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Theorem umgrwwlks2on 29742
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 = 𝐡 and/or 𝐡 = 𝐢 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 = (Vtxβ€˜πΊ)
usgrwwlks2on.e 𝐸 = (Edgβ€˜πΊ)
Assertion
Ref Expression
umgrwwlks2on ((𝐺 ∈ UMGraph ∧ (𝐴 ∈ 𝑉 ∧ 𝐡 ∈ 𝑉 ∧ 𝐢 ∈ 𝑉)) β†’ (βŸ¨β€œπ΄π΅πΆβ€βŸ© ∈ (𝐴(2 WWalksNOn 𝐺)𝐢) ↔ ({𝐴, 𝐡} ∈ 𝐸 ∧ {𝐡, 𝐢} ∈ 𝐸)))

Proof of Theorem umgrwwlks2on
Dummy variables 𝑓 𝑝 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 umgrupgr 28890 . . . 4 (𝐺 ∈ UMGraph β†’ 𝐺 ∈ UPGraph)
21adantr 480 . . 3 ((𝐺 ∈ UMGraph ∧ (𝐴 ∈ 𝑉 ∧ 𝐡 ∈ 𝑉 ∧ 𝐢 ∈ 𝑉)) β†’ 𝐺 ∈ UPGraph)
3 simp1 1134 . . . 4 ((𝐴 ∈ 𝑉 ∧ 𝐡 ∈ 𝑉 ∧ 𝐢 ∈ 𝑉) β†’ 𝐴 ∈ 𝑉)
43adantl 481 . . 3 ((𝐺 ∈ UMGraph ∧ (𝐴 ∈ 𝑉 ∧ 𝐡 ∈ 𝑉 ∧ 𝐢 ∈ 𝑉)) β†’ 𝐴 ∈ 𝑉)
5 simpr3 1194 . . 3 ((𝐺 ∈ UMGraph ∧ (𝐴 ∈ 𝑉 ∧ 𝐡 ∈ 𝑉 ∧ 𝐢 ∈ 𝑉)) β†’ 𝐢 ∈ 𝑉)
6 s3wwlks2on.v . . . 4 𝑉 = (Vtxβ€˜πΊ)
76s3wwlks2on 29741 . . 3 ((𝐺 ∈ UPGraph ∧ 𝐴 ∈ 𝑉 ∧ 𝐢 ∈ 𝑉) β†’ (βŸ¨β€œπ΄π΅πΆβ€βŸ© ∈ (𝐴(2 WWalksNOn 𝐺)𝐢) ↔ βˆƒπ‘“(𝑓(Walksβ€˜πΊ)βŸ¨β€œπ΄π΅πΆβ€βŸ© ∧ (β™―β€˜π‘“) = 2)))
82, 4, 5, 7syl3anc 1369 . 2 ((𝐺 ∈ UMGraph ∧ (𝐴 ∈ 𝑉 ∧ 𝐡 ∈ 𝑉 ∧ 𝐢 ∈ 𝑉)) β†’ (βŸ¨β€œπ΄π΅πΆβ€βŸ© ∈ (𝐴(2 WWalksNOn 𝐺)𝐢) ↔ βˆƒπ‘“(𝑓(Walksβ€˜πΊ)βŸ¨β€œπ΄π΅πΆβ€βŸ© ∧ (β™―β€˜π‘“) = 2)))
9 eqid 2727 . . . . . . . 8 (iEdgβ€˜πΊ) = (iEdgβ€˜πΊ)
106, 9upgr2wlk 29456 . . . . . . 7 (𝐺 ∈ UPGraph β†’ ((𝑓(Walksβ€˜πΊ)βŸ¨β€œπ΄π΅πΆβ€βŸ© ∧ (β™―β€˜π‘“) = 2) ↔ (𝑓:(0..^2)⟢dom (iEdgβ€˜πΊ) ∧ βŸ¨β€œπ΄π΅πΆβ€βŸ©:(0...2)βŸΆπ‘‰ ∧ (((iEdgβ€˜πΊ)β€˜(π‘“β€˜0)) = {(βŸ¨β€œπ΄π΅πΆβ€βŸ©β€˜0), (βŸ¨β€œπ΄π΅πΆβ€βŸ©β€˜1)} ∧ ((iEdgβ€˜πΊ)β€˜(π‘“β€˜1)) = {(βŸ¨β€œπ΄π΅πΆβ€βŸ©β€˜1), (βŸ¨β€œπ΄π΅πΆβ€βŸ©β€˜2)}))))
111, 10syl 17 . . . . . 6 (𝐺 ∈ UMGraph β†’ ((𝑓(Walksβ€˜πΊ)βŸ¨β€œπ΄π΅πΆβ€βŸ© ∧ (β™―β€˜π‘“) = 2) ↔ (𝑓:(0..^2)⟢dom (iEdgβ€˜πΊ) ∧ βŸ¨β€œπ΄π΅πΆβ€βŸ©:(0...2)βŸΆπ‘‰ ∧ (((iEdgβ€˜πΊ)β€˜(π‘“β€˜0)) = {(βŸ¨β€œπ΄π΅πΆβ€βŸ©β€˜0), (βŸ¨β€œπ΄π΅πΆβ€βŸ©β€˜1)} ∧ ((iEdgβ€˜πΊ)β€˜(π‘“β€˜1)) = {(βŸ¨β€œπ΄π΅πΆβ€βŸ©β€˜1), (βŸ¨β€œπ΄π΅πΆβ€βŸ©β€˜2)}))))
1211adantr 480 . . . . 5 ((𝐺 ∈ UMGraph ∧ (𝐴 ∈ 𝑉 ∧ 𝐡 ∈ 𝑉 ∧ 𝐢 ∈ 𝑉)) β†’ ((𝑓(Walksβ€˜πΊ)βŸ¨β€œπ΄π΅πΆβ€βŸ© ∧ (β™―β€˜π‘“) = 2) ↔ (𝑓:(0..^2)⟢dom (iEdgβ€˜πΊ) ∧ βŸ¨β€œπ΄π΅πΆβ€βŸ©:(0...2)βŸΆπ‘‰ ∧ (((iEdgβ€˜πΊ)β€˜(π‘“β€˜0)) = {(βŸ¨β€œπ΄π΅πΆβ€βŸ©β€˜0), (βŸ¨β€œπ΄π΅πΆβ€βŸ©β€˜1)} ∧ ((iEdgβ€˜πΊ)β€˜(π‘“β€˜1)) = {(βŸ¨β€œπ΄π΅πΆβ€βŸ©β€˜1), (βŸ¨β€œπ΄π΅πΆβ€βŸ©β€˜2)}))))
13 s3fv0 14860 . . . . . . . . . . . 12 (𝐴 ∈ 𝑉 β†’ (βŸ¨β€œπ΄π΅πΆβ€βŸ©β€˜0) = 𝐴)
14133ad2ant1 1131 . . . . . . . . . . 11 ((𝐴 ∈ 𝑉 ∧ 𝐡 ∈ 𝑉 ∧ 𝐢 ∈ 𝑉) β†’ (βŸ¨β€œπ΄π΅πΆβ€βŸ©β€˜0) = 𝐴)
15 s3fv1 14861 . . . . . . . . . . . 12 (𝐡 ∈ 𝑉 β†’ (βŸ¨β€œπ΄π΅πΆβ€βŸ©β€˜1) = 𝐡)
16153ad2ant2 1132 . . . . . . . . . . 11 ((𝐴 ∈ 𝑉 ∧ 𝐡 ∈ 𝑉 ∧ 𝐢 ∈ 𝑉) β†’ (βŸ¨β€œπ΄π΅πΆβ€βŸ©β€˜1) = 𝐡)
1714, 16preq12d 4741 . . . . . . . . . 10 ((𝐴 ∈ 𝑉 ∧ 𝐡 ∈ 𝑉 ∧ 𝐢 ∈ 𝑉) β†’ {(βŸ¨β€œπ΄π΅πΆβ€βŸ©β€˜0), (βŸ¨β€œπ΄π΅πΆβ€βŸ©β€˜1)} = {𝐴, 𝐡})
1817eqeq2d 2738 . . . . . . . . 9 ((𝐴 ∈ 𝑉 ∧ 𝐡 ∈ 𝑉 ∧ 𝐢 ∈ 𝑉) β†’ (((iEdgβ€˜πΊ)β€˜(π‘“β€˜0)) = {(βŸ¨β€œπ΄π΅πΆβ€βŸ©β€˜0), (βŸ¨β€œπ΄π΅πΆβ€βŸ©β€˜1)} ↔ ((iEdgβ€˜πΊ)β€˜(π‘“β€˜0)) = {𝐴, 𝐡}))
19 s3fv2 14862 . . . . . . . . . . . 12 (𝐢 ∈ 𝑉 β†’ (βŸ¨β€œπ΄π΅πΆβ€βŸ©β€˜2) = 𝐢)
20193ad2ant3 1133 . . . . . . . . . . 11 ((𝐴 ∈ 𝑉 ∧ 𝐡 ∈ 𝑉 ∧ 𝐢 ∈ 𝑉) β†’ (βŸ¨β€œπ΄π΅πΆβ€βŸ©β€˜2) = 𝐢)
2116, 20preq12d 4741 . . . . . . . . . 10 ((𝐴 ∈ 𝑉 ∧ 𝐡 ∈ 𝑉 ∧ 𝐢 ∈ 𝑉) β†’ {(βŸ¨β€œπ΄π΅πΆβ€βŸ©β€˜1), (βŸ¨β€œπ΄π΅πΆβ€βŸ©β€˜2)} = {𝐡, 𝐢})
2221eqeq2d 2738 . . . . . . . . 9 ((𝐴 ∈ 𝑉 ∧ 𝐡 ∈ 𝑉 ∧ 𝐢 ∈ 𝑉) β†’ (((iEdgβ€˜πΊ)β€˜(π‘“β€˜1)) = {(βŸ¨β€œπ΄π΅πΆβ€βŸ©β€˜1), (βŸ¨β€œπ΄π΅πΆβ€βŸ©β€˜2)} ↔ ((iEdgβ€˜πΊ)β€˜(π‘“β€˜1)) = {𝐡, 𝐢}))
2318, 22anbi12d 630 . . . . . . . 8 ((𝐴 ∈ 𝑉 ∧ 𝐡 ∈ 𝑉 ∧ 𝐢 ∈ 𝑉) β†’ ((((iEdgβ€˜πΊ)β€˜(π‘“β€˜0)) = {(βŸ¨β€œπ΄π΅πΆβ€βŸ©β€˜0), (βŸ¨β€œπ΄π΅πΆβ€βŸ©β€˜1)} ∧ ((iEdgβ€˜πΊ)β€˜(π‘“β€˜1)) = {(βŸ¨β€œπ΄π΅πΆβ€βŸ©β€˜1), (βŸ¨β€œπ΄π΅πΆβ€βŸ©β€˜2)}) ↔ (((iEdgβ€˜πΊ)β€˜(π‘“β€˜0)) = {𝐴, 𝐡} ∧ ((iEdgβ€˜πΊ)β€˜(π‘“β€˜1)) = {𝐡, 𝐢})))
2423adantl 481 . . . . . . 7 ((𝐺 ∈ UMGraph ∧ (𝐴 ∈ 𝑉 ∧ 𝐡 ∈ 𝑉 ∧ 𝐢 ∈ 𝑉)) β†’ ((((iEdgβ€˜πΊ)β€˜(π‘“β€˜0)) = {(βŸ¨β€œπ΄π΅πΆβ€βŸ©β€˜0), (βŸ¨β€œπ΄π΅πΆβ€βŸ©β€˜1)} ∧ ((iEdgβ€˜πΊ)β€˜(π‘“β€˜1)) = {(βŸ¨β€œπ΄π΅πΆβ€βŸ©β€˜1), (βŸ¨β€œπ΄π΅πΆβ€βŸ©β€˜2)}) ↔ (((iEdgβ€˜πΊ)β€˜(π‘“β€˜0)) = {𝐴, 𝐡} ∧ ((iEdgβ€˜πΊ)β€˜(π‘“β€˜1)) = {𝐡, 𝐢})))
25243anbi3d 1439 . . . . . 6 ((𝐺 ∈ UMGraph ∧ (𝐴 ∈ 𝑉 ∧ 𝐡 ∈ 𝑉 ∧ 𝐢 ∈ 𝑉)) β†’ ((𝑓:(0..^2)⟢dom (iEdgβ€˜πΊ) ∧ βŸ¨β€œπ΄π΅πΆβ€βŸ©:(0...2)βŸΆπ‘‰ ∧ (((iEdgβ€˜πΊ)β€˜(π‘“β€˜0)) = {(βŸ¨β€œπ΄π΅πΆβ€βŸ©β€˜0), (βŸ¨β€œπ΄π΅πΆβ€βŸ©β€˜1)} ∧ ((iEdgβ€˜πΊ)β€˜(π‘“β€˜1)) = {(βŸ¨β€œπ΄π΅πΆβ€βŸ©β€˜1), (βŸ¨β€œπ΄π΅πΆβ€βŸ©β€˜2)})) ↔ (𝑓:(0..^2)⟢dom (iEdgβ€˜πΊ) ∧ βŸ¨β€œπ΄π΅πΆβ€βŸ©:(0...2)βŸΆπ‘‰ ∧ (((iEdgβ€˜πΊ)β€˜(π‘“β€˜0)) = {𝐴, 𝐡} ∧ ((iEdgβ€˜πΊ)β€˜(π‘“β€˜1)) = {𝐡, 𝐢}))))
26 umgruhgr 28891 . . . . . . . . . . 11 (𝐺 ∈ UMGraph β†’ 𝐺 ∈ UHGraph)
279uhgrfun 28853 . . . . . . . . . . 11 (𝐺 ∈ UHGraph β†’ Fun (iEdgβ€˜πΊ))
28 fdmrn 6749 . . . . . . . . . . . 12 (Fun (iEdgβ€˜πΊ) ↔ (iEdgβ€˜πΊ):dom (iEdgβ€˜πΊ)⟢ran (iEdgβ€˜πΊ))
29 simpr 484 . . . . . . . . . . . . . . . . 17 ((𝑓:(0..^2)⟢dom (iEdgβ€˜πΊ) ∧ (iEdgβ€˜πΊ):dom (iEdgβ€˜πΊ)⟢ran (iEdgβ€˜πΊ)) β†’ (iEdgβ€˜πΊ):dom (iEdgβ€˜πΊ)⟢ran (iEdgβ€˜πΊ))
30 id 22 . . . . . . . . . . . . . . . . . . 19 (𝑓:(0..^2)⟢dom (iEdgβ€˜πΊ) β†’ 𝑓:(0..^2)⟢dom (iEdgβ€˜πΊ))
31 c0ex 11224 . . . . . . . . . . . . . . . . . . . . . 22 0 ∈ V
3231prid1 4762 . . . . . . . . . . . . . . . . . . . . 21 0 ∈ {0, 1}
33 fzo0to2pr 13735 . . . . . . . . . . . . . . . . . . . . 21 (0..^2) = {0, 1}
3432, 33eleqtrri 2827 . . . . . . . . . . . . . . . . . . . 20 0 ∈ (0..^2)
3534a1i 11 . . . . . . . . . . . . . . . . . . 19 (𝑓:(0..^2)⟢dom (iEdgβ€˜πΊ) β†’ 0 ∈ (0..^2))
3630, 35ffvelcdmd 7089 . . . . . . . . . . . . . . . . . 18 (𝑓:(0..^2)⟢dom (iEdgβ€˜πΊ) β†’ (π‘“β€˜0) ∈ dom (iEdgβ€˜πΊ))
3736adantr 480 . . . . . . . . . . . . . . . . 17 ((𝑓:(0..^2)⟢dom (iEdgβ€˜πΊ) ∧ (iEdgβ€˜πΊ):dom (iEdgβ€˜πΊ)⟢ran (iEdgβ€˜πΊ)) β†’ (π‘“β€˜0) ∈ dom (iEdgβ€˜πΊ))
3829, 37ffvelcdmd 7089 . . . . . . . . . . . . . . . 16 ((𝑓:(0..^2)⟢dom (iEdgβ€˜πΊ) ∧ (iEdgβ€˜πΊ):dom (iEdgβ€˜πΊ)⟢ran (iEdgβ€˜πΊ)) β†’ ((iEdgβ€˜πΊ)β€˜(π‘“β€˜0)) ∈ ran (iEdgβ€˜πΊ))
39 1ex 11226 . . . . . . . . . . . . . . . . . . . . . 22 1 ∈ V
4039prid2 4763 . . . . . . . . . . . . . . . . . . . . 21 1 ∈ {0, 1}
4140, 33eleqtrri 2827 . . . . . . . . . . . . . . . . . . . 20 1 ∈ (0..^2)
4241a1i 11 . . . . . . . . . . . . . . . . . . 19 (𝑓:(0..^2)⟢dom (iEdgβ€˜πΊ) β†’ 1 ∈ (0..^2))
4330, 42ffvelcdmd 7089 . . . . . . . . . . . . . . . . . 18 (𝑓:(0..^2)⟢dom (iEdgβ€˜πΊ) β†’ (π‘“β€˜1) ∈ dom (iEdgβ€˜πΊ))
4443adantr 480 . . . . . . . . . . . . . . . . 17 ((𝑓:(0..^2)⟢dom (iEdgβ€˜πΊ) ∧ (iEdgβ€˜πΊ):dom (iEdgβ€˜πΊ)⟢ran (iEdgβ€˜πΊ)) β†’ (π‘“β€˜1) ∈ dom (iEdgβ€˜πΊ))
4529, 44ffvelcdmd 7089 . . . . . . . . . . . . . . . 16 ((𝑓:(0..^2)⟢dom (iEdgβ€˜πΊ) ∧ (iEdgβ€˜πΊ):dom (iEdgβ€˜πΊ)⟢ran (iEdgβ€˜πΊ)) β†’ ((iEdgβ€˜πΊ)β€˜(π‘“β€˜1)) ∈ ran (iEdgβ€˜πΊ))
4638, 45jca 511 . . . . . . . . . . . . . . 15 ((𝑓:(0..^2)⟢dom (iEdgβ€˜πΊ) ∧ (iEdgβ€˜πΊ):dom (iEdgβ€˜πΊ)⟢ran (iEdgβ€˜πΊ)) β†’ (((iEdgβ€˜πΊ)β€˜(π‘“β€˜0)) ∈ ran (iEdgβ€˜πΊ) ∧ ((iEdgβ€˜πΊ)β€˜(π‘“β€˜1)) ∈ ran (iEdgβ€˜πΊ)))
4746ex 412 . . . . . . . . . . . . . 14 (𝑓:(0..^2)⟢dom (iEdgβ€˜πΊ) β†’ ((iEdgβ€˜πΊ):dom (iEdgβ€˜πΊ)⟢ran (iEdgβ€˜πΊ) β†’ (((iEdgβ€˜πΊ)β€˜(π‘“β€˜0)) ∈ ran (iEdgβ€˜πΊ) ∧ ((iEdgβ€˜πΊ)β€˜(π‘“β€˜1)) ∈ ran (iEdgβ€˜πΊ))))
48473ad2ant1 1131 . . . . . . . . . . . . 13 ((𝑓:(0..^2)⟢dom (iEdgβ€˜πΊ) ∧ βŸ¨β€œπ΄π΅πΆβ€βŸ©:(0...2)βŸΆπ‘‰ ∧ (((iEdgβ€˜πΊ)β€˜(π‘“β€˜0)) = {𝐴, 𝐡} ∧ ((iEdgβ€˜πΊ)β€˜(π‘“β€˜1)) = {𝐡, 𝐢})) β†’ ((iEdgβ€˜πΊ):dom (iEdgβ€˜πΊ)⟢ran (iEdgβ€˜πΊ) β†’ (((iEdgβ€˜πΊ)β€˜(π‘“β€˜0)) ∈ ran (iEdgβ€˜πΊ) ∧ ((iEdgβ€˜πΊ)β€˜(π‘“β€˜1)) ∈ ran (iEdgβ€˜πΊ))))
4948com12 32 . . . . . . . . . . . 12 ((iEdgβ€˜πΊ):dom (iEdgβ€˜πΊ)⟢ran (iEdgβ€˜πΊ) β†’ ((𝑓:(0..^2)⟢dom (iEdgβ€˜πΊ) ∧ βŸ¨β€œπ΄π΅πΆβ€βŸ©:(0...2)βŸΆπ‘‰ ∧ (((iEdgβ€˜πΊ)β€˜(π‘“β€˜0)) = {𝐴, 𝐡} ∧ ((iEdgβ€˜πΊ)β€˜(π‘“β€˜1)) = {𝐡, 𝐢})) β†’ (((iEdgβ€˜πΊ)β€˜(π‘“β€˜0)) ∈ ran (iEdgβ€˜πΊ) ∧ ((iEdgβ€˜πΊ)β€˜(π‘“β€˜1)) ∈ ran (iEdgβ€˜πΊ))))
5028, 49sylbi 216 . . . . . . . . . . 11 (Fun (iEdgβ€˜πΊ) β†’ ((𝑓:(0..^2)⟢dom (iEdgβ€˜πΊ) ∧ βŸ¨β€œπ΄π΅πΆβ€βŸ©:(0...2)βŸΆπ‘‰ ∧ (((iEdgβ€˜πΊ)β€˜(π‘“β€˜0)) = {𝐴, 𝐡} ∧ ((iEdgβ€˜πΊ)β€˜(π‘“β€˜1)) = {𝐡, 𝐢})) β†’ (((iEdgβ€˜πΊ)β€˜(π‘“β€˜0)) ∈ ran (iEdgβ€˜πΊ) ∧ ((iEdgβ€˜πΊ)β€˜(π‘“β€˜1)) ∈ ran (iEdgβ€˜πΊ))))
5126, 27, 503syl 18 . . . . . . . . . 10 (𝐺 ∈ UMGraph β†’ ((𝑓:(0..^2)⟢dom (iEdgβ€˜πΊ) ∧ βŸ¨β€œπ΄π΅πΆβ€βŸ©:(0...2)βŸΆπ‘‰ ∧ (((iEdgβ€˜πΊ)β€˜(π‘“β€˜0)) = {𝐴, 𝐡} ∧ ((iEdgβ€˜πΊ)β€˜(π‘“β€˜1)) = {𝐡, 𝐢})) β†’ (((iEdgβ€˜πΊ)β€˜(π‘“β€˜0)) ∈ ran (iEdgβ€˜πΊ) ∧ ((iEdgβ€˜πΊ)β€˜(π‘“β€˜1)) ∈ ran (iEdgβ€˜πΊ))))
5251imp 406 . . . . . . . . 9 ((𝐺 ∈ UMGraph ∧ (𝑓:(0..^2)⟢dom (iEdgβ€˜πΊ) ∧ βŸ¨β€œπ΄π΅πΆβ€βŸ©:(0...2)βŸΆπ‘‰ ∧ (((iEdgβ€˜πΊ)β€˜(π‘“β€˜0)) = {𝐴, 𝐡} ∧ ((iEdgβ€˜πΊ)β€˜(π‘“β€˜1)) = {𝐡, 𝐢}))) β†’ (((iEdgβ€˜πΊ)β€˜(π‘“β€˜0)) ∈ ran (iEdgβ€˜πΊ) ∧ ((iEdgβ€˜πΊ)β€˜(π‘“β€˜1)) ∈ ran (iEdgβ€˜πΊ)))
53 eqcom 2734 . . . . . . . . . . . . . . 15 (((iEdgβ€˜πΊ)β€˜(π‘“β€˜0)) = {𝐴, 𝐡} ↔ {𝐴, 𝐡} = ((iEdgβ€˜πΊ)β€˜(π‘“β€˜0)))
5453biimpi 215 . . . . . . . . . . . . . 14 (((iEdgβ€˜πΊ)β€˜(π‘“β€˜0)) = {𝐴, 𝐡} β†’ {𝐴, 𝐡} = ((iEdgβ€˜πΊ)β€˜(π‘“β€˜0)))
5554adantr 480 . . . . . . . . . . . . 13 ((((iEdgβ€˜πΊ)β€˜(π‘“β€˜0)) = {𝐴, 𝐡} ∧ ((iEdgβ€˜πΊ)β€˜(π‘“β€˜1)) = {𝐡, 𝐢}) β†’ {𝐴, 𝐡} = ((iEdgβ€˜πΊ)β€˜(π‘“β€˜0)))
56553ad2ant3 1133 . . . . . . . . . . . 12 ((𝑓:(0..^2)⟢dom (iEdgβ€˜πΊ) ∧ βŸ¨β€œπ΄π΅πΆβ€βŸ©:(0...2)βŸΆπ‘‰ ∧ (((iEdgβ€˜πΊ)β€˜(π‘“β€˜0)) = {𝐴, 𝐡} ∧ ((iEdgβ€˜πΊ)β€˜(π‘“β€˜1)) = {𝐡, 𝐢})) β†’ {𝐴, 𝐡} = ((iEdgβ€˜πΊ)β€˜(π‘“β€˜0)))
5756adantl 481 . . . . . . . . . . 11 ((𝐺 ∈ UMGraph ∧ (𝑓:(0..^2)⟢dom (iEdgβ€˜πΊ) ∧ βŸ¨β€œπ΄π΅πΆβ€βŸ©:(0...2)βŸΆπ‘‰ ∧ (((iEdgβ€˜πΊ)β€˜(π‘“β€˜0)) = {𝐴, 𝐡} ∧ ((iEdgβ€˜πΊ)β€˜(π‘“β€˜1)) = {𝐡, 𝐢}))) β†’ {𝐴, 𝐡} = ((iEdgβ€˜πΊ)β€˜(π‘“β€˜0)))
58 usgrwwlks2on.e . . . . . . . . . . . . 13 𝐸 = (Edgβ€˜πΊ)
59 edgval 28836 . . . . . . . . . . . . 13 (Edgβ€˜πΊ) = ran (iEdgβ€˜πΊ)
6058, 59eqtri 2755 . . . . . . . . . . . 12 𝐸 = ran (iEdgβ€˜πΊ)
6160a1i 11 . . . . . . . . . . 11 ((𝐺 ∈ UMGraph ∧ (𝑓:(0..^2)⟢dom (iEdgβ€˜πΊ) ∧ βŸ¨β€œπ΄π΅πΆβ€βŸ©:(0...2)βŸΆπ‘‰ ∧ (((iEdgβ€˜πΊ)β€˜(π‘“β€˜0)) = {𝐴, 𝐡} ∧ ((iEdgβ€˜πΊ)β€˜(π‘“β€˜1)) = {𝐡, 𝐢}))) β†’ 𝐸 = ran (iEdgβ€˜πΊ))
6257, 61eleq12d 2822 . . . . . . . . . 10 ((𝐺 ∈ UMGraph ∧ (𝑓:(0..^2)⟢dom (iEdgβ€˜πΊ) ∧ βŸ¨β€œπ΄π΅πΆβ€βŸ©:(0...2)βŸΆπ‘‰ ∧ (((iEdgβ€˜πΊ)β€˜(π‘“β€˜0)) = {𝐴, 𝐡} ∧ ((iEdgβ€˜πΊ)β€˜(π‘“β€˜1)) = {𝐡, 𝐢}))) β†’ ({𝐴, 𝐡} ∈ 𝐸 ↔ ((iEdgβ€˜πΊ)β€˜(π‘“β€˜0)) ∈ ran (iEdgβ€˜πΊ)))
63 eqcom 2734 . . . . . . . . . . . . . . 15 (((iEdgβ€˜πΊ)β€˜(π‘“β€˜1)) = {𝐡, 𝐢} ↔ {𝐡, 𝐢} = ((iEdgβ€˜πΊ)β€˜(π‘“β€˜1)))
6463biimpi 215 . . . . . . . . . . . . . 14 (((iEdgβ€˜πΊ)β€˜(π‘“β€˜1)) = {𝐡, 𝐢} β†’ {𝐡, 𝐢} = ((iEdgβ€˜πΊ)β€˜(π‘“β€˜1)))
6564adantl 481 . . . . . . . . . . . . 13 ((((iEdgβ€˜πΊ)β€˜(π‘“β€˜0)) = {𝐴, 𝐡} ∧ ((iEdgβ€˜πΊ)β€˜(π‘“β€˜1)) = {𝐡, 𝐢}) β†’ {𝐡, 𝐢} = ((iEdgβ€˜πΊ)β€˜(π‘“β€˜1)))
66653ad2ant3 1133 . . . . . . . . . . . 12 ((𝑓:(0..^2)⟢dom (iEdgβ€˜πΊ) ∧ βŸ¨β€œπ΄π΅πΆβ€βŸ©:(0...2)βŸΆπ‘‰ ∧ (((iEdgβ€˜πΊ)β€˜(π‘“β€˜0)) = {𝐴, 𝐡} ∧ ((iEdgβ€˜πΊ)β€˜(π‘“β€˜1)) = {𝐡, 𝐢})) β†’ {𝐡, 𝐢} = ((iEdgβ€˜πΊ)β€˜(π‘“β€˜1)))
6766adantl 481 . . . . . . . . . . 11 ((𝐺 ∈ UMGraph ∧ (𝑓:(0..^2)⟢dom (iEdgβ€˜πΊ) ∧ βŸ¨β€œπ΄π΅πΆβ€βŸ©:(0...2)βŸΆπ‘‰ ∧ (((iEdgβ€˜πΊ)β€˜(π‘“β€˜0)) = {𝐴, 𝐡} ∧ ((iEdgβ€˜πΊ)β€˜(π‘“β€˜1)) = {𝐡, 𝐢}))) β†’ {𝐡, 𝐢} = ((iEdgβ€˜πΊ)β€˜(π‘“β€˜1)))
6867, 61eleq12d 2822 . . . . . . . . . 10 ((𝐺 ∈ UMGraph ∧ (𝑓:(0..^2)⟢dom (iEdgβ€˜πΊ) ∧ βŸ¨β€œπ΄π΅πΆβ€βŸ©:(0...2)βŸΆπ‘‰ ∧ (((iEdgβ€˜πΊ)β€˜(π‘“β€˜0)) = {𝐴, 𝐡} ∧ ((iEdgβ€˜πΊ)β€˜(π‘“β€˜1)) = {𝐡, 𝐢}))) β†’ ({𝐡, 𝐢} ∈ 𝐸 ↔ ((iEdgβ€˜πΊ)β€˜(π‘“β€˜1)) ∈ ran (iEdgβ€˜πΊ)))
6962, 68anbi12d 630 . . . . . . . . 9 ((𝐺 ∈ UMGraph ∧ (𝑓:(0..^2)⟢dom (iEdgβ€˜πΊ) ∧ βŸ¨β€œπ΄π΅πΆβ€βŸ©:(0...2)βŸΆπ‘‰ ∧ (((iEdgβ€˜πΊ)β€˜(π‘“β€˜0)) = {𝐴, 𝐡} ∧ ((iEdgβ€˜πΊ)β€˜(π‘“β€˜1)) = {𝐡, 𝐢}))) β†’ (({𝐴, 𝐡} ∈ 𝐸 ∧ {𝐡, 𝐢} ∈ 𝐸) ↔ (((iEdgβ€˜πΊ)β€˜(π‘“β€˜0)) ∈ ran (iEdgβ€˜πΊ) ∧ ((iEdgβ€˜πΊ)β€˜(π‘“β€˜1)) ∈ ran (iEdgβ€˜πΊ))))
7052, 69mpbird 257 . . . . . . . 8 ((𝐺 ∈ UMGraph ∧ (𝑓:(0..^2)⟢dom (iEdgβ€˜πΊ) ∧ βŸ¨β€œπ΄π΅πΆβ€βŸ©:(0...2)βŸΆπ‘‰ ∧ (((iEdgβ€˜πΊ)β€˜(π‘“β€˜0)) = {𝐴, 𝐡} ∧ ((iEdgβ€˜πΊ)β€˜(π‘“β€˜1)) = {𝐡, 𝐢}))) β†’ ({𝐴, 𝐡} ∈ 𝐸 ∧ {𝐡, 𝐢} ∈ 𝐸))
7170ex 412 . . . . . . 7 (𝐺 ∈ UMGraph β†’ ((𝑓:(0..^2)⟢dom (iEdgβ€˜πΊ) ∧ βŸ¨β€œπ΄π΅πΆβ€βŸ©:(0...2)βŸΆπ‘‰ ∧ (((iEdgβ€˜πΊ)β€˜(π‘“β€˜0)) = {𝐴, 𝐡} ∧ ((iEdgβ€˜πΊ)β€˜(π‘“β€˜1)) = {𝐡, 𝐢})) β†’ ({𝐴, 𝐡} ∈ 𝐸 ∧ {𝐡, 𝐢} ∈ 𝐸)))
7271adantr 480 . . . . . 6 ((𝐺 ∈ UMGraph ∧ (𝐴 ∈ 𝑉 ∧ 𝐡 ∈ 𝑉 ∧ 𝐢 ∈ 𝑉)) β†’ ((𝑓:(0..^2)⟢dom (iEdgβ€˜πΊ) ∧ βŸ¨β€œπ΄π΅πΆβ€βŸ©:(0...2)βŸΆπ‘‰ ∧ (((iEdgβ€˜πΊ)β€˜(π‘“β€˜0)) = {𝐴, 𝐡} ∧ ((iEdgβ€˜πΊ)β€˜(π‘“β€˜1)) = {𝐡, 𝐢})) β†’ ({𝐴, 𝐡} ∈ 𝐸 ∧ {𝐡, 𝐢} ∈ 𝐸)))
7325, 72sylbid 239 . . . . 5 ((𝐺 ∈ UMGraph ∧ (𝐴 ∈ 𝑉 ∧ 𝐡 ∈ 𝑉 ∧ 𝐢 ∈ 𝑉)) β†’ ((𝑓:(0..^2)⟢dom (iEdgβ€˜πΊ) ∧ βŸ¨β€œπ΄π΅πΆβ€βŸ©:(0...2)βŸΆπ‘‰ ∧ (((iEdgβ€˜πΊ)β€˜(π‘“β€˜0)) = {(βŸ¨β€œπ΄π΅πΆβ€βŸ©β€˜0), (βŸ¨β€œπ΄π΅πΆβ€βŸ©β€˜1)} ∧ ((iEdgβ€˜πΊ)β€˜(π‘“β€˜1)) = {(βŸ¨β€œπ΄π΅πΆβ€βŸ©β€˜1), (βŸ¨β€œπ΄π΅πΆβ€βŸ©β€˜2)})) β†’ ({𝐴, 𝐡} ∈ 𝐸 ∧ {𝐡, 𝐢} ∈ 𝐸)))
7412, 73sylbid 239 . . . 4 ((𝐺 ∈ UMGraph ∧ (𝐴 ∈ 𝑉 ∧ 𝐡 ∈ 𝑉 ∧ 𝐢 ∈ 𝑉)) β†’ ((𝑓(Walksβ€˜πΊ)βŸ¨β€œπ΄π΅πΆβ€βŸ© ∧ (β™―β€˜π‘“) = 2) β†’ ({𝐴, 𝐡} ∈ 𝐸 ∧ {𝐡, 𝐢} ∈ 𝐸)))
7574exlimdv 1929 . . 3 ((𝐺 ∈ UMGraph ∧ (𝐴 ∈ 𝑉 ∧ 𝐡 ∈ 𝑉 ∧ 𝐢 ∈ 𝑉)) β†’ (βˆƒπ‘“(𝑓(Walksβ€˜πΊ)βŸ¨β€œπ΄π΅πΆβ€βŸ© ∧ (β™―β€˜π‘“) = 2) β†’ ({𝐴, 𝐡} ∈ 𝐸 ∧ {𝐡, 𝐢} ∈ 𝐸)))
7658umgr2wlk 29734 . . . . . . 7 ((𝐺 ∈ UMGraph ∧ {𝐴, 𝐡} ∈ 𝐸 ∧ {𝐡, 𝐢} ∈ 𝐸) β†’ βˆƒπ‘“βˆƒπ‘(𝑓(Walksβ€˜πΊ)𝑝 ∧ (β™―β€˜π‘“) = 2 ∧ (𝐴 = (π‘β€˜0) ∧ 𝐡 = (π‘β€˜1) ∧ 𝐢 = (π‘β€˜2))))
77 wlklenvp1 29406 . . . . . . . . . . . . . . . . . . . 20 (𝑓(Walksβ€˜πΊ)𝑝 β†’ (β™―β€˜π‘) = ((β™―β€˜π‘“) + 1))
78 oveq1 7421 . . . . . . . . . . . . . . . . . . . . . 22 ((β™―β€˜π‘“) = 2 β†’ ((β™―β€˜π‘“) + 1) = (2 + 1))
79 2p1e3 12370 . . . . . . . . . . . . . . . . . . . . . 22 (2 + 1) = 3
8078, 79eqtrdi 2783 . . . . . . . . . . . . . . . . . . . . 21 ((β™―β€˜π‘“) = 2 β†’ ((β™―β€˜π‘“) + 1) = 3)
8180adantr 480 . . . . . . . . . . . . . . . . . . . 20 (((β™―β€˜π‘“) = 2 ∧ (𝐴 = (π‘β€˜0) ∧ 𝐡 = (π‘β€˜1) ∧ 𝐢 = (π‘β€˜2))) β†’ ((β™―β€˜π‘“) + 1) = 3)
8277, 81sylan9eq 2787 . . . . . . . . . . . . . . . . . . 19 ((𝑓(Walksβ€˜πΊ)𝑝 ∧ ((β™―β€˜π‘“) = 2 ∧ (𝐴 = (π‘β€˜0) ∧ 𝐡 = (π‘β€˜1) ∧ 𝐢 = (π‘β€˜2)))) β†’ (β™―β€˜π‘) = 3)
83 eqcom 2734 . . . . . . . . . . . . . . . . . . . . . . 23 (𝐴 = (π‘β€˜0) ↔ (π‘β€˜0) = 𝐴)
84 eqcom 2734 . . . . . . . . . . . . . . . . . . . . . . 23 (𝐡 = (π‘β€˜1) ↔ (π‘β€˜1) = 𝐡)
85 eqcom 2734 . . . . . . . . . . . . . . . . . . . . . . 23 (𝐢 = (π‘β€˜2) ↔ (π‘β€˜2) = 𝐢)
8683, 84, 853anbi123i 1153 . . . . . . . . . . . . . . . . . . . . . 22 ((𝐴 = (π‘β€˜0) ∧ 𝐡 = (π‘β€˜1) ∧ 𝐢 = (π‘β€˜2)) ↔ ((π‘β€˜0) = 𝐴 ∧ (π‘β€˜1) = 𝐡 ∧ (π‘β€˜2) = 𝐢))
8786biimpi 215 . . . . . . . . . . . . . . . . . . . . 21 ((𝐴 = (π‘β€˜0) ∧ 𝐡 = (π‘β€˜1) ∧ 𝐢 = (π‘β€˜2)) β†’ ((π‘β€˜0) = 𝐴 ∧ (π‘β€˜1) = 𝐡 ∧ (π‘β€˜2) = 𝐢))
8887adantl 481 . . . . . . . . . . . . . . . . . . . 20 (((β™―β€˜π‘“) = 2 ∧ (𝐴 = (π‘β€˜0) ∧ 𝐡 = (π‘β€˜1) ∧ 𝐢 = (π‘β€˜2))) β†’ ((π‘β€˜0) = 𝐴 ∧ (π‘β€˜1) = 𝐡 ∧ (π‘β€˜2) = 𝐢))
8988adantl 481 . . . . . . . . . . . . . . . . . . 19 ((𝑓(Walksβ€˜πΊ)𝑝 ∧ ((β™―β€˜π‘“) = 2 ∧ (𝐴 = (π‘β€˜0) ∧ 𝐡 = (π‘β€˜1) ∧ 𝐢 = (π‘β€˜2)))) β†’ ((π‘β€˜0) = 𝐴 ∧ (π‘β€˜1) = 𝐡 ∧ (π‘β€˜2) = 𝐢))
9082, 89jca 511 . . . . . . . . . . . . . . . . . 18 ((𝑓(Walksβ€˜πΊ)𝑝 ∧ ((β™―β€˜π‘“) = 2 ∧ (𝐴 = (π‘β€˜0) ∧ 𝐡 = (π‘β€˜1) ∧ 𝐢 = (π‘β€˜2)))) β†’ ((β™―β€˜π‘) = 3 ∧ ((π‘β€˜0) = 𝐴 ∧ (π‘β€˜1) = 𝐡 ∧ (π‘β€˜2) = 𝐢)))
916wlkpwrd 29405 . . . . . . . . . . . . . . . . . . . . 21 (𝑓(Walksβ€˜πΊ)𝑝 β†’ 𝑝 ∈ Word 𝑉)
9280eqeq2d 2738 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((β™―β€˜π‘“) = 2 β†’ ((β™―β€˜π‘) = ((β™―β€˜π‘“) + 1) ↔ (β™―β€˜π‘) = 3))
9392adantl 481 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑝 ∈ Word 𝑉 ∧ (β™―β€˜π‘“) = 2) β†’ ((β™―β€˜π‘) = ((β™―β€˜π‘“) + 1) ↔ (β™―β€˜π‘) = 3))
94 simp1 1134 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑝 ∈ Word 𝑉 ∧ (β™―β€˜π‘) = 3 ∧ (𝐴 = (π‘β€˜0) ∧ 𝐡 = (π‘β€˜1) ∧ 𝐢 = (π‘β€˜2))) β†’ 𝑝 ∈ Word 𝑉)
95 oveq2 7422 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((β™―β€˜π‘) = 3 β†’ (0..^(β™―β€˜π‘)) = (0..^3))
96 fzo0to3tp 13736 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (0..^3) = {0, 1, 2}
9795, 96eqtrdi 2783 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((β™―β€˜π‘) = 3 β†’ (0..^(β™―β€˜π‘)) = {0, 1, 2})
9831tpid1 4768 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 0 ∈ {0, 1, 2}
99 eleq2 2817 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((0..^(β™―β€˜π‘)) = {0, 1, 2} β†’ (0 ∈ (0..^(β™―β€˜π‘)) ↔ 0 ∈ {0, 1, 2}))
10098, 99mpbiri 258 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((0..^(β™―β€˜π‘)) = {0, 1, 2} β†’ 0 ∈ (0..^(β™―β€˜π‘)))
101 wrdsymbcl 14495 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((𝑝 ∈ Word 𝑉 ∧ 0 ∈ (0..^(β™―β€˜π‘))) β†’ (π‘β€˜0) ∈ 𝑉)
102100, 101sylan2 592 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((𝑝 ∈ Word 𝑉 ∧ (0..^(β™―β€˜π‘)) = {0, 1, 2}) β†’ (π‘β€˜0) ∈ 𝑉)
10339tpid2 4770 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 1 ∈ {0, 1, 2}
104 eleq2 2817 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((0..^(β™―β€˜π‘)) = {0, 1, 2} β†’ (1 ∈ (0..^(β™―β€˜π‘)) ↔ 1 ∈ {0, 1, 2}))
105103, 104mpbiri 258 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((0..^(β™―β€˜π‘)) = {0, 1, 2} β†’ 1 ∈ (0..^(β™―β€˜π‘)))
106 wrdsymbcl 14495 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((𝑝 ∈ Word 𝑉 ∧ 1 ∈ (0..^(β™―β€˜π‘))) β†’ (π‘β€˜1) ∈ 𝑉)
107105, 106sylan2 592 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((𝑝 ∈ Word 𝑉 ∧ (0..^(β™―β€˜π‘)) = {0, 1, 2}) β†’ (π‘β€˜1) ∈ 𝑉)
108 2ex 12305 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 2 ∈ V
109108tpid3 4773 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 2 ∈ {0, 1, 2}
110 eleq2 2817 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((0..^(β™―β€˜π‘)) = {0, 1, 2} β†’ (2 ∈ (0..^(β™―β€˜π‘)) ↔ 2 ∈ {0, 1, 2}))
111109, 110mpbiri 258 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((0..^(β™―β€˜π‘)) = {0, 1, 2} β†’ 2 ∈ (0..^(β™―β€˜π‘)))
112 wrdsymbcl 14495 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((𝑝 ∈ Word 𝑉 ∧ 2 ∈ (0..^(β™―β€˜π‘))) β†’ (π‘β€˜2) ∈ 𝑉)
113111, 112sylan2 592 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((𝑝 ∈ Word 𝑉 ∧ (0..^(β™―β€˜π‘)) = {0, 1, 2}) β†’ (π‘β€˜2) ∈ 𝑉)
114102, 107, 1133jca 1126 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((𝑝 ∈ Word 𝑉 ∧ (0..^(β™―β€˜π‘)) = {0, 1, 2}) β†’ ((π‘β€˜0) ∈ 𝑉 ∧ (π‘β€˜1) ∈ 𝑉 ∧ (π‘β€˜2) ∈ 𝑉))
11597, 114sylan2 592 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝑝 ∈ Word 𝑉 ∧ (β™―β€˜π‘) = 3) β†’ ((π‘β€˜0) ∈ 𝑉 ∧ (π‘β€˜1) ∈ 𝑉 ∧ (π‘β€˜2) ∈ 𝑉))
1161153adant3 1130 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝑝 ∈ Word 𝑉 ∧ (β™―β€˜π‘) = 3 ∧ (𝐴 = (π‘β€˜0) ∧ 𝐡 = (π‘β€˜1) ∧ 𝐢 = (π‘β€˜2))) β†’ ((π‘β€˜0) ∈ 𝑉 ∧ (π‘β€˜1) ∈ 𝑉 ∧ (π‘β€˜2) ∈ 𝑉))
117 eleq1 2816 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝐴 = (π‘β€˜0) β†’ (𝐴 ∈ 𝑉 ↔ (π‘β€˜0) ∈ 𝑉))
1181173ad2ant1 1131 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((𝐴 = (π‘β€˜0) ∧ 𝐡 = (π‘β€˜1) ∧ 𝐢 = (π‘β€˜2)) β†’ (𝐴 ∈ 𝑉 ↔ (π‘β€˜0) ∈ 𝑉))
119 eleq1 2816 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝐡 = (π‘β€˜1) β†’ (𝐡 ∈ 𝑉 ↔ (π‘β€˜1) ∈ 𝑉))
1201193ad2ant2 1132 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((𝐴 = (π‘β€˜0) ∧ 𝐡 = (π‘β€˜1) ∧ 𝐢 = (π‘β€˜2)) β†’ (𝐡 ∈ 𝑉 ↔ (π‘β€˜1) ∈ 𝑉))
121 eleq1 2816 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝐢 = (π‘β€˜2) β†’ (𝐢 ∈ 𝑉 ↔ (π‘β€˜2) ∈ 𝑉))
1221213ad2ant3 1133 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((𝐴 = (π‘β€˜0) ∧ 𝐡 = (π‘β€˜1) ∧ 𝐢 = (π‘β€˜2)) β†’ (𝐢 ∈ 𝑉 ↔ (π‘β€˜2) ∈ 𝑉))
123118, 120, 1223anbi123d 1433 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝐴 = (π‘β€˜0) ∧ 𝐡 = (π‘β€˜1) ∧ 𝐢 = (π‘β€˜2)) β†’ ((𝐴 ∈ 𝑉 ∧ 𝐡 ∈ 𝑉 ∧ 𝐢 ∈ 𝑉) ↔ ((π‘β€˜0) ∈ 𝑉 ∧ (π‘β€˜1) ∈ 𝑉 ∧ (π‘β€˜2) ∈ 𝑉)))
1241233ad2ant3 1133 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝑝 ∈ Word 𝑉 ∧ (β™―β€˜π‘) = 3 ∧ (𝐴 = (π‘β€˜0) ∧ 𝐡 = (π‘β€˜1) ∧ 𝐢 = (π‘β€˜2))) β†’ ((𝐴 ∈ 𝑉 ∧ 𝐡 ∈ 𝑉 ∧ 𝐢 ∈ 𝑉) ↔ ((π‘β€˜0) ∈ 𝑉 ∧ (π‘β€˜1) ∈ 𝑉 ∧ (π‘β€˜2) ∈ 𝑉)))
125116, 124mpbird 257 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑝 ∈ Word 𝑉 ∧ (β™―β€˜π‘) = 3 ∧ (𝐴 = (π‘β€˜0) ∧ 𝐡 = (π‘β€˜1) ∧ 𝐢 = (π‘β€˜2))) β†’ (𝐴 ∈ 𝑉 ∧ 𝐡 ∈ 𝑉 ∧ 𝐢 ∈ 𝑉))
12694, 125jca 511 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑝 ∈ Word 𝑉 ∧ (β™―β€˜π‘) = 3 ∧ (𝐴 = (π‘β€˜0) ∧ 𝐡 = (π‘β€˜1) ∧ 𝐢 = (π‘β€˜2))) β†’ (𝑝 ∈ Word 𝑉 ∧ (𝐴 ∈ 𝑉 ∧ 𝐡 ∈ 𝑉 ∧ 𝐢 ∈ 𝑉)))
1271263exp 1117 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑝 ∈ Word 𝑉 β†’ ((β™―β€˜π‘) = 3 β†’ ((𝐴 = (π‘β€˜0) ∧ 𝐡 = (π‘β€˜1) ∧ 𝐢 = (π‘β€˜2)) β†’ (𝑝 ∈ Word 𝑉 ∧ (𝐴 ∈ 𝑉 ∧ 𝐡 ∈ 𝑉 ∧ 𝐢 ∈ 𝑉)))))
128127adantr 480 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑝 ∈ Word 𝑉 ∧ (β™―β€˜π‘“) = 2) β†’ ((β™―β€˜π‘) = 3 β†’ ((𝐴 = (π‘β€˜0) ∧ 𝐡 = (π‘β€˜1) ∧ 𝐢 = (π‘β€˜2)) β†’ (𝑝 ∈ Word 𝑉 ∧ (𝐴 ∈ 𝑉 ∧ 𝐡 ∈ 𝑉 ∧ 𝐢 ∈ 𝑉)))))
12993, 128sylbid 239 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑝 ∈ Word 𝑉 ∧ (β™―β€˜π‘“) = 2) β†’ ((β™―β€˜π‘) = ((β™―β€˜π‘“) + 1) β†’ ((𝐴 = (π‘β€˜0) ∧ 𝐡 = (π‘β€˜1) ∧ 𝐢 = (π‘β€˜2)) β†’ (𝑝 ∈ Word 𝑉 ∧ (𝐴 ∈ 𝑉 ∧ 𝐡 ∈ 𝑉 ∧ 𝐢 ∈ 𝑉)))))
130129impancom 451 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑝 ∈ Word 𝑉 ∧ (β™―β€˜π‘) = ((β™―β€˜π‘“) + 1)) β†’ ((β™―β€˜π‘“) = 2 β†’ ((𝐴 = (π‘β€˜0) ∧ 𝐡 = (π‘β€˜1) ∧ 𝐢 = (π‘β€˜2)) β†’ (𝑝 ∈ Word 𝑉 ∧ (𝐴 ∈ 𝑉 ∧ 𝐡 ∈ 𝑉 ∧ 𝐢 ∈ 𝑉)))))
131130impd 410 . . . . . . . . . . . . . . . . . . . . 21 ((𝑝 ∈ Word 𝑉 ∧ (β™―β€˜π‘) = ((β™―β€˜π‘“) + 1)) β†’ (((β™―β€˜π‘“) = 2 ∧ (𝐴 = (π‘β€˜0) ∧ 𝐡 = (π‘β€˜1) ∧ 𝐢 = (π‘β€˜2))) β†’ (𝑝 ∈ Word 𝑉 ∧ (𝐴 ∈ 𝑉 ∧ 𝐡 ∈ 𝑉 ∧ 𝐢 ∈ 𝑉))))
13291, 77, 131syl2anc 583 . . . . . . . . . . . . . . . . . . . 20 (𝑓(Walksβ€˜πΊ)𝑝 β†’ (((β™―β€˜π‘“) = 2 ∧ (𝐴 = (π‘β€˜0) ∧ 𝐡 = (π‘β€˜1) ∧ 𝐢 = (π‘β€˜2))) β†’ (𝑝 ∈ Word 𝑉 ∧ (𝐴 ∈ 𝑉 ∧ 𝐡 ∈ 𝑉 ∧ 𝐢 ∈ 𝑉))))
133132imp 406 . . . . . . . . . . . . . . . . . . 19 ((𝑓(Walksβ€˜πΊ)𝑝 ∧ ((β™―β€˜π‘“) = 2 ∧ (𝐴 = (π‘β€˜0) ∧ 𝐡 = (π‘β€˜1) ∧ 𝐢 = (π‘β€˜2)))) β†’ (𝑝 ∈ Word 𝑉 ∧ (𝐴 ∈ 𝑉 ∧ 𝐡 ∈ 𝑉 ∧ 𝐢 ∈ 𝑉)))
134 eqwrds3 14930 . . . . . . . . . . . . . . . . . . 19 ((𝑝 ∈ Word 𝑉 ∧ (𝐴 ∈ 𝑉 ∧ 𝐡 ∈ 𝑉 ∧ 𝐢 ∈ 𝑉)) β†’ (𝑝 = βŸ¨β€œπ΄π΅πΆβ€βŸ© ↔ ((β™―β€˜π‘) = 3 ∧ ((π‘β€˜0) = 𝐴 ∧ (π‘β€˜1) = 𝐡 ∧ (π‘β€˜2) = 𝐢))))
135133, 134syl 17 . . . . . . . . . . . . . . . . . 18 ((𝑓(Walksβ€˜πΊ)𝑝 ∧ ((β™―β€˜π‘“) = 2 ∧ (𝐴 = (π‘β€˜0) ∧ 𝐡 = (π‘β€˜1) ∧ 𝐢 = (π‘β€˜2)))) β†’ (𝑝 = βŸ¨β€œπ΄π΅πΆβ€βŸ© ↔ ((β™―β€˜π‘) = 3 ∧ ((π‘β€˜0) = 𝐴 ∧ (π‘β€˜1) = 𝐡 ∧ (π‘β€˜2) = 𝐢))))
13690, 135mpbird 257 . . . . . . . . . . . . . . . . 17 ((𝑓(Walksβ€˜πΊ)𝑝 ∧ ((β™―β€˜π‘“) = 2 ∧ (𝐴 = (π‘β€˜0) ∧ 𝐡 = (π‘β€˜1) ∧ 𝐢 = (π‘β€˜2)))) β†’ 𝑝 = βŸ¨β€œπ΄π΅πΆβ€βŸ©)
137136breq2d 5154 . . . . . . . . . . . . . . . 16 ((𝑓(Walksβ€˜πΊ)𝑝 ∧ ((β™―β€˜π‘“) = 2 ∧ (𝐴 = (π‘β€˜0) ∧ 𝐡 = (π‘β€˜1) ∧ 𝐢 = (π‘β€˜2)))) β†’ (𝑓(Walksβ€˜πΊ)𝑝 ↔ 𝑓(Walksβ€˜πΊ)βŸ¨β€œπ΄π΅πΆβ€βŸ©))
138137biimpd 228 . . . . . . . . . . . . . . 15 ((𝑓(Walksβ€˜πΊ)𝑝 ∧ ((β™―β€˜π‘“) = 2 ∧ (𝐴 = (π‘β€˜0) ∧ 𝐡 = (π‘β€˜1) ∧ 𝐢 = (π‘β€˜2)))) β†’ (𝑓(Walksβ€˜πΊ)𝑝 β†’ 𝑓(Walksβ€˜πΊ)βŸ¨β€œπ΄π΅πΆβ€βŸ©))
139138ex 412 . . . . . . . . . . . . . 14 (𝑓(Walksβ€˜πΊ)𝑝 β†’ (((β™―β€˜π‘“) = 2 ∧ (𝐴 = (π‘β€˜0) ∧ 𝐡 = (π‘β€˜1) ∧ 𝐢 = (π‘β€˜2))) β†’ (𝑓(Walksβ€˜πΊ)𝑝 β†’ 𝑓(Walksβ€˜πΊ)βŸ¨β€œπ΄π΅πΆβ€βŸ©)))
140139pm2.43a 54 . . . . . . . . . . . . 13 (𝑓(Walksβ€˜πΊ)𝑝 β†’ (((β™―β€˜π‘“) = 2 ∧ (𝐴 = (π‘β€˜0) ∧ 𝐡 = (π‘β€˜1) ∧ 𝐢 = (π‘β€˜2))) β†’ 𝑓(Walksβ€˜πΊ)βŸ¨β€œπ΄π΅πΆβ€βŸ©))
1411403impib 1114 . . . . . . . . . . . 12 ((𝑓(Walksβ€˜πΊ)𝑝 ∧ (β™―β€˜π‘“) = 2 ∧ (𝐴 = (π‘β€˜0) ∧ 𝐡 = (π‘β€˜1) ∧ 𝐢 = (π‘β€˜2))) β†’ 𝑓(Walksβ€˜πΊ)βŸ¨β€œπ΄π΅πΆβ€βŸ©)
142141adantl 481 . . . . . . . . . . 11 (((𝐴 ∈ 𝑉 ∧ 𝐡 ∈ 𝑉 ∧ 𝐢 ∈ 𝑉) ∧ (𝑓(Walksβ€˜πΊ)𝑝 ∧ (β™―β€˜π‘“) = 2 ∧ (𝐴 = (π‘β€˜0) ∧ 𝐡 = (π‘β€˜1) ∧ 𝐢 = (π‘β€˜2)))) β†’ 𝑓(Walksβ€˜πΊ)βŸ¨β€œπ΄π΅πΆβ€βŸ©)
143 simpr2 1193 . . . . . . . . . . 11 (((𝐴 ∈ 𝑉 ∧ 𝐡 ∈ 𝑉 ∧ 𝐢 ∈ 𝑉) ∧ (𝑓(Walksβ€˜πΊ)𝑝 ∧ (β™―β€˜π‘“) = 2 ∧ (𝐴 = (π‘β€˜0) ∧ 𝐡 = (π‘β€˜1) ∧ 𝐢 = (π‘β€˜2)))) β†’ (β™―β€˜π‘“) = 2)
144142, 143jca 511 . . . . . . . . . 10 (((𝐴 ∈ 𝑉 ∧ 𝐡 ∈ 𝑉 ∧ 𝐢 ∈ 𝑉) ∧ (𝑓(Walksβ€˜πΊ)𝑝 ∧ (β™―β€˜π‘“) = 2 ∧ (𝐴 = (π‘β€˜0) ∧ 𝐡 = (π‘β€˜1) ∧ 𝐢 = (π‘β€˜2)))) β†’ (𝑓(Walksβ€˜πΊ)βŸ¨β€œπ΄π΅πΆβ€βŸ© ∧ (β™―β€˜π‘“) = 2))
145144ex 412 . . . . . . . . 9 ((𝐴 ∈ 𝑉 ∧ 𝐡 ∈ 𝑉 ∧ 𝐢 ∈ 𝑉) β†’ ((𝑓(Walksβ€˜πΊ)𝑝 ∧ (β™―β€˜π‘“) = 2 ∧ (𝐴 = (π‘β€˜0) ∧ 𝐡 = (π‘β€˜1) ∧ 𝐢 = (π‘β€˜2))) β†’ (𝑓(Walksβ€˜πΊ)βŸ¨β€œπ΄π΅πΆβ€βŸ© ∧ (β™―β€˜π‘“) = 2)))
146145exlimdv 1929 . . . . . . . 8 ((𝐴 ∈ 𝑉 ∧ 𝐡 ∈ 𝑉 ∧ 𝐢 ∈ 𝑉) β†’ (βˆƒπ‘(𝑓(Walksβ€˜πΊ)𝑝 ∧ (β™―β€˜π‘“) = 2 ∧ (𝐴 = (π‘β€˜0) ∧ 𝐡 = (π‘β€˜1) ∧ 𝐢 = (π‘β€˜2))) β†’ (𝑓(Walksβ€˜πΊ)βŸ¨β€œπ΄π΅πΆβ€βŸ© ∧ (β™―β€˜π‘“) = 2)))
147146eximdv 1913 . . . . . . 7 ((𝐴 ∈ 𝑉 ∧ 𝐡 ∈ 𝑉 ∧ 𝐢 ∈ 𝑉) β†’ (βˆƒπ‘“βˆƒπ‘(𝑓(Walksβ€˜πΊ)𝑝 ∧ (β™―β€˜π‘“) = 2 ∧ (𝐴 = (π‘β€˜0) ∧ 𝐡 = (π‘β€˜1) ∧ 𝐢 = (π‘β€˜2))) β†’ βˆƒπ‘“(𝑓(Walksβ€˜πΊ)βŸ¨β€œπ΄π΅πΆβ€βŸ© ∧ (β™―β€˜π‘“) = 2)))
14876, 147syl5com 31 . . . . . 6 ((𝐺 ∈ UMGraph ∧ {𝐴, 𝐡} ∈ 𝐸 ∧ {𝐡, 𝐢} ∈ 𝐸) β†’ ((𝐴 ∈ 𝑉 ∧ 𝐡 ∈ 𝑉 ∧ 𝐢 ∈ 𝑉) β†’ βˆƒπ‘“(𝑓(Walksβ€˜πΊ)βŸ¨β€œπ΄π΅πΆβ€βŸ© ∧ (β™―β€˜π‘“) = 2)))
1491483expib 1120 . . . . 5 (𝐺 ∈ UMGraph β†’ (({𝐴, 𝐡} ∈ 𝐸 ∧ {𝐡, 𝐢} ∈ 𝐸) β†’ ((𝐴 ∈ 𝑉 ∧ 𝐡 ∈ 𝑉 ∧ 𝐢 ∈ 𝑉) β†’ βˆƒπ‘“(𝑓(Walksβ€˜πΊ)βŸ¨β€œπ΄π΅πΆβ€βŸ© ∧ (β™―β€˜π‘“) = 2))))
150149com23 86 . . . 4 (𝐺 ∈ UMGraph β†’ ((𝐴 ∈ 𝑉 ∧ 𝐡 ∈ 𝑉 ∧ 𝐢 ∈ 𝑉) β†’ (({𝐴, 𝐡} ∈ 𝐸 ∧ {𝐡, 𝐢} ∈ 𝐸) β†’ βˆƒπ‘“(𝑓(Walksβ€˜πΊ)βŸ¨β€œπ΄π΅πΆβ€βŸ© ∧ (β™―β€˜π‘“) = 2))))
151150imp 406 . . 3 ((𝐺 ∈ UMGraph ∧ (𝐴 ∈ 𝑉 ∧ 𝐡 ∈ 𝑉 ∧ 𝐢 ∈ 𝑉)) β†’ (({𝐴, 𝐡} ∈ 𝐸 ∧ {𝐡, 𝐢} ∈ 𝐸) β†’ βˆƒπ‘“(𝑓(Walksβ€˜πΊ)βŸ¨β€œπ΄π΅πΆβ€βŸ© ∧ (β™―β€˜π‘“) = 2)))
15275, 151impbid 211 . 2 ((𝐺 ∈ UMGraph ∧ (𝐴 ∈ 𝑉 ∧ 𝐡 ∈ 𝑉 ∧ 𝐢 ∈ 𝑉)) β†’ (βˆƒπ‘“(𝑓(Walksβ€˜πΊ)βŸ¨β€œπ΄π΅πΆβ€βŸ© ∧ (β™―β€˜π‘“) = 2) ↔ ({𝐴, 𝐡} ∈ 𝐸 ∧ {𝐡, 𝐢} ∈ 𝐸)))
1538, 152bitrd 279 1 ((𝐺 ∈ UMGraph ∧ (𝐴 ∈ 𝑉 ∧ 𝐡 ∈ 𝑉 ∧ 𝐢 ∈ 𝑉)) β†’ (βŸ¨β€œπ΄π΅πΆβ€βŸ© ∈ (𝐴(2 WWalksNOn 𝐺)𝐢) ↔ ({𝐴, 𝐡} ∈ 𝐸 ∧ {𝐡, 𝐢} ∈ 𝐸)))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 395   ∧ w3a 1085   = wceq 1534  βˆƒwex 1774   ∈ wcel 2099  {cpr 4626  {ctp 4628   class class class wbr 5142  dom cdm 5672  ran crn 5673  Fun wfun 6536  βŸΆwf 6538  β€˜cfv 6542  (class class class)co 7414  0cc0 11124  1c1 11125   + caddc 11127  2c2 12283  3c3 12284  ...cfz 13502  ..^cfzo 13645  β™―chash 14307  Word cword 14482  βŸ¨β€œcs3 14811  Vtxcvtx 28783  iEdgciedg 28784  Edgcedg 28834  UHGraphcuhgr 28843  UPGraphcupgr 28867  UMGraphcumgr 28868  Walkscwlks 29384   WWalksNOn cwwlksnon 29612
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-rep 5279  ax-sep 5293  ax-nul 5300  ax-pow 5359  ax-pr 5423  ax-un 7732  ax-ac2 10472  ax-cnex 11180  ax-resscn 11181  ax-1cn 11182  ax-icn 11183  ax-addcl 11184  ax-addrcl 11185  ax-mulcl 11186  ax-mulrcl 11187  ax-mulcom 11188  ax-addass 11189  ax-mulass 11190  ax-distr 11191  ax-i2m1 11192  ax-1ne0 11193  ax-1rid 11194  ax-rnegex 11195  ax-rrecex 11196  ax-cnre 11197  ax-pre-lttri 11198  ax-pre-lttrn 11199  ax-pre-ltadd 11200  ax-pre-mulgt0 11201
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-ifp 1062  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2936  df-nel 3042  df-ral 3057  df-rex 3066  df-rmo 3371  df-reu 3372  df-rab 3428  df-v 3471  df-sbc 3775  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3963  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-tp 4629  df-op 4631  df-uni 4904  df-int 4945  df-iun 4993  df-br 5143  df-opab 5205  df-mpt 5226  df-tr 5260  df-id 5570  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-se 5628  df-we 5629  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-pred 6299  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-isom 6551  df-riota 7370  df-ov 7417  df-oprab 7418  df-mpo 7419  df-om 7863  df-1st 7985  df-2nd 7986  df-frecs 8278  df-wrecs 8309  df-recs 8383  df-rdg 8422  df-1o 8478  df-2o 8479  df-oadd 8482  df-er 8716  df-map 8836  df-pm 8837  df-en 8954  df-dom 8955  df-sdom 8956  df-fin 8957  df-dju 9910  df-card 9948  df-ac 10125  df-pnf 11266  df-mnf 11267  df-xr 11268  df-ltxr 11269  df-le 11270  df-sub 11462  df-neg 11463  df-nn 12229  df-2 12291  df-3 12292  df-n0 12489  df-xnn0 12561  df-z 12575  df-uz 12839  df-fz 13503  df-fzo 13646  df-hash 14308  df-word 14483  df-concat 14539  df-s1 14564  df-s2 14817  df-s3 14818  df-edg 28835  df-uhgr 28845  df-upgr 28869  df-umgr 28870  df-wlks 29387  df-wwlks 29615  df-wwlksn 29616  df-wwlksnon 29617
This theorem is referenced by:  wwlks2onsym  29743  usgr2wspthons3  29749  frgr2wwlkeu  30111
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