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Theorem umgrwwlks2on 30045
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 29190 . . . 4 (𝐺 ∈ UMGraph → 𝐺 ∈ UPGraph)
21adantr 481 . . 3 ((𝐺 ∈ UMGraph ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) → 𝐺 ∈ UPGraph)
3 simp1 1142 . . . 4 ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) → 𝐴 ∈ 𝑉)
43adantl 482 . . 3 ((𝐺 ∈ UMGraph ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) → 𝐴 ∈ 𝑉)
5 simpr3 1203 . . 3 ((𝐺 ∈ UMGraph ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) → 𝐶 ∈ 𝑉)
6 s3wwlks2on.v . . . 4 𝑉 = (Vtx‘𝐺)
76s3wwlks2on 30042 . . 3 ((𝐺 ∈ UPGraph ∧ 𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) → (⟨“𝐴𝐵𝐶”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐶) ↔ ∃𝑓(𝑓(Walks‘𝐺)⟨“𝐴𝐵𝐶”⟩ ∧ (♯‘𝑓) = 2)))
82, 4, 5, 7syl3anc 1379 . 2 ((𝐺 ∈ UMGraph ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) → (⟨“𝐴𝐵𝐶”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐶) ↔ ∃𝑓(𝑓(Walks‘𝐺)⟨“𝐴𝐵𝐶”⟩ ∧ (♯‘𝑓) = 2)))
9 eqid 2739 . . . . . . . 8 (iEdg‘𝐺) = (iEdg‘𝐺)
106, 9upgr2wlk 29753 . . . . . . 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 481 . . . . 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 14844 . . . . . . . . . . . 12 (𝐴 ∈ 𝑉 → (⟨“𝐴𝐵𝐶”⟩‘0) = 𝐴)
14133ad2ant1 1139 . . . . . . . . . . 11 ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) → (⟨“𝐴𝐵𝐶”⟩‘0) = 𝐴)
15 s3fv1 14845 . . . . . . . . . . . 12 (𝐵 ∈ 𝑉 → (⟨“𝐴𝐵𝐶”⟩‘1) = 𝐵)
16153ad2ant2 1140 . . . . . . . . . . 11 ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) → (⟨“𝐴𝐵𝐶”⟩‘1) = 𝐵)
1714, 16preq12d 4673 . . . . . . . . . 10 ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) → {(⟨“𝐴𝐵𝐶”⟩‘0), (⟨“𝐴𝐵𝐶”⟩‘1)} = {𝐴, 𝐵})
1817eqeq2d 2750 . . . . . . . . 9 ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) → (((iEdg‘𝐺)‘(𝑓‘0)) = {(⟨“𝐴𝐵𝐶”⟩‘0), (⟨“𝐴𝐵𝐶”⟩‘1)} ↔ ((iEdg‘𝐺)‘(𝑓‘0)) = {𝐴, 𝐵}))
19 s3fv2 14846 . . . . . . . . . . . 12 (𝐶 ∈ 𝑉 → (⟨“𝐴𝐵𝐶”⟩‘2) = 𝐶)
20193ad2ant3 1141 . . . . . . . . . . 11 ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) → (⟨“𝐴𝐵𝐶”⟩‘2) = 𝐶)
2116, 20preq12d 4673 . . . . . . . . . 10 ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) → {(⟨“𝐴𝐵𝐶”⟩‘1), (⟨“𝐴𝐵𝐶”⟩‘2)} = {𝐵, 𝐶})
2221eqeq2d 2750 . . . . . . . . 9 ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) → (((iEdg‘𝐺)‘(𝑓‘1)) = {(⟨“𝐴𝐵𝐶”⟩‘1), (⟨“𝐴𝐵𝐶”⟩‘2)} ↔ ((iEdg‘𝐺)‘(𝑓‘1)) = {𝐵, 𝐶}))
2318, 22anbi12d 638 . . . . . . . 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 482 . . . . . . 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 1450 . . . . . 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 29191 . . . . . . . . . . 11 (𝐺 ∈ UMGraph → 𝐺 ∈ UHGraph)
279uhgrfun 29153 . . . . . . . . . . 11 (𝐺 ∈ UHGraph → Fun (iEdg‘𝐺))
28 fdmrn 6686 . . . . . . . . . . . 12 (Fun (iEdg‘𝐺) ↔ (iEdg‘𝐺):dom (iEdg‘𝐺)⟶ran (iEdg‘𝐺))
29 simpr 485 . . . . . . . . . . . . . . . . 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 11129 . . . . . . . . . . . . . . . . . . . . . 22 0 ∈ V
3231prid1 4694 . . . . . . . . . . . . . . . . . . . . 21 0 ∈ {0, 1}
33 fzo0to2pr 13696 . . . . . . . . . . . . . . . . . . . . 21 (0..^2) = {0, 1}
3432, 33eleqtrri 2838 . . . . . . . . . . . . . . . . . . . 20 0 ∈ (0..^2)
3534a1i 11 . . . . . . . . . . . . . . . . . . 19 (𝑓:(0..^2)⟶dom (iEdg‘𝐺) → 0 ∈ (0..^2))
3630, 35ffvelcdmd 7026 . . . . . . . . . . . . . . . . . 18 (𝑓:(0..^2)⟶dom (iEdg‘𝐺) → (𝑓‘0) ∈ dom (iEdg‘𝐺))
3736adantr 481 . . . . . . . . . . . . . . . . 17 ((𝑓:(0..^2)⟶dom (iEdg‘𝐺) ∧ (iEdg‘𝐺):dom (iEdg‘𝐺)⟶ran (iEdg‘𝐺)) → (𝑓‘0) ∈ dom (iEdg‘𝐺))
3829, 37ffvelcdmd 7026 . . . . . . . . . . . . . . . 16 ((𝑓:(0..^2)⟶dom (iEdg‘𝐺) ∧ (iEdg‘𝐺):dom (iEdg‘𝐺)⟶ran (iEdg‘𝐺)) → ((iEdg‘𝐺)‘(𝑓‘0)) ∈ ran (iEdg‘𝐺))
39 1ex 11131 . . . . . . . . . . . . . . . . . . . . . 22 1 ∈ V
4039prid2 4695 . . . . . . . . . . . . . . . . . . . . 21 1 ∈ {0, 1}
4140, 33eleqtrri 2838 . . . . . . . . . . . . . . . . . . . 20 1 ∈ (0..^2)
4241a1i 11 . . . . . . . . . . . . . . . . . . 19 (𝑓:(0..^2)⟶dom (iEdg‘𝐺) → 1 ∈ (0..^2))
4330, 42ffvelcdmd 7026 . . . . . . . . . . . . . . . . . 18 (𝑓:(0..^2)⟶dom (iEdg‘𝐺) → (𝑓‘1) ∈ dom (iEdg‘𝐺))
4443adantr 481 . . . . . . . . . . . . . . . . 17 ((𝑓:(0..^2)⟶dom (iEdg‘𝐺) ∧ (iEdg‘𝐺):dom (iEdg‘𝐺)⟶ran (iEdg‘𝐺)) → (𝑓‘1) ∈ dom (iEdg‘𝐺))
4529, 44ffvelcdmd 7026 . . . . . . . . . . . . . . . 16 ((𝑓:(0..^2)⟶dom (iEdg‘𝐺) ∧ (iEdg‘𝐺):dom (iEdg‘𝐺)⟶ran (iEdg‘𝐺)) → ((iEdg‘𝐺)‘(𝑓‘1)) ∈ ran (iEdg‘𝐺))
4638, 45jca 516 . . . . . . . . . . . . . . 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 413 . . . . . . . . . . . . . 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 1139 . . . . . . . . . . . . 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 218 . . . . . . . . . . 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 407 . . . . . . . . 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 2746 . . . . . . . . . . . . . 14 (((iEdg‘𝐺)‘(𝑓‘0)) = {𝐴, 𝐵} ↔ {𝐴, 𝐵} = ((iEdg‘𝐺)‘(𝑓‘0)))
5453birani 504 . . . . . . . . . . . . 13 ((((iEdg‘𝐺)‘(𝑓‘0)) = {𝐴, 𝐵} ∧ ((iEdg‘𝐺)‘(𝑓‘1)) = {𝐵, 𝐶}) → {𝐴, 𝐵} = ((iEdg‘𝐺)‘(𝑓‘0)))
55543ad2ant3 1141 . . . . . . . . . . . 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)))
5655adantl 482 . . . . . . . . . . 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)))
57 usgrwwlks2on.e . . . . . . . . . . . . 13 𝐸 = (Edg‘𝐺)
58 edgval 29136 . . . . . . . . . . . . 13 (Edg‘𝐺) = ran (iEdg‘𝐺)
5957, 58eqtri 2762 . . . . . . . . . . . 12 𝐸 = ran (iEdg‘𝐺)
6059a1i 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))
6156, 60eleq12d 2833 . . . . . . . . . 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)))
62 eqcom 2746 . . . . . . . . . . . . . 14 (((iEdg‘𝐺)‘(𝑓‘1)) = {𝐵, 𝐶} ↔ {𝐵, 𝐶} = ((iEdg‘𝐺)‘(𝑓‘1)))
6362bilani 505 . . . . . . . . . . . . 13 ((((iEdg‘𝐺)‘(𝑓‘0)) = {𝐴, 𝐵} ∧ ((iEdg‘𝐺)‘(𝑓‘1)) = {𝐵, 𝐶}) → {𝐵, 𝐶} = ((iEdg‘𝐺)‘(𝑓‘1)))
64633ad2ant3 1141 . . . . . . . . . . . 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)))
6564adantl 482 . . . . . . . . . . 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)))
6665, 60eleq12d 2833 . . . . . . . . . 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)))
6761, 66anbi12d 638 . . . . . . . . 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))))
6852, 67mpbird 258 . . . . . . . 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))
6968ex 413 . . . . . . 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)))
7069adantr 481 . . . . . 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)))
7125, 70sylbid 241 . . . . 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)))
7212, 71sylbid 241 . . . 4 ((𝐺 ∈ UMGraph ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) → ((𝑓(Walks‘𝐺)⟨“𝐴𝐵𝐶”⟩ ∧ (♯‘𝑓) = 2) → ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸)))
7372exlimdv 1940 . . 3 ((𝐺 ∈ UMGraph ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) → (∃𝑓(𝑓(Walks‘𝐺)⟨“𝐴𝐵𝐶”⟩ ∧ (♯‘𝑓) = 2) → ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸)))
7457umgr2wlk 30035 . . . . . . 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))))
75 wlklenvp1 29705 . . . . . . . . . . . . . . . . . . . 20 (𝑓(Walks‘𝐺)𝑝 → (♯‘𝑝) = ((♯‘𝑓) + 1))
76 oveq1 7363 . . . . . . . . . . . . . . . . . . . . . 22 ((♯‘𝑓) = 2 → ((♯‘𝑓) + 1) = (2 + 1))
77 2p1e3 12309 . . . . . . . . . . . . . . . . . . . . . 22 (2 + 1) = 3
7876, 77eqtrdi 2790 . . . . . . . . . . . . . . . . . . . . 21 ((♯‘𝑓) = 2 → ((♯‘𝑓) + 1) = 3)
7978adantr 481 . . . . . . . . . . . . . . . . . . . 20 (((♯‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2))) → ((♯‘𝑓) + 1) = 3)
8075, 79sylan9eq 2794 . . . . . . . . . . . . . . . . . . 19 ((𝑓(Walks‘𝐺)𝑝 ∧ ((♯‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)))) → (♯‘𝑝) = 3)
81 eqcom 2746 . . . . . . . . . . . . . . . . . . . . . 22 (𝐴 = (𝑝‘0) ↔ (𝑝‘0) = 𝐴)
82 eqcom 2746 . . . . . . . . . . . . . . . . . . . . . 22 (𝐵 = (𝑝‘1) ↔ (𝑝‘1) = 𝐵)
83 eqcom 2746 . . . . . . . . . . . . . . . . . . . . . 22 (𝐶 = (𝑝‘2) ↔ (𝑝‘2) = 𝐶)
8481, 82, 833anbi123i 1161 . . . . . . . . . . . . . . . . . . . . 21 ((𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)) ↔ ((𝑝‘0) = 𝐴 ∧ (𝑝‘1) = 𝐵 ∧ (𝑝‘2) = 𝐶))
8584bilani 505 . . . . . . . . . . . . . . . . . . . 20 (((♯‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2))) → ((𝑝‘0) = 𝐴 ∧ (𝑝‘1) = 𝐵 ∧ (𝑝‘2) = 𝐶))
8685adantl 482 . . . . . . . . . . . . . . . . . . 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))
8780, 86jca 516 . . . . . . . . . . . . . . . . . 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)))
886wlkpwrd 29704 . . . . . . . . . . . . . . . . . . . . 21 (𝑓(Walks‘𝐺)𝑝 → 𝑝 ∈ Word 𝑉)
8978eqeq2d 2750 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((♯‘𝑓) = 2 → ((♯‘𝑝) = ((♯‘𝑓) + 1) ↔ (♯‘𝑝) = 3))
9089adantl 482 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑝 ∈ Word 𝑉 ∧ (♯‘𝑓) = 2) → ((♯‘𝑝) = ((♯‘𝑓) + 1) ↔ (♯‘𝑝) = 3))
91 simp1 1142 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑝 ∈ Word 𝑉 ∧ (♯‘𝑝) = 3 ∧ (𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2))) → 𝑝 ∈ Word 𝑉)
92 oveq2 7364 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((♯‘𝑝) = 3 → (0..^(♯‘𝑝)) = (0..^3))
93 fzo0to3tp 13698 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (0..^3) = {0, 1, 2}
9492, 93eqtrdi 2790 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((♯‘𝑝) = 3 → (0..^(♯‘𝑝)) = {0, 1, 2})
9531tpid1 4700 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 0 ∈ {0, 1, 2}
96 eleq2 2828 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((0..^(♯‘𝑝)) = {0, 1, 2} → (0 ∈ (0..^(♯‘𝑝)) ↔ 0 ∈ {0, 1, 2}))
9795, 96mpbiri 259 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((0..^(♯‘𝑝)) = {0, 1, 2} → 0 ∈ (0..^(♯‘𝑝)))
98 wrdsymbcl 14480 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((𝑝 ∈ Word 𝑉 ∧ 0 ∈ (0..^(♯‘𝑝))) → (𝑝‘0) ∈ 𝑉)
9997, 98sylan2 599 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((𝑝 ∈ Word 𝑉 ∧ (0..^(♯‘𝑝)) = {0, 1, 2}) → (𝑝‘0) ∈ 𝑉)
10039tpid2 4702 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 1 ∈ {0, 1, 2}
101 eleq2 2828 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((0..^(♯‘𝑝)) = {0, 1, 2} → (1 ∈ (0..^(♯‘𝑝)) ↔ 1 ∈ {0, 1, 2}))
102100, 101mpbiri 259 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((0..^(♯‘𝑝)) = {0, 1, 2} → 1 ∈ (0..^(♯‘𝑝)))
103 wrdsymbcl 14480 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((𝑝 ∈ Word 𝑉 ∧ 1 ∈ (0..^(♯‘𝑝))) → (𝑝‘1) ∈ 𝑉)
104102, 103sylan2 599 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((𝑝 ∈ Word 𝑉 ∧ (0..^(♯‘𝑝)) = {0, 1, 2}) → (𝑝‘1) ∈ 𝑉)
105 2ex 12249 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 2 ∈ V
106105tpid3 4705 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 2 ∈ {0, 1, 2}
107 eleq2 2828 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((0..^(♯‘𝑝)) = {0, 1, 2} → (2 ∈ (0..^(♯‘𝑝)) ↔ 2 ∈ {0, 1, 2}))
108106, 107mpbiri 259 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((0..^(♯‘𝑝)) = {0, 1, 2} → 2 ∈ (0..^(♯‘𝑝)))
109 wrdsymbcl 14480 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((𝑝 ∈ Word 𝑉 ∧ 2 ∈ (0..^(♯‘𝑝))) → (𝑝‘2) ∈ 𝑉)
110108, 109sylan2 599 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((𝑝 ∈ Word 𝑉 ∧ (0..^(♯‘𝑝)) = {0, 1, 2}) → (𝑝‘2) ∈ 𝑉)
11199, 104, 1103jca 1134 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((𝑝 ∈ Word 𝑉 ∧ (0..^(♯‘𝑝)) = {0, 1, 2}) → ((𝑝‘0) ∈ 𝑉 ∧ (𝑝‘1) ∈ 𝑉 ∧ (𝑝‘2) ∈ 𝑉))
11294, 111sylan2 599 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝑝 ∈ Word 𝑉 ∧ (♯‘𝑝) = 3) → ((𝑝‘0) ∈ 𝑉 ∧ (𝑝‘1) ∈ 𝑉 ∧ (𝑝‘2) ∈ 𝑉))
1131123adant3 1138 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝑝 ∈ Word 𝑉 ∧ (♯‘𝑝) = 3 ∧ (𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2))) → ((𝑝‘0) ∈ 𝑉 ∧ (𝑝‘1) ∈ 𝑉 ∧ (𝑝‘2) ∈ 𝑉))
114 eleq1 2827 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝐴 = (𝑝‘0) → (𝐴 ∈ 𝑉 ↔ (𝑝‘0) ∈ 𝑉))
1151143ad2ant1 1139 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)) → (𝐴 ∈ 𝑉 ↔ (𝑝‘0) ∈ 𝑉))
116 eleq1 2827 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝐵 = (𝑝‘1) → (𝐵 ∈ 𝑉 ↔ (𝑝‘1) ∈ 𝑉))
1171163ad2ant2 1140 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)) → (𝐵 ∈ 𝑉 ↔ (𝑝‘1) ∈ 𝑉))
118 eleq1 2827 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝐶 = (𝑝‘2) → (𝐶 ∈ 𝑉 ↔ (𝑝‘2) ∈ 𝑉))
1191183ad2ant3 1141 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)) → (𝐶 ∈ 𝑉 ↔ (𝑝‘2) ∈ 𝑉))
120115, 117, 1193anbi123d 1444 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)))
1211203ad2ant3 1141 . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)))
122113, 121mpbird 258 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑝 ∈ Word 𝑉 ∧ (♯‘𝑝) = 3 ∧ (𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2))) → (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉))
12391, 122jca 516 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑝 ∈ Word 𝑉 ∧ (♯‘𝑝) = 3 ∧ (𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2))) → (𝑝 ∈ Word 𝑉 ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)))
1241233exp 1125 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑝 ∈ Word 𝑉 → ((♯‘𝑝) = 3 → ((𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)) → (𝑝 ∈ Word 𝑉 ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)))))
125124adantr 481 . . . . . . . . . . . . . . . . . . . . . . . 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)))))
12690, 125sylbid 241 . . . . . . . . . . . . . . . . . . . . . . 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)))))
127126impancom 452 . . . . . . . . . . . . . . . . . . . . . 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)))))
128127impd 411 . . . . . . . . . . . . . . . . . . . . 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))))
12988, 75, 128syl2anc 590 . . . . . . . . . . . . . . . . . . . 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))))
130129imp 407 . . . . . . . . . . . . . . . . . . 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)))
131 eqwrds3 14914 . . . . . . . . . . . . . . . . . . 19 ((𝑝 ∈ Word 𝑉 ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) → (𝑝 = ⟨“𝐴𝐵𝐶”⟩ ↔ ((♯‘𝑝) = 3 ∧ ((𝑝‘0) = 𝐴 ∧ (𝑝‘1) = 𝐵 ∧ (𝑝‘2) = 𝐶))))
132130, 131syl 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))))
13387, 132mpbird 258 . . . . . . . . . . . . . . . . 17 ((𝑓(Walks‘𝐺)𝑝 ∧ ((♯‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)))) → 𝑝 = ⟨“𝐴𝐵𝐶”⟩)
134133breq2d 5084 . . . . . . . . . . . . . . . 16 ((𝑓(Walks‘𝐺)𝑝 ∧ ((♯‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)))) → (𝑓(Walks‘𝐺)𝑝 ↔ 𝑓(Walks‘𝐺)⟨“𝐴𝐵𝐶”⟩))
135134biimpd 230 . . . . . . . . . . . . . . 15 ((𝑓(Walks‘𝐺)𝑝 ∧ ((♯‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)))) → (𝑓(Walks‘𝐺)𝑝 → 𝑓(Walks‘𝐺)⟨“𝐴𝐵𝐶”⟩))
136135ex 413 . . . . . . . . . . . . . 14 (𝑓(Walks‘𝐺)𝑝 → (((♯‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2))) → (𝑓(Walks‘𝐺)𝑝 → 𝑓(Walks‘𝐺)⟨“𝐴𝐵𝐶”⟩)))
137136pm2.43a 54 . . . . . . . . . . . . 13 (𝑓(Walks‘𝐺)𝑝 → (((♯‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2))) → 𝑓(Walks‘𝐺)⟨“𝐴𝐵𝐶”⟩))
1381373impib 1122 . . . . . . . . . . . 12 ((𝑓(Walks‘𝐺)𝑝 ∧ (♯‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2))) → 𝑓(Walks‘𝐺)⟨“𝐴𝐵𝐶”⟩)
139138adantl 482 . . . . . . . . . . 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”⟩)
140 simpr2 1202 . . . . . . . . . . 11 (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) ∧ (𝑓(Walks‘𝐺)𝑝 ∧ (♯‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)))) → (♯‘𝑓) = 2)
141139, 140jca 516 . . . . . . . . . 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))
142141ex 413 . . . . . . . . 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)))
143142exlimdv 1940 . . . . . . . 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)))
144143eximdv 1924 . . . . . . 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)))
14574, 144syl5com 31 . . . . . 6 ((𝐺 ∈ UMGraph ∧ {𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸) → ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) → ∃𝑓(𝑓(Walks‘𝐺)⟨“𝐴𝐵𝐶”⟩ ∧ (♯‘𝑓) = 2)))
1461453expib 1128 . . . . 5 (𝐺 ∈ UMGraph → (({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸) → ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) → ∃𝑓(𝑓(Walks‘𝐺)⟨“𝐴𝐵𝐶”⟩ ∧ (♯‘𝑓) = 2))))
147146com23 86 . . . 4 (𝐺 ∈ UMGraph → ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) → (({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸) → ∃𝑓(𝑓(Walks‘𝐺)⟨“𝐴𝐵𝐶”⟩ ∧ (♯‘𝑓) = 2))))
148147imp 407 . . 3 ((𝐺 ∈ UMGraph ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) → (({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸) → ∃𝑓(𝑓(Walks‘𝐺)⟨“𝐴𝐵𝐶”⟩ ∧ (♯‘𝑓) = 2)))
14973, 148impbid 213 . 2 ((𝐺 ∈ UMGraph ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) → (∃𝑓(𝑓(Walks‘𝐺)⟨“𝐴𝐵𝐶”⟩ ∧ (♯‘𝑓) = 2) ↔ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸)))
1508, 149bitrd 280 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 207   ∧ wa 396   ∧ w3a 1092   = wceq 1547  ∃wex 1786   ∈ wcel 2119  {cpr 4557  {ctp 4559   class class class wbr 5072  dom cdm 5618  ran crn 5619  Fun wfun 6479  ⟶wf 6481  ‘cfv 6485  (class class class)co 7356  0cc0 11029  1c1 11030   + caddc 11032  2c2 12227  3c3 12228  ...cfz 13452  ..^cfzo 13599  ♯chash 14283  Word cword 14466  ⟨“cs3 14795  Vtxcvtx 29083  iEdgciedg 29084  Edgcedg 29134  UHGraphcuhgr 29143  UPGraphcupgr 29167  UMGraphcumgr 29168  Walkscwlks 29683   WWalksNOn cwwlksnon 29913
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-rep 5199  ax-sep 5218  ax-nul 5228  ax-pow 5294  ax-pr 5362  ax-un 7678  ax-ac2 10376  ax-cnex 11085  ax-resscn 11086  ax-1cn 11087  ax-icn 11088  ax-addcl 11089  ax-addrcl 11090  ax-mulcl 11091  ax-mulrcl 11092  ax-mulcom 11093  ax-addass 11094  ax-mulass 11095  ax-distr 11096  ax-i2m1 11097  ax-1ne0 11098  ax-1rid 11099  ax-rnegex 11100  ax-rrecex 11101  ax-cnre 11102  ax-pre-lttri 11103  ax-pre-lttrn 11104  ax-pre-ltadd 11105  ax-pre-mulgt0 11106
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-ifp 1069  df-3or 1093  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-nel 3039  df-ral 3054  df-rex 3064  df-rmo 3344  df-reu 3345  df-rab 3392  df-v 3433  df-sbc 3724  df-csb 3832  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3903  df-nul 4262  df-if 4455  df-pw 4531  df-sn 4556  df-pr 4558  df-tp 4560  df-op 4562  df-uni 4839  df-int 4878  df-iun 4923  df-br 5073  df-opab 5135  df-mpt 5154  df-tr 5180  df-id 5513  df-eprel 5518  df-po 5526  df-so 5527  df-fr 5571  df-se 5572  df-we 5573  df-xp 5624  df-rel 5625  df-cnv 5626  df-co 5627  df-dm 5628  df-rn 5629  df-res 5630  df-ima 5631  df-pred 6252  df-ord 6313  df-on 6314  df-lim 6315  df-suc 6316  df-iota 6441  df-fun 6487  df-fn 6488  df-f 6489  df-f1 6490  df-fo 6491  df-f1o 6492  df-fv 6493  df-isom 6494  df-riota 7313  df-ov 7359  df-oprab 7360  df-mpo 7361  df-om 7807  df-1st 7931  df-2nd 7932  df-frecs 8221  df-wrecs 8252  df-recs 8301  df-rdg 8339  df-1o 8395  df-2o 8396  df-oadd 8399  df-er 8633  df-map 8765  df-pm 8766  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-dju 9816  df-card 9854  df-ac 10029  df-pnf 11172  df-mnf 11173  df-xr 11174  df-ltxr 11175  df-le 11176  df-sub 11370  df-neg 11371  df-nn 12166  df-2 12235  df-3 12236  df-n0 12429  df-xnn0 12502  df-z 12516  df-uz 12780  df-fz 13453  df-fzo 13600  df-hash 14284  df-word 14467  df-concat 14524  df-s1 14550  df-s2 14801  df-s3 14802  df-edg 29135  df-uhgr 29145  df-upgr 29169  df-umgr 29170  df-wlks 29686  df-wwlks 29916  df-wwlksn 29917  df-wwlksnon 29918
This theorem is referenced by:  wwlks2onsym  30046
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