ax-icn 11097  ax-addcl 11098  ax-addrcl 11099  ax-mulcl 11100  ax-mulrcl 11101  ax-mulcom 11102  ax-addass 11103  ax-mulass 11104  ax-distr 11105  ax-i2m1 11106  ax-1ne0 11107  ax-1rid 11108  ax-rnegex 11109  ax-rrecex 11110  ax-cnre 11111  ax-pre-lttri 11112  ax-pre-lttrn 11113  ax-pre-ltadd 11114  ax-pre-mulgt0 11115
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-ifp 1064  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3062  df-rmo 3342  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-pss 3909  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-tp 4572  df-op 4574  df-uni 4851  df-int 4890  df-iun 4935  df-br 5086  df-opab 5148  df-mpt 5167  df-tr 5193  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-se 5585  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6265  df-ord 6326  df-on 6327  df-lim 6328  df-suc 6329  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506  df-isom 6507  df-riota 7324  df-ov 7370  df-oprab 7371  df-mpo 7372  df-om 7818  df-1st 7942  df-2nd 7943  df-frecs 8231  df-wrecs 8262  df-recs 8311  df-rdg 8349  df-1o 8405  df-2o 8406  df-oadd 8409  df-er 8643  df-map 8775  df-pm 8776  df-en 8894  df-dom 8895  df-sdom 8896  df-fin 8897  df-dju 9825  df-card 9863  df-ac 10038  df-pnf 11181  df-mnf 11182  df-xr 11183  df-ltxr 11184  df-le 11185  df-sub 11379  df-neg 11380  df-nn 12175  df-2 12244  df-3 12245  df-n0 12438  df-xnn0 12511  df-z 12525  df-uz 12789  df-fz 13462  df-fzo 13609  df-hash 14293  df-word 14476  df-concat 14533  df-s1 14559  df-s2 14810  df-s3 14811  df-edg 29117  df-uhgr 29127  df-upgr 29151  df-umgr 29152  df-wlks 29668  df-wwlks 29898  df-wwlksn 29899  df-wwlksnon 29900
This theorem is referenced by:  wwlks2onsym  30028
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