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Theorem umgrwwlks2on 30032
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 29176 . . . 4 (𝐺 ∈ UMGraph → 𝐺 ∈ UPGraph)
21adantr 480 . . 3 ((𝐺 ∈ UMGraph ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) → 𝐺 ∈ UPGraph)
3 simp1 1136 . . . 4 ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) → 𝐴 ∈ 𝑉)
43adantl 481 . . 3 ((𝐺 ∈ UMGraph ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) → 𝐴 ∈ 𝑉)
5 simpr3 1197 . . 3 ((𝐺 ∈ UMGraph ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) → 𝐶 ∈ 𝑉)
6 s3wwlks2on.v . . . 4 𝑉 = (Vtx‘𝐺)
76s3wwlks2on 30029 . . 3 ((𝐺 ∈ UPGraph ∧ 𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) → (⟨“𝐴𝐵𝐶”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐶) ↔ ∃𝑓(𝑓(Walks‘𝐺)⟨“𝐴𝐵𝐶”⟩ ∧ (♯‘𝑓) = 2)))
82, 4, 5, 7syl3anc 1373 . 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 29740 . . . . . . 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 14814 . . . . . . . . . . . 12 (𝐴 ∈ 𝑉 → (⟨“𝐴𝐵𝐶”⟩‘0) = 𝐴)
14133ad2ant1 1133 . . . . . . . . . . 11 ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) → (⟨“𝐴𝐵𝐶”⟩‘0) = 𝐴)
15 s3fv1 14815 . . . . . . . . . . . 12 (𝐵 ∈ 𝑉 → (⟨“𝐴𝐵𝐶”⟩‘1) = 𝐵)
16153ad2ant2 1134 . . . . . . . . . . 11 ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) → (⟨“𝐴𝐵𝐶”⟩‘1) = 𝐵)
1714, 16preq12d 4698 . . . . . . . . . 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 14816 . . . . . . . . . . . 12 (𝐶 ∈ 𝑉 → (⟨“𝐴𝐵𝐶”⟩‘2) = 𝐶)
20193ad2ant3 1135 . . . . . . . . . . 11 ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) → (⟨“𝐴𝐵𝐶”⟩‘2) = 𝐶)
2116, 20preq12d 4698 . . . . . . . . . 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 632 . . . . . . . 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 1444 . . . . . 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 29177 . . . . . . . . . . 11 (𝐺 ∈ UMGraph → 𝐺 ∈ UHGraph)
279uhgrfun 29139 . . . . . . . . . . 11 (𝐺 ∈ UHGraph → Fun (iEdg‘𝐺))
28 fdmrn 6693 . . . . . . . . . . . 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 11126 . . . . . . . . . . . . . . . . . . . . . 22 0 ∈ V
3231prid1 4719 . . . . . . . . . . . . . . . . . . . . 21 0 ∈ {0, 1}
33 fzo0to2pr 13666 . . . . . . . . . . . . . . . . . . . . 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 7030 . . . . . . . . . . . . . . . . . 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 7030 . . . . . . . . . . . . . . . 16 ((𝑓:(0..^2)⟶dom (iEdg‘𝐺) ∧ (iEdg‘𝐺):dom (iEdg‘𝐺)⟶ran (iEdg‘𝐺)) → ((iEdg‘𝐺)‘(𝑓‘0)) ∈ ran (iEdg‘𝐺))
39 1ex 11128 . . . . . . . . . . . . . . . . . . . . . 22 1 ∈ V
4039prid2 4720 . . . . . . . . . . . . . . . . . . . . 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 7030 . . . . . . . . . . . . . . . . . 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 7030 . . . . . . . . . . . . . . . 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 1133 . . . . . . . . . . . . 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 1135 . . . . . . . . . . . 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 29122 . . . . . . . . . . . . 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 1135 . . . . . . . . . . . 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 632 . . . . . . . . 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 1934 . . 3 ((𝐺 ∈ UMGraph ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) → (∃𝑓(𝑓(Walks‘𝐺)⟨“𝐴𝐵𝐶”⟩ ∧ (♯‘𝑓) = 2) → ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸)))
7658umgr2wlk 30022 . . . . . . 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 29692 . . . . . . . . . . . . . . . . . . . 20 (𝑓(Walks‘𝐺)𝑝 → (♯‘𝑝) = ((♯‘𝑓) + 1))
78 oveq1 7365 . . . . . . . . . . . . . . . . . . . . . 22 ((♯‘𝑓) = 2 → ((♯‘𝑓) + 1) = (2 + 1))
79 2p1e3 12282 . . . . . . . . . . . . . . . . . . . . . 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 1155 . . . . . . . . . . . . . . . . . . . . . 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 29691 . . . . . . . . . . . . . . . . . . . . 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 1136 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑝 ∈ Word 𝑉 ∧ (♯‘𝑝) = 3 ∧ (𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2))) → 𝑝 ∈ Word 𝑉)
95 oveq2 7366 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((♯‘𝑝) = 3 → (0..^(♯‘𝑝)) = (0..^3))
96 fzo0to3tp 13668 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (0..^3) = {0, 1, 2}
9795, 96eqtrdi 2787 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((♯‘𝑝) = 3 → (0..^(♯‘𝑝)) = {0, 1, 2})
9831tpid1 4725 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 14450 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((𝑝 ∈ Word 𝑉 ∧ 0 ∈ (0..^(♯‘𝑝))) → (𝑝‘0) ∈ 𝑉)
102100, 101sylan2 593 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((𝑝 ∈ Word 𝑉 ∧ (0..^(♯‘𝑝)) = {0, 1, 2}) → (𝑝‘0) ∈ 𝑉)
10339tpid2 4727 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 14450 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((𝑝 ∈ Word 𝑉 ∧ 1 ∈ (0..^(♯‘𝑝))) → (𝑝‘1) ∈ 𝑉)
107105, 106sylan2 593 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((𝑝 ∈ Word 𝑉 ∧ (0..^(♯‘𝑝)) = {0, 1, 2}) → (𝑝‘1) ∈ 𝑉)
108 2ex 12222 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 2 ∈ V
109108tpid3 4730 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 14450 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((𝑝 ∈ Word 𝑉 ∧ 2 ∈ (0..^(♯‘𝑝))) → (𝑝‘2) ∈ 𝑉)
113111, 112sylan2 593 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((𝑝 ∈ Word 𝑉 ∧ (0..^(♯‘𝑝)) = {0, 1, 2}) → (𝑝‘2) ∈ 𝑉)
114102, 107, 1133jca 1128 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((𝑝 ∈ Word 𝑉 ∧ (0..^(♯‘𝑝)) = {0, 1, 2}) → ((𝑝‘0) ∈ 𝑉 ∧ (𝑝‘1) ∈ 𝑉 ∧ (𝑝‘2) ∈ 𝑉))
11597, 114sylan2 593 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝑝 ∈ Word 𝑉 ∧ (♯‘𝑝) = 3) → ((𝑝‘0) ∈ 𝑉 ∧ (𝑝‘1) ∈ 𝑉 ∧ (𝑝‘2) ∈ 𝑉))
1161153adant3 1132 . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 1133 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 1134 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 1135 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)) → (𝐶 ∈ 𝑉 ↔ (𝑝‘2) ∈ 𝑉))
123118, 120, 1223anbi123d 1438 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 1135 . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 1119 . . . . . . . . . . . . . . . . . . . . . . . . 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 584 . . . . . . . . . . . . . . . . . . . 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 14884 . . . . . . . . . . . . . . . . . . 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 5110 . . . . . . . . . . . . . . . 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 1116 . . . . . . . . . . . 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 1196 . . . . . . . . . . 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 1934 . . . . . . . 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 1918 . . . . . . 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 1122 . . . . 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 1086   = wceq 1541  ∃wex 1780   ∈ wcel 2113  {cpr 4582  {ctp 4584   class class class wbr 5098  dom cdm 5624  ran crn 5625  Fun wfun 6486  ⟶wf 6488  ‘cfv 6492  (class class class)co 7358  0cc0 11026  1c1 11027   + caddc 11029  2c2 12200  3c3 12201  ...cfz 13423  ..^cfzo 13570  ♯chash 14253  Word cword 14436  ⟨“cs3 14765  Vtxcvtx 29069  iEdgciedg 29070  Edgcedg 29120  UHGraphcuhgr 29129  UPGraphcupgr 29153  UMGraphcumgr 29154  Walkscwlks 29670   WWalksNOn cwwlksnon 29900
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-rep 5224  ax-sep 5241  ax-nul 5251  ax-pow 5310  ax-pr 5377  ax-un 7680  ax-ac2 10373  ax-cnex 11082  ax-resscn 11083  ax-1cn 11084  ax-icn 11085  ax-addcl 11086  ax-addrcl 11087  ax-mulcl 11088  ax-mulrcl 11089  ax-mulcom 11090  ax-addass 11091  ax-mulass 11092  ax-distr 11093  ax-i2m1 11094  ax-1ne0 11095  ax-1rid 11096  ax-rnegex 11097  ax-rrecex 11098  ax-cnre 11099  ax-pre-lttri 11100  ax-pre-lttrn 11101  ax-pre-ltadd 11102  ax-pre-mulgt0 11103
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-ifp 1063  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  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 3061  df-rmo 3350  df-reu 3351  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-pss 3921  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-tp 4585  df-op 4587  df-uni 4864  df-int 4903  df-iun 4948  df-br 5099  df-opab 5161  df-mpt 5180  df-tr 5206  df-id 5519  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-se 5578  df-we 5579  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-pred 6259  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-isom 6501  df-riota 7315  df-ov 7361  df-oprab 7362  df-mpo 7363  df-om 7809  df-1st 7933  df-2nd 7934  df-frecs 8223  df-wrecs 8254  df-recs 8303  df-rdg 8341  df-1o 8397  df-2o 8398  df-oadd 8401  df-er 8635  df-map 8765  df-pm 8766  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-dju 9813  df-card 9851  df-ac 10026  df-pnf 11168  df-mnf 11169  df-xr 11170  df-ltxr 11171  df-le 11172  df-sub 11366  df-neg 11367  df-nn 12146  df-2 12208  df-3 12209  df-n0 12402  df-xnn0 12475  df-z 12489  df-uz 12752  df-fz 13424  df-fzo 13571  df-hash 14254  df-word 14437  df-concat 14494  df-s1 14520  df-s2 14771  df-s3 14772  df-edg 29121  df-uhgr 29131  df-upgr 29155  df-umgr 29156  df-wlks 29673  df-wwlks 29903  df-wwlksn 29904  df-wwlksnon 29905
This theorem is referenced by:  wwlks2onsym  30033
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