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Theorem umgrwwlks2on 27120
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 26234 . . . 4 (𝐺 ∈ UMGraph → 𝐺 ∈ UPGraph)
21adantr 468 . . 3 ((𝐺 ∈ UMGraph ∧ (𝐴𝑉𝐵𝑉𝐶𝑉)) → 𝐺 ∈ UPGraph)
3 simp1 1159 . . . 4 ((𝐴𝑉𝐵𝑉𝐶𝑉) → 𝐴𝑉)
43adantl 469 . . 3 ((𝐺 ∈ UMGraph ∧ (𝐴𝑉𝐵𝑉𝐶𝑉)) → 𝐴𝑉)
5 simpr3 1245 . . 3 ((𝐺 ∈ UMGraph ∧ (𝐴𝑉𝐵𝑉𝐶𝑉)) → 𝐶𝑉)
6 s3wwlks2on.v . . . 4 𝑉 = (Vtx‘𝐺)
76s3wwlks2on 27119 . . 3 ((𝐺 ∈ UPGraph ∧ 𝐴𝑉𝐶𝑉) → (⟨“𝐴𝐵𝐶”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐶) ↔ ∃𝑓(𝑓(Walks‘𝐺)⟨“𝐴𝐵𝐶”⟩ ∧ (♯‘𝑓) = 2)))
82, 4, 5, 7syl3anc 1483 . 2 ((𝐺 ∈ UMGraph ∧ (𝐴𝑉𝐵𝑉𝐶𝑉)) → (⟨“𝐴𝐵𝐶”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐶) ↔ ∃𝑓(𝑓(Walks‘𝐺)⟨“𝐴𝐵𝐶”⟩ ∧ (♯‘𝑓) = 2)))
9 eqid 2817 . . . . . . . 8 (iEdg‘𝐺) = (iEdg‘𝐺)
106, 9upgr2wlk 26814 . . . . . . 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 468 . . . . 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 13879 . . . . . . . . . . . 12 (𝐴𝑉 → (⟨“𝐴𝐵𝐶”⟩‘0) = 𝐴)
14133ad2ant1 1156 . . . . . . . . . . 11 ((𝐴𝑉𝐵𝑉𝐶𝑉) → (⟨“𝐴𝐵𝐶”⟩‘0) = 𝐴)
15 s3fv1 13880 . . . . . . . . . . . 12 (𝐵𝑉 → (⟨“𝐴𝐵𝐶”⟩‘1) = 𝐵)
16153ad2ant2 1157 . . . . . . . . . . 11 ((𝐴𝑉𝐵𝑉𝐶𝑉) → (⟨“𝐴𝐵𝐶”⟩‘1) = 𝐵)
1714, 16preq12d 4478 . . . . . . . . . 10 ((𝐴𝑉𝐵𝑉𝐶𝑉) → {(⟨“𝐴𝐵𝐶”⟩‘0), (⟨“𝐴𝐵𝐶”⟩‘1)} = {𝐴, 𝐵})
1817eqeq2d 2827 . . . . . . . . 9 ((𝐴𝑉𝐵𝑉𝐶𝑉) → (((iEdg‘𝐺)‘(𝑓‘0)) = {(⟨“𝐴𝐵𝐶”⟩‘0), (⟨“𝐴𝐵𝐶”⟩‘1)} ↔ ((iEdg‘𝐺)‘(𝑓‘0)) = {𝐴, 𝐵}))
19 s3fv2 13881 . . . . . . . . . . . 12 (𝐶𝑉 → (⟨“𝐴𝐵𝐶”⟩‘2) = 𝐶)
20193ad2ant3 1158 . . . . . . . . . . 11 ((𝐴𝑉𝐵𝑉𝐶𝑉) → (⟨“𝐴𝐵𝐶”⟩‘2) = 𝐶)
2116, 20preq12d 4478 . . . . . . . . . 10 ((𝐴𝑉𝐵𝑉𝐶𝑉) → {(⟨“𝐴𝐵𝐶”⟩‘1), (⟨“𝐴𝐵𝐶”⟩‘2)} = {𝐵, 𝐶})
2221eqeq2d 2827 . . . . . . . . 9 ((𝐴𝑉𝐵𝑉𝐶𝑉) → (((iEdg‘𝐺)‘(𝑓‘1)) = {(⟨“𝐴𝐵𝐶”⟩‘1), (⟨“𝐴𝐵𝐶”⟩‘2)} ↔ ((iEdg‘𝐺)‘(𝑓‘1)) = {𝐵, 𝐶}))
2318, 22anbi12d 618 . . . . . . . 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 469 . . . . . . 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 1559 . . . . . 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 26235 . . . . . . . . . . 11 (𝐺 ∈ UMGraph → 𝐺 ∈ UHGraph)
279uhgrfun 26197 . . . . . . . . . . 11 (𝐺 ∈ UHGraph → Fun (iEdg‘𝐺))
28 fdmrn 6288 . . . . . . . . . . . 12 (Fun (iEdg‘𝐺) ↔ (iEdg‘𝐺):dom (iEdg‘𝐺)⟶ran (iEdg‘𝐺))
29 simpr 473 . . . . . . . . . . . . . . . . 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 10328 . . . . . . . . . . . . . . . . . . . . . 22 0 ∈ V
3231prid1 4499 . . . . . . . . . . . . . . . . . . . . 21 0 ∈ {0, 1}
33 fzo0to2pr 12796 . . . . . . . . . . . . . . . . . . . . 21 (0..^2) = {0, 1}
3432, 33eleqtrri 2895 . . . . . . . . . . . . . . . . . . . 20 0 ∈ (0..^2)
3534a1i 11 . . . . . . . . . . . . . . . . . . 19 (𝑓:(0..^2)⟶dom (iEdg‘𝐺) → 0 ∈ (0..^2))
3630, 35ffvelrnd 6591 . . . . . . . . . . . . . . . . . 18 (𝑓:(0..^2)⟶dom (iEdg‘𝐺) → (𝑓‘0) ∈ dom (iEdg‘𝐺))
3736adantr 468 . . . . . . . . . . . . . . . . 17 ((𝑓:(0..^2)⟶dom (iEdg‘𝐺) ∧ (iEdg‘𝐺):dom (iEdg‘𝐺)⟶ran (iEdg‘𝐺)) → (𝑓‘0) ∈ dom (iEdg‘𝐺))
3829, 37ffvelrnd 6591 . . . . . . . . . . . . . . . 16 ((𝑓:(0..^2)⟶dom (iEdg‘𝐺) ∧ (iEdg‘𝐺):dom (iEdg‘𝐺)⟶ran (iEdg‘𝐺)) → ((iEdg‘𝐺)‘(𝑓‘0)) ∈ ran (iEdg‘𝐺))
39 1ex 10330 . . . . . . . . . . . . . . . . . . . . . 22 1 ∈ V
4039prid2 4500 . . . . . . . . . . . . . . . . . . . . 21 1 ∈ {0, 1}
4140, 33eleqtrri 2895 . . . . . . . . . . . . . . . . . . . 20 1 ∈ (0..^2)
4241a1i 11 . . . . . . . . . . . . . . . . . . 19 (𝑓:(0..^2)⟶dom (iEdg‘𝐺) → 1 ∈ (0..^2))
4330, 42ffvelrnd 6591 . . . . . . . . . . . . . . . . . 18 (𝑓:(0..^2)⟶dom (iEdg‘𝐺) → (𝑓‘1) ∈ dom (iEdg‘𝐺))
4443adantr 468 . . . . . . . . . . . . . . . . 17 ((𝑓:(0..^2)⟶dom (iEdg‘𝐺) ∧ (iEdg‘𝐺):dom (iEdg‘𝐺)⟶ran (iEdg‘𝐺)) → (𝑓‘1) ∈ dom (iEdg‘𝐺))
4529, 44ffvelrnd 6591 . . . . . . . . . . . . . . . 16 ((𝑓:(0..^2)⟶dom (iEdg‘𝐺) ∧ (iEdg‘𝐺):dom (iEdg‘𝐺)⟶ran (iEdg‘𝐺)) → ((iEdg‘𝐺)‘(𝑓‘1)) ∈ ran (iEdg‘𝐺))
4638, 45jca 503 . . . . . . . . . . . . . . 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 399 . . . . . . . . . . . . . 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 1156 . . . . . . . . . . . . 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 208 . . . . . . . . . . 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 395 . . . . . . . . 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 2824 . . . . . . . . . . . . . . 15 (((iEdg‘𝐺)‘(𝑓‘0)) = {𝐴, 𝐵} ↔ {𝐴, 𝐵} = ((iEdg‘𝐺)‘(𝑓‘0)))
5453biimpi 207 . . . . . . . . . . . . . 14 (((iEdg‘𝐺)‘(𝑓‘0)) = {𝐴, 𝐵} → {𝐴, 𝐵} = ((iEdg‘𝐺)‘(𝑓‘0)))
5554adantr 468 . . . . . . . . . . . . 13 ((((iEdg‘𝐺)‘(𝑓‘0)) = {𝐴, 𝐵} ∧ ((iEdg‘𝐺)‘(𝑓‘1)) = {𝐵, 𝐶}) → {𝐴, 𝐵} = ((iEdg‘𝐺)‘(𝑓‘0)))
56553ad2ant3 1158 . . . . . . . . . . . 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 469 . . . . . . . . . . 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 26177 . . . . . . . . . . . . 13 (Edg‘𝐺) = ran (iEdg‘𝐺)
6058, 59eqtri 2839 . . . . . . . . . . . 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 2890 . . . . . . . . . 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 2824 . . . . . . . . . . . . . . 15 (((iEdg‘𝐺)‘(𝑓‘1)) = {𝐵, 𝐶} ↔ {𝐵, 𝐶} = ((iEdg‘𝐺)‘(𝑓‘1)))
6463biimpi 207 . . . . . . . . . . . . . 14 (((iEdg‘𝐺)‘(𝑓‘1)) = {𝐵, 𝐶} → {𝐵, 𝐶} = ((iEdg‘𝐺)‘(𝑓‘1)))
6564adantl 469 . . . . . . . . . . . . 13 ((((iEdg‘𝐺)‘(𝑓‘0)) = {𝐴, 𝐵} ∧ ((iEdg‘𝐺)‘(𝑓‘1)) = {𝐵, 𝐶}) → {𝐵, 𝐶} = ((iEdg‘𝐺)‘(𝑓‘1)))
66653ad2ant3 1158 . . . . . . . . . . . 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 469 . . . . . . . . . . 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 2890 . . . . . . . . . 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 618 . . . . . . . . 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 248 . . . . . . . 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 399 . . . . . . 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 468 . . . . . 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 231 . . . . 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 231 . . . 4 ((𝐺 ∈ UMGraph ∧ (𝐴𝑉𝐵𝑉𝐶𝑉)) → ((𝑓(Walks‘𝐺)⟨“𝐴𝐵𝐶”⟩ ∧ (♯‘𝑓) = 2) → ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸)))
7574exlimdv 2024 . . 3 ((𝐺 ∈ UMGraph ∧ (𝐴𝑉𝐵𝑉𝐶𝑉)) → (∃𝑓(𝑓(Walks‘𝐺)⟨“𝐴𝐵𝐶”⟩ ∧ (♯‘𝑓) = 2) → ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸)))
7658umgr2wlk 27111 . . . . . . 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 26764 . . . . . . . . . . . . . . . . . . . 20 (𝑓(Walks‘𝐺)𝑝 → (♯‘𝑝) = ((♯‘𝑓) + 1))
78 oveq1 6890 . . . . . . . . . . . . . . . . . . . . . 22 ((♯‘𝑓) = 2 → ((♯‘𝑓) + 1) = (2 + 1))
79 2p1e3 11461 . . . . . . . . . . . . . . . . . . . . . 22 (2 + 1) = 3
8078, 79syl6eq 2867 . . . . . . . . . . . . . . . . . . . . 21 ((♯‘𝑓) = 2 → ((♯‘𝑓) + 1) = 3)
8180adantr 468 . . . . . . . . . . . . . . . . . . . 20 (((♯‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2))) → ((♯‘𝑓) + 1) = 3)
8277, 81sylan9eq 2871 . . . . . . . . . . . . . . . . . . 19 ((𝑓(Walks‘𝐺)𝑝 ∧ ((♯‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)))) → (♯‘𝑝) = 3)
83 eqcom 2824 . . . . . . . . . . . . . . . . . . . . . . 23 (𝐴 = (𝑝‘0) ↔ (𝑝‘0) = 𝐴)
84 eqcom 2824 . . . . . . . . . . . . . . . . . . . . . . 23 (𝐵 = (𝑝‘1) ↔ (𝑝‘1) = 𝐵)
85 eqcom 2824 . . . . . . . . . . . . . . . . . . . . . . 23 (𝐶 = (𝑝‘2) ↔ (𝑝‘2) = 𝐶)
8683, 84, 853anbi123i 1187 . . . . . . . . . . . . . . . . . . . . . 22 ((𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)) ↔ ((𝑝‘0) = 𝐴 ∧ (𝑝‘1) = 𝐵 ∧ (𝑝‘2) = 𝐶))
8786biimpi 207 . . . . . . . . . . . . . . . . . . . . 21 ((𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)) → ((𝑝‘0) = 𝐴 ∧ (𝑝‘1) = 𝐵 ∧ (𝑝‘2) = 𝐶))
8887adantl 469 . . . . . . . . . . . . . . . . . . . 20 (((♯‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2))) → ((𝑝‘0) = 𝐴 ∧ (𝑝‘1) = 𝐵 ∧ (𝑝‘2) = 𝐶))
8988adantl 469 . . . . . . . . . . . . . . . . . . 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 503 . . . . . . . . . . . . . . . . . 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 26763 . . . . . . . . . . . . . . . . . . . . 21 (𝑓(Walks‘𝐺)𝑝𝑝 ∈ Word 𝑉)
9280eqeq2d 2827 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((♯‘𝑓) = 2 → ((♯‘𝑝) = ((♯‘𝑓) + 1) ↔ (♯‘𝑝) = 3))
9392adantl 469 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑝 ∈ Word 𝑉 ∧ (♯‘𝑓) = 2) → ((♯‘𝑝) = ((♯‘𝑓) + 1) ↔ (♯‘𝑝) = 3))
94 simp1 1159 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑝 ∈ Word 𝑉 ∧ (♯‘𝑝) = 3 ∧ (𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2))) → 𝑝 ∈ Word 𝑉)
95 oveq2 6891 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((♯‘𝑝) = 3 → (0..^(♯‘𝑝)) = (0..^3))
96 fzo0to3tp 12797 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (0..^3) = {0, 1, 2}
9795, 96syl6eq 2867 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((♯‘𝑝) = 3 → (0..^(♯‘𝑝)) = {0, 1, 2})
9831tpid1 4505 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 0 ∈ {0, 1, 2}
99 eleq2 2885 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((0..^(♯‘𝑝)) = {0, 1, 2} → (0 ∈ (0..^(♯‘𝑝)) ↔ 0 ∈ {0, 1, 2}))
10098, 99mpbiri 249 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((0..^(♯‘𝑝)) = {0, 1, 2} → 0 ∈ (0..^(♯‘𝑝)))
101 wrdsymbcl 13548 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((𝑝 ∈ Word 𝑉 ∧ 0 ∈ (0..^(♯‘𝑝))) → (𝑝‘0) ∈ 𝑉)
102100, 101sylan2 582 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((𝑝 ∈ Word 𝑉 ∧ (0..^(♯‘𝑝)) = {0, 1, 2}) → (𝑝‘0) ∈ 𝑉)
10339tpid2 4506 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 1 ∈ {0, 1, 2}
104 eleq2 2885 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((0..^(♯‘𝑝)) = {0, 1, 2} → (1 ∈ (0..^(♯‘𝑝)) ↔ 1 ∈ {0, 1, 2}))
105103, 104mpbiri 249 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((0..^(♯‘𝑝)) = {0, 1, 2} → 1 ∈ (0..^(♯‘𝑝)))
106 wrdsymbcl 13548 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((𝑝 ∈ Word 𝑉 ∧ 1 ∈ (0..^(♯‘𝑝))) → (𝑝‘1) ∈ 𝑉)
107105, 106sylan2 582 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((𝑝 ∈ Word 𝑉 ∧ (0..^(♯‘𝑝)) = {0, 1, 2}) → (𝑝‘1) ∈ 𝑉)
108 2ex 11389 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 2 ∈ V
109108tpid3 4508 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 2 ∈ {0, 1, 2}
110 eleq2 2885 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((0..^(♯‘𝑝)) = {0, 1, 2} → (2 ∈ (0..^(♯‘𝑝)) ↔ 2 ∈ {0, 1, 2}))
111109, 110mpbiri 249 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((0..^(♯‘𝑝)) = {0, 1, 2} → 2 ∈ (0..^(♯‘𝑝)))
112 wrdsymbcl 13548 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((𝑝 ∈ Word 𝑉 ∧ 2 ∈ (0..^(♯‘𝑝))) → (𝑝‘2) ∈ 𝑉)
113111, 112sylan2 582 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((𝑝 ∈ Word 𝑉 ∧ (0..^(♯‘𝑝)) = {0, 1, 2}) → (𝑝‘2) ∈ 𝑉)
114102, 107, 1133jca 1151 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((𝑝 ∈ Word 𝑉 ∧ (0..^(♯‘𝑝)) = {0, 1, 2}) → ((𝑝‘0) ∈ 𝑉 ∧ (𝑝‘1) ∈ 𝑉 ∧ (𝑝‘2) ∈ 𝑉))
11597, 114sylan2 582 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝑝 ∈ Word 𝑉 ∧ (♯‘𝑝) = 3) → ((𝑝‘0) ∈ 𝑉 ∧ (𝑝‘1) ∈ 𝑉 ∧ (𝑝‘2) ∈ 𝑉))
1161153adant3 1155 . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 2884 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝐴 = (𝑝‘0) → (𝐴𝑉 ↔ (𝑝‘0) ∈ 𝑉))
1181173ad2ant1 1156 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)) → (𝐴𝑉 ↔ (𝑝‘0) ∈ 𝑉))
119 eleq1 2884 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝐵 = (𝑝‘1) → (𝐵𝑉 ↔ (𝑝‘1) ∈ 𝑉))
1201193ad2ant2 1157 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)) → (𝐵𝑉 ↔ (𝑝‘1) ∈ 𝑉))
121 eleq1 2884 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝐶 = (𝑝‘2) → (𝐶𝑉 ↔ (𝑝‘2) ∈ 𝑉))
1221213ad2ant3 1158 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)) → (𝐶𝑉 ↔ (𝑝‘2) ∈ 𝑉))
123118, 120, 1223anbi123d 1553 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)) → ((𝐴𝑉𝐵𝑉𝐶𝑉) ↔ ((𝑝‘0) ∈ 𝑉 ∧ (𝑝‘1) ∈ 𝑉 ∧ (𝑝‘2) ∈ 𝑉)))
1241233ad2ant3 1158 . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 248 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑝 ∈ Word 𝑉 ∧ (♯‘𝑝) = 3 ∧ (𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2))) → (𝐴𝑉𝐵𝑉𝐶𝑉))
12694, 125jca 503 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑝 ∈ Word 𝑉 ∧ (♯‘𝑝) = 3 ∧ (𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2))) → (𝑝 ∈ Word 𝑉 ∧ (𝐴𝑉𝐵𝑉𝐶𝑉)))
1271263exp 1141 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑝 ∈ Word 𝑉 → ((♯‘𝑝) = 3 → ((𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)) → (𝑝 ∈ Word 𝑉 ∧ (𝐴𝑉𝐵𝑉𝐶𝑉)))))
128127adantr 468 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑝 ∈ Word 𝑉 ∧ (♯‘𝑓) = 2) → ((♯‘𝑝) = 3 → ((𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)) → (𝑝 ∈ Word 𝑉 ∧ (𝐴𝑉𝐵𝑉𝐶𝑉)))))
12993, 128sylbid 231 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑝 ∈ Word 𝑉 ∧ (♯‘𝑓) = 2) → ((♯‘𝑝) = ((♯‘𝑓) + 1) → ((𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)) → (𝑝 ∈ Word 𝑉 ∧ (𝐴𝑉𝐵𝑉𝐶𝑉)))))
130129impancom 441 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑝 ∈ Word 𝑉 ∧ (♯‘𝑝) = ((♯‘𝑓) + 1)) → ((♯‘𝑓) = 2 → ((𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)) → (𝑝 ∈ Word 𝑉 ∧ (𝐴𝑉𝐵𝑉𝐶𝑉)))))
131130impd 398 . . . . . . . . . . . . . . . . . . . . 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 575 . . . . . . . . . . . . . . . . . . . 20 (𝑓(Walks‘𝐺)𝑝 → (((♯‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2))) → (𝑝 ∈ Word 𝑉 ∧ (𝐴𝑉𝐵𝑉𝐶𝑉))))
133132imp 395 . . . . . . . . . . . . . . . . . . 19 ((𝑓(Walks‘𝐺)𝑝 ∧ ((♯‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)))) → (𝑝 ∈ Word 𝑉 ∧ (𝐴𝑉𝐵𝑉𝐶𝑉)))
134 eqwrds3 13948 . . . . . . . . . . . . . . . . . . 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 248 . . . . . . . . . . . . . . . . 17 ((𝑓(Walks‘𝐺)𝑝 ∧ ((♯‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)))) → 𝑝 = ⟨“𝐴𝐵𝐶”⟩)
137136breq2d 4867 . . . . . . . . . . . . . . . 16 ((𝑓(Walks‘𝐺)𝑝 ∧ ((♯‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)))) → (𝑓(Walks‘𝐺)𝑝𝑓(Walks‘𝐺)⟨“𝐴𝐵𝐶”⟩))
138137biimpd 220 . . . . . . . . . . . . . . 15 ((𝑓(Walks‘𝐺)𝑝 ∧ ((♯‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)))) → (𝑓(Walks‘𝐺)𝑝𝑓(Walks‘𝐺)⟨“𝐴𝐵𝐶”⟩))
139138ex 399 . . . . . . . . . . . . . 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 1137 . . . . . . . . . . . 12 ((𝑓(Walks‘𝐺)𝑝 ∧ (♯‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2))) → 𝑓(Walks‘𝐺)⟨“𝐴𝐵𝐶”⟩)
142141adantl 469 . . . . . . . . . . 11 (((𝐴𝑉𝐵𝑉𝐶𝑉) ∧ (𝑓(Walks‘𝐺)𝑝 ∧ (♯‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)))) → 𝑓(Walks‘𝐺)⟨“𝐴𝐵𝐶”⟩)
143 simpr2 1243 . . . . . . . . . . 11 (((𝐴𝑉𝐵𝑉𝐶𝑉) ∧ (𝑓(Walks‘𝐺)𝑝 ∧ (♯‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)))) → (♯‘𝑓) = 2)
144142, 143jca 503 . . . . . . . . . 10 (((𝐴𝑉𝐵𝑉𝐶𝑉) ∧ (𝑓(Walks‘𝐺)𝑝 ∧ (♯‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)))) → (𝑓(Walks‘𝐺)⟨“𝐴𝐵𝐶”⟩ ∧ (♯‘𝑓) = 2))
145144ex 399 . . . . . . . . 9 ((𝐴𝑉𝐵𝑉𝐶𝑉) → ((𝑓(Walks‘𝐺)𝑝 ∧ (♯‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2))) → (𝑓(Walks‘𝐺)⟨“𝐴𝐵𝐶”⟩ ∧ (♯‘𝑓) = 2)))
146145exlimdv 2024 . . . . . . . 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 2008 . . . . . . 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 1145 . . . . 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 395 . . 3 ((𝐺 ∈ UMGraph ∧ (𝐴𝑉𝐵𝑉𝐶𝑉)) → (({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸) → ∃𝑓(𝑓(Walks‘𝐺)⟨“𝐴𝐵𝐶”⟩ ∧ (♯‘𝑓) = 2)))
15275, 151impbid 203 . 2 ((𝐺 ∈ UMGraph ∧ (𝐴𝑉𝐵𝑉𝐶𝑉)) → (∃𝑓(𝑓(Walks‘𝐺)⟨“𝐴𝐵𝐶”⟩ ∧ (♯‘𝑓) = 2) ↔ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸)))
1538, 152bitrd 270 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 197  wa 384  w3a 1100   = wceq 1637  wex 1859  wcel 2157  {cpr 4383  {ctp 4385   class class class wbr 4855  dom cdm 5324  ran crn 5325  Fun wfun 6104  wf 6106  cfv 6110  (class class class)co 6883  0cc0 10230  1c1 10231   + caddc 10233  2c2 11367  3c3 11368  ...cfz 12568  ..^cfzo 12708  chash 13356  Word cword 13521  ⟨“cs3 13830  Vtxcvtx 26110  iEdgciedg 26111  Edgcedg 26175  UHGraphcuhgr 26187  UPGraphcupgr 26211  UMGraphcumgr 26212  Walkscwlks 26742   WWalksNOn cwwlksnon 26970
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2069  ax-7 2105  ax-8 2159  ax-9 2166  ax-10 2186  ax-11 2202  ax-12 2215  ax-13 2422  ax-ext 2795  ax-rep 4977  ax-sep 4988  ax-nul 4996  ax-pow 5048  ax-pr 5109  ax-un 7188  ax-ac2 9579  ax-cnex 10286  ax-resscn 10287  ax-1cn 10288  ax-icn 10289  ax-addcl 10290  ax-addrcl 10291  ax-mulcl 10292  ax-mulrcl 10293  ax-mulcom 10294  ax-addass 10295  ax-mulass 10296  ax-distr 10297  ax-i2m1 10298  ax-1ne0 10299  ax-1rid 10300  ax-rnegex 10301  ax-rrecex 10302  ax-cnre 10303  ax-pre-lttri 10304  ax-pre-lttrn 10305  ax-pre-ltadd 10306  ax-pre-mulgt0 10307
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-ifp 1079  df-3or 1101  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2062  df-mo 2635  df-eu 2642  df-clab 2804  df-cleq 2810  df-clel 2813  df-nfc 2948  df-ne 2990  df-nel 3093  df-ral 3112  df-rex 3113  df-reu 3114  df-rmo 3115  df-rab 3116  df-v 3404  df-sbc 3645  df-csb 3740  df-dif 3783  df-un 3785  df-in 3787  df-ss 3794  df-pss 3796  df-nul 4128  df-if 4291  df-pw 4364  df-sn 4382  df-pr 4384  df-tp 4386  df-op 4388  df-uni 4642  df-int 4681  df-iun 4725  df-br 4856  df-opab 4918  df-mpt 4935  df-tr 4958  df-id 5232  df-eprel 5237  df-po 5245  df-so 5246  df-fr 5283  df-se 5284  df-we 5285  df-xp 5330  df-rel 5331  df-cnv 5332  df-co 5333  df-dm 5334  df-rn 5335  df-res 5336  df-ima 5337  df-pred 5906  df-ord 5952  df-on 5953  df-lim 5954  df-suc 5955  df-iota 6073  df-fun 6112  df-fn 6113  df-f 6114  df-f1 6115  df-fo 6116  df-f1o 6117  df-fv 6118  df-isom 6119  df-riota 6844  df-ov 6886  df-oprab 6887  df-mpt2 6888  df-om 7305  df-1st 7407  df-2nd 7408  df-wrecs 7651  df-recs 7713  df-rdg 7751  df-1o 7805  df-2o 7806  df-oadd 7809  df-er 7988  df-map 8103  df-pm 8104  df-en 8202  df-dom 8203  df-sdom 8204  df-fin 8205  df-card 9057  df-ac 9231  df-cda 9284  df-pnf 10370  df-mnf 10371  df-xr 10372  df-ltxr 10373  df-le 10374  df-sub 10562  df-neg 10563  df-nn 11315  df-2 11375  df-3 11376  df-n0 11579  df-xnn0 11649  df-z 11663  df-uz 11924  df-fz 12569  df-fzo 12709  df-hash 13357  df-word 13529  df-concat 13531  df-s1 13532  df-s2 13836  df-s3 13837  df-edg 26176  df-uhgr 26189  df-upgr 26213  df-umgr 26214  df-wlks 26745  df-wwlks 26973  df-wwlksn 26974  df-wwlksnon 26975
This theorem is referenced by:  wwlks2onsym  27121  usgr2wspthons3  27128  frgr2wwlkeu  27524
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