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Theorem umgrwwlks2on 29987
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 ∧ (𝐴𝑉𝐵𝑉𝐶𝑉)) → (⟨“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 29135 . . . 4 (𝐺 ∈ UMGraph → 𝐺 ∈ UPGraph)
21adantr 480 . . 3 ((𝐺 ∈ UMGraph ∧ (𝐴𝑉𝐵𝑉𝐶𝑉)) → 𝐺 ∈ UPGraph)
3 simp1 1135 . . . 4 ((𝐴𝑉𝐵𝑉𝐶𝑉) → 𝐴𝑉)
43adantl 481 . . 3 ((𝐺 ∈ UMGraph ∧ (𝐴𝑉𝐵𝑉𝐶𝑉)) → 𝐴𝑉)
5 simpr3 1195 . . 3 ((𝐺 ∈ UMGraph ∧ (𝐴𝑉𝐵𝑉𝐶𝑉)) → 𝐶𝑉)
6 s3wwlks2on.v . . . 4 𝑉 = (Vtx‘𝐺)
76s3wwlks2on 29986 . . 3 ((𝐺 ∈ UPGraph ∧ 𝐴𝑉𝐶𝑉) → (⟨“𝐴𝐵𝐶”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐶) ↔ ∃𝑓(𝑓(Walks‘𝐺)⟨“𝐴𝐵𝐶”⟩ ∧ (♯‘𝑓) = 2)))
82, 4, 5, 7syl3anc 1370 . 2 ((𝐺 ∈ UMGraph ∧ (𝐴𝑉𝐵𝑉𝐶𝑉)) → (⟨“𝐴𝐵𝐶”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐶) ↔ ∃𝑓(𝑓(Walks‘𝐺)⟨“𝐴𝐵𝐶”⟩ ∧ (♯‘𝑓) = 2)))
9 eqid 2735 . . . . . . . 8 (iEdg‘𝐺) = (iEdg‘𝐺)
106, 9upgr2wlk 29701 . . . . . . 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 ∧ (𝐴𝑉𝐵𝑉𝐶𝑉)) → ((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 14927 . . . . . . . . . . . 12 (𝐴𝑉 → (⟨“𝐴𝐵𝐶”⟩‘0) = 𝐴)
14133ad2ant1 1132 . . . . . . . . . . 11 ((𝐴𝑉𝐵𝑉𝐶𝑉) → (⟨“𝐴𝐵𝐶”⟩‘0) = 𝐴)
15 s3fv1 14928 . . . . . . . . . . . 12 (𝐵𝑉 → (⟨“𝐴𝐵𝐶”⟩‘1) = 𝐵)
16153ad2ant2 1133 . . . . . . . . . . 11 ((𝐴𝑉𝐵𝑉𝐶𝑉) → (⟨“𝐴𝐵𝐶”⟩‘1) = 𝐵)
1714, 16preq12d 4746 . . . . . . . . . 10 ((𝐴𝑉𝐵𝑉𝐶𝑉) → {(⟨“𝐴𝐵𝐶”⟩‘0), (⟨“𝐴𝐵𝐶”⟩‘1)} = {𝐴, 𝐵})
1817eqeq2d 2746 . . . . . . . . 9 ((𝐴𝑉𝐵𝑉𝐶𝑉) → (((iEdg‘𝐺)‘(𝑓‘0)) = {(⟨“𝐴𝐵𝐶”⟩‘0), (⟨“𝐴𝐵𝐶”⟩‘1)} ↔ ((iEdg‘𝐺)‘(𝑓‘0)) = {𝐴, 𝐵}))
19 s3fv2 14929 . . . . . . . . . . . 12 (𝐶𝑉 → (⟨“𝐴𝐵𝐶”⟩‘2) = 𝐶)
20193ad2ant3 1134 . . . . . . . . . . 11 ((𝐴𝑉𝐵𝑉𝐶𝑉) → (⟨“𝐴𝐵𝐶”⟩‘2) = 𝐶)
2116, 20preq12d 4746 . . . . . . . . . 10 ((𝐴𝑉𝐵𝑉𝐶𝑉) → {(⟨“𝐴𝐵𝐶”⟩‘1), (⟨“𝐴𝐵𝐶”⟩‘2)} = {𝐵, 𝐶})
2221eqeq2d 2746 . . . . . . . . 9 ((𝐴𝑉𝐵𝑉𝐶𝑉) → (((iEdg‘𝐺)‘(𝑓‘1)) = {(⟨“𝐴𝐵𝐶”⟩‘1), (⟨“𝐴𝐵𝐶”⟩‘2)} ↔ ((iEdg‘𝐺)‘(𝑓‘1)) = {𝐵, 𝐶}))
2318, 22anbi12d 632 . . . . . . . 8 ((𝐴𝑉𝐵𝑉𝐶𝑉) → ((((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 ∧ (𝐴𝑉𝐵𝑉𝐶𝑉)) → ((((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 1441 . . . . . 6 ((𝐺 ∈ UMGraph ∧ (𝐴𝑉𝐵𝑉𝐶𝑉)) → ((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 29136 . . . . . . . . . . 11 (𝐺 ∈ UMGraph → 𝐺 ∈ UHGraph)
279uhgrfun 29098 . . . . . . . . . . 11 (𝐺 ∈ UHGraph → Fun (iEdg‘𝐺))
28 fdmrn 6768 . . . . . . . . . . . 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 11253 . . . . . . . . . . . . . . . . . . . . . 22 0 ∈ V
3231prid1 4767 . . . . . . . . . . . . . . . . . . . . 21 0 ∈ {0, 1}
33 fzo0to2pr 13786 . . . . . . . . . . . . . . . . . . . . 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 7105 . . . . . . . . . . . . . . . . . 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 7105 . . . . . . . . . . . . . . . 16 ((𝑓:(0..^2)⟶dom (iEdg‘𝐺) ∧ (iEdg‘𝐺):dom (iEdg‘𝐺)⟶ran (iEdg‘𝐺)) → ((iEdg‘𝐺)‘(𝑓‘0)) ∈ ran (iEdg‘𝐺))
39 1ex 11255 . . . . . . . . . . . . . . . . . . . . . 22 1 ∈ V
4039prid2 4768 . . . . . . . . . . . . . . . . . . . . 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 7105 . . . . . . . . . . . . . . . . . 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 7105 . . . . . . . . . . . . . . . 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 1132 . . . . . . . . . . . . 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 2742 . . . . . . . . . . . . . . 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 1134 . . . . . . . . . . . 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 29081 . . . . . . . . . . . . 13 (Edg‘𝐺) = ran (iEdg‘𝐺)
6058, 59eqtri 2763 . . . . . . . . . . . 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 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)))
63 eqcom 2742 . . . . . . . . . . . . . . 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 1134 . . . . . . . . . . . 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 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)))
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 ∧ (𝐴𝑉𝐵𝑉𝐶𝑉)) → ((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 ∧ (𝐴𝑉𝐵𝑉𝐶𝑉)) → ((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 ∧ (𝐴𝑉𝐵𝑉𝐶𝑉)) → ((𝑓(Walks‘𝐺)⟨“𝐴𝐵𝐶”⟩ ∧ (♯‘𝑓) = 2) → ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸)))
7574exlimdv 1931 . . 3 ((𝐺 ∈ UMGraph ∧ (𝐴𝑉𝐵𝑉𝐶𝑉)) → (∃𝑓(𝑓(Walks‘𝐺)⟨“𝐴𝐵𝐶”⟩ ∧ (♯‘𝑓) = 2) → ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸)))
7658umgr2wlk 29979 . . . . . . 7 ((𝐺 ∈ UMGraph ∧ {𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸) → ∃𝑓𝑝(𝑓(Walks‘𝐺)𝑝 ∧ (♯‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2))))
77 wlklenvp1 29651 . . . . . . . . . . . . . . . . . . . 20 (𝑓(Walks‘𝐺)𝑝 → (♯‘𝑝) = ((♯‘𝑓) + 1))
78 oveq1 7438 . . . . . . . . . . . . . . . . . . . . . 22 ((♯‘𝑓) = 2 → ((♯‘𝑓) + 1) = (2 + 1))
79 2p1e3 12406 . . . . . . . . . . . . . . . . . . . . . 22 (2 + 1) = 3
8078, 79eqtrdi 2791 . . . . . . . . . . . . . . . . . . . . 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 2795 . . . . . . . . . . . . . . . . . . 19 ((𝑓(Walks‘𝐺)𝑝 ∧ ((♯‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)))) → (♯‘𝑝) = 3)
83 eqcom 2742 . . . . . . . . . . . . . . . . . . . . . . 23 (𝐴 = (𝑝‘0) ↔ (𝑝‘0) = 𝐴)
84 eqcom 2742 . . . . . . . . . . . . . . . . . . . . . . 23 (𝐵 = (𝑝‘1) ↔ (𝑝‘1) = 𝐵)
85 eqcom 2742 . . . . . . . . . . . . . . . . . . . . . . 23 (𝐶 = (𝑝‘2) ↔ (𝑝‘2) = 𝐶)
8683, 84, 853anbi123i 1154 . . . . . . . . . . . . . . . . . . . . . 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 29650 . . . . . . . . . . . . . . . . . . . . 21 (𝑓(Walks‘𝐺)𝑝𝑝 ∈ Word 𝑉)
9280eqeq2d 2746 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((♯‘𝑓) = 2 → ((♯‘𝑝) = ((♯‘𝑓) + 1) ↔ (♯‘𝑝) = 3))
9392adantl 481 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑝 ∈ Word 𝑉 ∧ (♯‘𝑓) = 2) → ((♯‘𝑝) = ((♯‘𝑓) + 1) ↔ (♯‘𝑝) = 3))
94 simp1 1135 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑝 ∈ Word 𝑉 ∧ (♯‘𝑝) = 3 ∧ (𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2))) → 𝑝 ∈ Word 𝑉)
95 oveq2 7439 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((♯‘𝑝) = 3 → (0..^(♯‘𝑝)) = (0..^3))
96 fzo0to3tp 13788 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (0..^3) = {0, 1, 2}
9795, 96eqtrdi 2791 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((♯‘𝑝) = 3 → (0..^(♯‘𝑝)) = {0, 1, 2})
9831tpid1 4773 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 0 ∈ {0, 1, 2}
99 eleq2 2828 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 14562 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 4775 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 1 ∈ {0, 1, 2}
104 eleq2 2828 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 14562 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 12341 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 2 ∈ V
109108tpid3 4778 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 2 ∈ {0, 1, 2}
110 eleq2 2828 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 14562 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 1127 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 1131 . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 2827 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝐴 = (𝑝‘0) → (𝐴𝑉 ↔ (𝑝‘0) ∈ 𝑉))
1181173ad2ant1 1132 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)) → (𝐴𝑉 ↔ (𝑝‘0) ∈ 𝑉))
119 eleq1 2827 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝐵 = (𝑝‘1) → (𝐵𝑉 ↔ (𝑝‘1) ∈ 𝑉))
1201193ad2ant2 1133 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)) → (𝐵𝑉 ↔ (𝑝‘1) ∈ 𝑉))
121 eleq1 2827 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝐶 = (𝑝‘2) → (𝐶𝑉 ↔ (𝑝‘2) ∈ 𝑉))
1221213ad2ant3 1134 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)) → (𝐶𝑉 ↔ (𝑝‘2) ∈ 𝑉))
123118, 120, 1223anbi123d 1435 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)) → ((𝐴𝑉𝐵𝑉𝐶𝑉) ↔ ((𝑝‘0) ∈ 𝑉 ∧ (𝑝‘1) ∈ 𝑉 ∧ (𝑝‘2) ∈ 𝑉)))
1241233ad2ant3 1134 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝑝 ∈ Word 𝑉 ∧ (♯‘𝑝) = 3 ∧ (𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2))) → ((𝐴𝑉𝐵𝑉𝐶𝑉) ↔ ((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))) → (𝐴𝑉𝐵𝑉𝐶𝑉))
12694, 125jca 511 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑝 ∈ Word 𝑉 ∧ (♯‘𝑝) = 3 ∧ (𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2))) → (𝑝 ∈ Word 𝑉 ∧ (𝐴𝑉𝐵𝑉𝐶𝑉)))
1271263exp 1118 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑝 ∈ Word 𝑉 → ((♯‘𝑝) = 3 → ((𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)) → (𝑝 ∈ Word 𝑉 ∧ (𝐴𝑉𝐵𝑉𝐶𝑉)))))
128127adantr 480 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑝 ∈ Word 𝑉 ∧ (♯‘𝑓) = 2) → ((♯‘𝑝) = 3 → ((𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)) → (𝑝 ∈ Word 𝑉 ∧ (𝐴𝑉𝐵𝑉𝐶𝑉)))))
12993, 128sylbid 240 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑝 ∈ Word 𝑉 ∧ (♯‘𝑓) = 2) → ((♯‘𝑝) = ((♯‘𝑓) + 1) → ((𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)) → (𝑝 ∈ Word 𝑉 ∧ (𝐴𝑉𝐵𝑉𝐶𝑉)))))
130129impancom 451 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑝 ∈ Word 𝑉 ∧ (♯‘𝑝) = ((♯‘𝑓) + 1)) → ((♯‘𝑓) = 2 → ((𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)) → (𝑝 ∈ Word 𝑉 ∧ (𝐴𝑉𝐵𝑉𝐶𝑉)))))
131130impd 410 . . . . . . . . . . . . . . . . . . . . 21 ((𝑝 ∈ Word 𝑉 ∧ (♯‘𝑝) = ((♯‘𝑓) + 1)) → (((♯‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2))) → (𝑝 ∈ Word 𝑉 ∧ (𝐴𝑉𝐵𝑉𝐶𝑉))))
13291, 77, 131syl2anc 584 . . . . . . . . . . . . . . . . . . . 20 (𝑓(Walks‘𝐺)𝑝 → (((♯‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2))) → (𝑝 ∈ Word 𝑉 ∧ (𝐴𝑉𝐵𝑉𝐶𝑉))))
133132imp 406 . . . . . . . . . . . . . . . . . . 19 ((𝑓(Walks‘𝐺)𝑝 ∧ ((♯‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)))) → (𝑝 ∈ Word 𝑉 ∧ (𝐴𝑉𝐵𝑉𝐶𝑉)))
134 eqwrds3 14997 . . . . . . . . . . . . . . . . . . 19 ((𝑝 ∈ Word 𝑉 ∧ (𝐴𝑉𝐵𝑉𝐶𝑉)) → (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 5160 . . . . . . . . . . . . . . . 16 ((𝑓(Walks‘𝐺)𝑝 ∧ ((♯‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)))) → (𝑓(Walks‘𝐺)𝑝𝑓(Walks‘𝐺)⟨“𝐴𝐵𝐶”⟩))
138137biimpd 229 . . . . . . . . . . . . . . 15 ((𝑓(Walks‘𝐺)𝑝 ∧ ((♯‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)))) → (𝑓(Walks‘𝐺)𝑝𝑓(Walks‘𝐺)⟨“𝐴𝐵𝐶”⟩))
139138ex 412 . . . . . . . . . . . . . 14 (𝑓(Walks‘𝐺)𝑝 → (((♯‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2))) → (𝑓(Walks‘𝐺)𝑝𝑓(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 1115 . . . . . . . . . . . 12 ((𝑓(Walks‘𝐺)𝑝 ∧ (♯‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2))) → 𝑓(Walks‘𝐺)⟨“𝐴𝐵𝐶”⟩)
142141adantl 481 . . . . . . . . . . 11 (((𝐴𝑉𝐵𝑉𝐶𝑉) ∧ (𝑓(Walks‘𝐺)𝑝 ∧ (♯‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)))) → 𝑓(Walks‘𝐺)⟨“𝐴𝐵𝐶”⟩)
143 simpr2 1194 . . . . . . . . . . 11 (((𝐴𝑉𝐵𝑉𝐶𝑉) ∧ (𝑓(Walks‘𝐺)𝑝 ∧ (♯‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)))) → (♯‘𝑓) = 2)
144142, 143jca 511 . . . . . . . . . 10 (((𝐴𝑉𝐵𝑉𝐶𝑉) ∧ (𝑓(Walks‘𝐺)𝑝 ∧ (♯‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)))) → (𝑓(Walks‘𝐺)⟨“𝐴𝐵𝐶”⟩ ∧ (♯‘𝑓) = 2))
145144ex 412 . . . . . . . . 9 ((𝐴𝑉𝐵𝑉𝐶𝑉) → ((𝑓(Walks‘𝐺)𝑝 ∧ (♯‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2))) → (𝑓(Walks‘𝐺)⟨“𝐴𝐵𝐶”⟩ ∧ (♯‘𝑓) = 2)))
146145exlimdv 1931 . . . . . . . 8 ((𝐴𝑉𝐵𝑉𝐶𝑉) → (∃𝑝(𝑓(Walks‘𝐺)𝑝 ∧ (♯‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2))) → (𝑓(Walks‘𝐺)⟨“𝐴𝐵𝐶”⟩ ∧ (♯‘𝑓) = 2)))
147146eximdv 1915 . . . . . . 7 ((𝐴𝑉𝐵𝑉𝐶𝑉) → (∃𝑓𝑝(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) → ((𝐴𝑉𝐵𝑉𝐶𝑉) → ∃𝑓(𝑓(Walks‘𝐺)⟨“𝐴𝐵𝐶”⟩ ∧ (♯‘𝑓) = 2)))
1491483expib 1121 . . . . 5 (𝐺 ∈ UMGraph → (({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸) → ((𝐴𝑉𝐵𝑉𝐶𝑉) → ∃𝑓(𝑓(Walks‘𝐺)⟨“𝐴𝐵𝐶”⟩ ∧ (♯‘𝑓) = 2))))
150149com23 86 . . . 4 (𝐺 ∈ UMGraph → ((𝐴𝑉𝐵𝑉𝐶𝑉) → (({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸) → ∃𝑓(𝑓(Walks‘𝐺)⟨“𝐴𝐵𝐶”⟩ ∧ (♯‘𝑓) = 2))))
151150imp 406 . . 3 ((𝐺 ∈ UMGraph ∧ (𝐴𝑉𝐵𝑉𝐶𝑉)) → (({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸) → ∃𝑓(𝑓(Walks‘𝐺)⟨“𝐴𝐵𝐶”⟩ ∧ (♯‘𝑓) = 2)))
15275, 151impbid 212 . 2 ((𝐺 ∈ UMGraph ∧ (𝐴𝑉𝐵𝑉𝐶𝑉)) → (∃𝑓(𝑓(Walks‘𝐺)⟨“𝐴𝐵𝐶”⟩ ∧ (♯‘𝑓) = 2) ↔ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸)))
1538, 152bitrd 279 1 ((𝐺 ∈ UMGraph ∧ (𝐴𝑉𝐵𝑉𝐶𝑉)) → (⟨“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 1537  wex 1776  wcel 2106  {cpr 4633  {ctp 4635   class class class wbr 5148  dom cdm 5689  ran crn 5690  Fun wfun 6557  wf 6559  cfv 6563  (class class class)co 7431  0cc0 11153  1c1 11154   + caddc 11156  2c2 12319  3c3 12320  ...cfz 13544  ..^cfzo 13691  chash 14366  Word cword 14549  ⟨“cs3 14878  Vtxcvtx 29028  iEdgciedg 29029  Edgcedg 29079  UHGraphcuhgr 29088  UPGraphcupgr 29112  UMGraphcumgr 29113  Walkscwlks 29629   WWalksNOn cwwlksnon 29857
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-ac2 10501  ax-cnex 11209  ax-resscn 11210  ax-1cn 11211  ax-icn 11212  ax-addcl 11213  ax-addrcl 11214  ax-mulcl 11215  ax-mulrcl 11216  ax-mulcom 11217  ax-addass 11218  ax-mulass 11219  ax-distr 11220  ax-i2m1 11221  ax-1ne0 11222  ax-1rid 11223  ax-rnegex 11224  ax-rrecex 11225  ax-cnre 11226  ax-pre-lttri 11227  ax-pre-lttrn 11228  ax-pre-ltadd 11229  ax-pre-mulgt0 11230
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 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-tp 4636  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-se 5642  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-isom 6572  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8013  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-1o 8505  df-2o 8506  df-oadd 8509  df-er 8744  df-map 8867  df-pm 8868  df-en 8985  df-dom 8986  df-sdom 8987  df-fin 8988  df-dju 9939  df-card 9977  df-ac 10154  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298  df-le 11299  df-sub 11492  df-neg 11493  df-nn 12265  df-2 12327  df-3 12328  df-n0 12525  df-xnn0 12598  df-z 12612  df-uz 12877  df-fz 13545  df-fzo 13692  df-hash 14367  df-word 14550  df-concat 14606  df-s1 14631  df-s2 14884  df-s3 14885  df-edg 29080  df-uhgr 29090  df-upgr 29114  df-umgr 29115  df-wlks 29632  df-wwlks 29860  df-wwlksn 29861  df-wwlksnon 29862
This theorem is referenced by:  wwlks2onsym  29988  usgr2wspthons3  29994  frgr2wwlkeu  30356
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