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Theorem umgr2wlk 29712
Description: In a multigraph, there is a walk of length 2 for each pair of adjacent edges. (Contributed by Alexander van der Vekens, 18-Feb-2018.) (Revised by AV, 30-Jan-2021.)
Hypothesis
Ref Expression
umgr2wlk.e 𝐸 = (Edgβ€˜πΊ)
Assertion
Ref Expression
umgr2wlk ((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))))
Distinct variable groups:   𝐴,𝑓,𝑝   𝐡,𝑓,𝑝   𝐢,𝑓,𝑝   𝑓,𝐺,𝑝
Allowed substitution hints:   𝐸(𝑓,𝑝)

Proof of Theorem umgr2wlk
Dummy variables 𝑖 𝑗 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 umgruhgr 28872 . . . . . 6 (𝐺 ∈ UMGraph β†’ 𝐺 ∈ UHGraph)
2 umgr2wlk.e . . . . . . . 8 𝐸 = (Edgβ€˜πΊ)
32eleq2i 2819 . . . . . . 7 ({𝐡, 𝐢} ∈ 𝐸 ↔ {𝐡, 𝐢} ∈ (Edgβ€˜πΊ))
4 eqid 2726 . . . . . . . 8 (iEdgβ€˜πΊ) = (iEdgβ€˜πΊ)
54uhgredgiedgb 28894 . . . . . . 7 (𝐺 ∈ UHGraph β†’ ({𝐡, 𝐢} ∈ (Edgβ€˜πΊ) ↔ βˆƒπ‘– ∈ dom (iEdgβ€˜πΊ){𝐡, 𝐢} = ((iEdgβ€˜πΊ)β€˜π‘–)))
63, 5bitrid 283 . . . . . 6 (𝐺 ∈ UHGraph β†’ ({𝐡, 𝐢} ∈ 𝐸 ↔ βˆƒπ‘– ∈ dom (iEdgβ€˜πΊ){𝐡, 𝐢} = ((iEdgβ€˜πΊ)β€˜π‘–)))
71, 6syl 17 . . . . 5 (𝐺 ∈ UMGraph β†’ ({𝐡, 𝐢} ∈ 𝐸 ↔ βˆƒπ‘– ∈ dom (iEdgβ€˜πΊ){𝐡, 𝐢} = ((iEdgβ€˜πΊ)β€˜π‘–)))
87biimpd 228 . . . 4 (𝐺 ∈ UMGraph β†’ ({𝐡, 𝐢} ∈ 𝐸 β†’ βˆƒπ‘– ∈ dom (iEdgβ€˜πΊ){𝐡, 𝐢} = ((iEdgβ€˜πΊ)β€˜π‘–)))
98a1d 25 . . 3 (𝐺 ∈ UMGraph β†’ ({𝐴, 𝐡} ∈ 𝐸 β†’ ({𝐡, 𝐢} ∈ 𝐸 β†’ βˆƒπ‘– ∈ dom (iEdgβ€˜πΊ){𝐡, 𝐢} = ((iEdgβ€˜πΊ)β€˜π‘–))))
1093imp 1108 . 2 ((𝐺 ∈ UMGraph ∧ {𝐴, 𝐡} ∈ 𝐸 ∧ {𝐡, 𝐢} ∈ 𝐸) β†’ βˆƒπ‘– ∈ dom (iEdgβ€˜πΊ){𝐡, 𝐢} = ((iEdgβ€˜πΊ)β€˜π‘–))
112eleq2i 2819 . . . . . . 7 ({𝐴, 𝐡} ∈ 𝐸 ↔ {𝐴, 𝐡} ∈ (Edgβ€˜πΊ))
124uhgredgiedgb 28894 . . . . . . 7 (𝐺 ∈ UHGraph β†’ ({𝐴, 𝐡} ∈ (Edgβ€˜πΊ) ↔ βˆƒπ‘— ∈ dom (iEdgβ€˜πΊ){𝐴, 𝐡} = ((iEdgβ€˜πΊ)β€˜π‘—)))
1311, 12bitrid 283 . . . . . 6 (𝐺 ∈ UHGraph β†’ ({𝐴, 𝐡} ∈ 𝐸 ↔ βˆƒπ‘— ∈ dom (iEdgβ€˜πΊ){𝐴, 𝐡} = ((iEdgβ€˜πΊ)β€˜π‘—)))
141, 13syl 17 . . . . 5 (𝐺 ∈ UMGraph β†’ ({𝐴, 𝐡} ∈ 𝐸 ↔ βˆƒπ‘— ∈ dom (iEdgβ€˜πΊ){𝐴, 𝐡} = ((iEdgβ€˜πΊ)β€˜π‘—)))
1514biimpd 228 . . . 4 (𝐺 ∈ UMGraph β†’ ({𝐴, 𝐡} ∈ 𝐸 β†’ βˆƒπ‘— ∈ dom (iEdgβ€˜πΊ){𝐴, 𝐡} = ((iEdgβ€˜πΊ)β€˜π‘—)))
1615a1dd 50 . . 3 (𝐺 ∈ UMGraph β†’ ({𝐴, 𝐡} ∈ 𝐸 β†’ ({𝐡, 𝐢} ∈ 𝐸 β†’ βˆƒπ‘— ∈ dom (iEdgβ€˜πΊ){𝐴, 𝐡} = ((iEdgβ€˜πΊ)β€˜π‘—))))
17163imp 1108 . 2 ((𝐺 ∈ UMGraph ∧ {𝐴, 𝐡} ∈ 𝐸 ∧ {𝐡, 𝐢} ∈ 𝐸) β†’ βˆƒπ‘— ∈ dom (iEdgβ€˜πΊ){𝐴, 𝐡} = ((iEdgβ€˜πΊ)β€˜π‘—))
18 s2cli 14837 . . . . . . . . . 10 βŸ¨β€œπ‘—π‘–β€βŸ© ∈ Word V
19 s3cli 14838 . . . . . . . . . 10 βŸ¨β€œπ΄π΅πΆβ€βŸ© ∈ Word V
2018, 19pm3.2i 470 . . . . . . . . 9 (βŸ¨β€œπ‘—π‘–β€βŸ© ∈ Word V ∧ βŸ¨β€œπ΄π΅πΆβ€βŸ© ∈ Word V)
21 eqid 2726 . . . . . . . . . 10 βŸ¨β€œπ‘—π‘–β€βŸ© = βŸ¨β€œπ‘—π‘–β€βŸ©
22 eqid 2726 . . . . . . . . . 10 βŸ¨β€œπ΄π΅πΆβ€βŸ© = βŸ¨β€œπ΄π΅πΆβ€βŸ©
23 simpl1 1188 . . . . . . . . . 10 (((𝐺 ∈ UMGraph ∧ {𝐴, 𝐡} ∈ 𝐸 ∧ {𝐡, 𝐢} ∈ 𝐸) ∧ ({𝐴, 𝐡} = ((iEdgβ€˜πΊ)β€˜π‘—) ∧ {𝐡, 𝐢} = ((iEdgβ€˜πΊ)β€˜π‘–))) β†’ 𝐺 ∈ UMGraph)
24 3simpc 1147 . . . . . . . . . . 11 ((𝐺 ∈ UMGraph ∧ {𝐴, 𝐡} ∈ 𝐸 ∧ {𝐡, 𝐢} ∈ 𝐸) β†’ ({𝐴, 𝐡} ∈ 𝐸 ∧ {𝐡, 𝐢} ∈ 𝐸))
2524adantr 480 . . . . . . . . . 10 (((𝐺 ∈ UMGraph ∧ {𝐴, 𝐡} ∈ 𝐸 ∧ {𝐡, 𝐢} ∈ 𝐸) ∧ ({𝐴, 𝐡} = ((iEdgβ€˜πΊ)β€˜π‘—) ∧ {𝐡, 𝐢} = ((iEdgβ€˜πΊ)β€˜π‘–))) β†’ ({𝐴, 𝐡} ∈ 𝐸 ∧ {𝐡, 𝐢} ∈ 𝐸))
26 simpl 482 . . . . . . . . . . . 12 (({𝐴, 𝐡} = ((iEdgβ€˜πΊ)β€˜π‘—) ∧ {𝐡, 𝐢} = ((iEdgβ€˜πΊ)β€˜π‘–)) β†’ {𝐴, 𝐡} = ((iEdgβ€˜πΊ)β€˜π‘—))
2726eqcomd 2732 . . . . . . . . . . 11 (({𝐴, 𝐡} = ((iEdgβ€˜πΊ)β€˜π‘—) ∧ {𝐡, 𝐢} = ((iEdgβ€˜πΊ)β€˜π‘–)) β†’ ((iEdgβ€˜πΊ)β€˜π‘—) = {𝐴, 𝐡})
2827adantl 481 . . . . . . . . . 10 (((𝐺 ∈ UMGraph ∧ {𝐴, 𝐡} ∈ 𝐸 ∧ {𝐡, 𝐢} ∈ 𝐸) ∧ ({𝐴, 𝐡} = ((iEdgβ€˜πΊ)β€˜π‘—) ∧ {𝐡, 𝐢} = ((iEdgβ€˜πΊ)β€˜π‘–))) β†’ ((iEdgβ€˜πΊ)β€˜π‘—) = {𝐴, 𝐡})
29 simpr 484 . . . . . . . . . . . 12 (({𝐴, 𝐡} = ((iEdgβ€˜πΊ)β€˜π‘—) ∧ {𝐡, 𝐢} = ((iEdgβ€˜πΊ)β€˜π‘–)) β†’ {𝐡, 𝐢} = ((iEdgβ€˜πΊ)β€˜π‘–))
3029eqcomd 2732 . . . . . . . . . . 11 (({𝐴, 𝐡} = ((iEdgβ€˜πΊ)β€˜π‘—) ∧ {𝐡, 𝐢} = ((iEdgβ€˜πΊ)β€˜π‘–)) β†’ ((iEdgβ€˜πΊ)β€˜π‘–) = {𝐡, 𝐢})
3130adantl 481 . . . . . . . . . 10 (((𝐺 ∈ UMGraph ∧ {𝐴, 𝐡} ∈ 𝐸 ∧ {𝐡, 𝐢} ∈ 𝐸) ∧ ({𝐴, 𝐡} = ((iEdgβ€˜πΊ)β€˜π‘—) ∧ {𝐡, 𝐢} = ((iEdgβ€˜πΊ)β€˜π‘–))) β†’ ((iEdgβ€˜πΊ)β€˜π‘–) = {𝐡, 𝐢})
322, 4, 21, 22, 23, 25, 28, 31umgr2adedgwlk 29708 . . . . . . . . 9 (((𝐺 ∈ UMGraph ∧ {𝐴, 𝐡} ∈ 𝐸 ∧ {𝐡, 𝐢} ∈ 𝐸) ∧ ({𝐴, 𝐡} = ((iEdgβ€˜πΊ)β€˜π‘—) ∧ {𝐡, 𝐢} = ((iEdgβ€˜πΊ)β€˜π‘–))) β†’ (βŸ¨β€œπ‘—π‘–β€βŸ©(Walksβ€˜πΊ)βŸ¨β€œπ΄π΅πΆβ€βŸ© ∧ (β™―β€˜βŸ¨β€œπ‘—π‘–β€βŸ©) = 2 ∧ (𝐴 = (βŸ¨β€œπ΄π΅πΆβ€βŸ©β€˜0) ∧ 𝐡 = (βŸ¨β€œπ΄π΅πΆβ€βŸ©β€˜1) ∧ 𝐢 = (βŸ¨β€œπ΄π΅πΆβ€βŸ©β€˜2))))
33 breq12 5146 . . . . . . . . . . 11 ((𝑓 = βŸ¨β€œπ‘—π‘–β€βŸ© ∧ 𝑝 = βŸ¨β€œπ΄π΅πΆβ€βŸ©) β†’ (𝑓(Walksβ€˜πΊ)𝑝 ↔ βŸ¨β€œπ‘—π‘–β€βŸ©(Walksβ€˜πΊ)βŸ¨β€œπ΄π΅πΆβ€βŸ©))
34 fveqeq2 6894 . . . . . . . . . . . 12 (𝑓 = βŸ¨β€œπ‘—π‘–β€βŸ© β†’ ((β™―β€˜π‘“) = 2 ↔ (β™―β€˜βŸ¨β€œπ‘—π‘–β€βŸ©) = 2))
3534adantr 480 . . . . . . . . . . 11 ((𝑓 = βŸ¨β€œπ‘—π‘–β€βŸ© ∧ 𝑝 = βŸ¨β€œπ΄π΅πΆβ€βŸ©) β†’ ((β™―β€˜π‘“) = 2 ↔ (β™―β€˜βŸ¨β€œπ‘—π‘–β€βŸ©) = 2))
36 fveq1 6884 . . . . . . . . . . . . . 14 (𝑝 = βŸ¨β€œπ΄π΅πΆβ€βŸ© β†’ (π‘β€˜0) = (βŸ¨β€œπ΄π΅πΆβ€βŸ©β€˜0))
3736eqeq2d 2737 . . . . . . . . . . . . 13 (𝑝 = βŸ¨β€œπ΄π΅πΆβ€βŸ© β†’ (𝐴 = (π‘β€˜0) ↔ 𝐴 = (βŸ¨β€œπ΄π΅πΆβ€βŸ©β€˜0)))
38 fveq1 6884 . . . . . . . . . . . . . 14 (𝑝 = βŸ¨β€œπ΄π΅πΆβ€βŸ© β†’ (π‘β€˜1) = (βŸ¨β€œπ΄π΅πΆβ€βŸ©β€˜1))
3938eqeq2d 2737 . . . . . . . . . . . . 13 (𝑝 = βŸ¨β€œπ΄π΅πΆβ€βŸ© β†’ (𝐡 = (π‘β€˜1) ↔ 𝐡 = (βŸ¨β€œπ΄π΅πΆβ€βŸ©β€˜1)))
40 fveq1 6884 . . . . . . . . . . . . . 14 (𝑝 = βŸ¨β€œπ΄π΅πΆβ€βŸ© β†’ (π‘β€˜2) = (βŸ¨β€œπ΄π΅πΆβ€βŸ©β€˜2))
4140eqeq2d 2737 . . . . . . . . . . . . 13 (𝑝 = βŸ¨β€œπ΄π΅πΆβ€βŸ© β†’ (𝐢 = (π‘β€˜2) ↔ 𝐢 = (βŸ¨β€œπ΄π΅πΆβ€βŸ©β€˜2)))
4237, 39, 413anbi123d 1432 . . . . . . . . . . . 12 (𝑝 = βŸ¨β€œπ΄π΅πΆβ€βŸ© β†’ ((𝐴 = (π‘β€˜0) ∧ 𝐡 = (π‘β€˜1) ∧ 𝐢 = (π‘β€˜2)) ↔ (𝐴 = (βŸ¨β€œπ΄π΅πΆβ€βŸ©β€˜0) ∧ 𝐡 = (βŸ¨β€œπ΄π΅πΆβ€βŸ©β€˜1) ∧ 𝐢 = (βŸ¨β€œπ΄π΅πΆβ€βŸ©β€˜2))))
4342adantl 481 . . . . . . . . . . 11 ((𝑓 = βŸ¨β€œπ‘—π‘–β€βŸ© ∧ 𝑝 = βŸ¨β€œπ΄π΅πΆβ€βŸ©) β†’ ((𝐴 = (π‘β€˜0) ∧ 𝐡 = (π‘β€˜1) ∧ 𝐢 = (π‘β€˜2)) ↔ (𝐴 = (βŸ¨β€œπ΄π΅πΆβ€βŸ©β€˜0) ∧ 𝐡 = (βŸ¨β€œπ΄π΅πΆβ€βŸ©β€˜1) ∧ 𝐢 = (βŸ¨β€œπ΄π΅πΆβ€βŸ©β€˜2))))
4433, 35, 433anbi123d 1432 . . . . . . . . . 10 ((𝑓 = βŸ¨β€œπ‘—π‘–β€βŸ© ∧ 𝑝 = βŸ¨β€œπ΄π΅πΆβ€βŸ©) β†’ ((𝑓(Walksβ€˜πΊ)𝑝 ∧ (β™―β€˜π‘“) = 2 ∧ (𝐴 = (π‘β€˜0) ∧ 𝐡 = (π‘β€˜1) ∧ 𝐢 = (π‘β€˜2))) ↔ (βŸ¨β€œπ‘—π‘–β€βŸ©(Walksβ€˜πΊ)βŸ¨β€œπ΄π΅πΆβ€βŸ© ∧ (β™―β€˜βŸ¨β€œπ‘—π‘–β€βŸ©) = 2 ∧ (𝐴 = (βŸ¨β€œπ΄π΅πΆβ€βŸ©β€˜0) ∧ 𝐡 = (βŸ¨β€œπ΄π΅πΆβ€βŸ©β€˜1) ∧ 𝐢 = (βŸ¨β€œπ΄π΅πΆβ€βŸ©β€˜2)))))
4544spc2egv 3583 . . . . . . . . 9 ((βŸ¨β€œπ‘—π‘–β€βŸ© ∈ Word V ∧ βŸ¨β€œπ΄π΅πΆβ€βŸ© ∈ Word V) β†’ ((βŸ¨β€œπ‘—π‘–β€βŸ©(Walksβ€˜πΊ)βŸ¨β€œπ΄π΅πΆβ€βŸ© ∧ (β™―β€˜βŸ¨β€œπ‘—π‘–β€βŸ©) = 2 ∧ (𝐴 = (βŸ¨β€œπ΄π΅πΆβ€βŸ©β€˜0) ∧ 𝐡 = (βŸ¨β€œπ΄π΅πΆβ€βŸ©β€˜1) ∧ 𝐢 = (βŸ¨β€œπ΄π΅πΆβ€βŸ©β€˜2))) β†’ βˆƒπ‘“βˆƒπ‘(𝑓(Walksβ€˜πΊ)𝑝 ∧ (β™―β€˜π‘“) = 2 ∧ (𝐴 = (π‘β€˜0) ∧ 𝐡 = (π‘β€˜1) ∧ 𝐢 = (π‘β€˜2)))))
4620, 32, 45mpsyl 68 . . . . . . . 8 (((𝐺 ∈ UMGraph ∧ {𝐴, 𝐡} ∈ 𝐸 ∧ {𝐡, 𝐢} ∈ 𝐸) ∧ ({𝐴, 𝐡} = ((iEdgβ€˜πΊ)β€˜π‘—) ∧ {𝐡, 𝐢} = ((iEdgβ€˜πΊ)β€˜π‘–))) β†’ βˆƒπ‘“βˆƒπ‘(𝑓(Walksβ€˜πΊ)𝑝 ∧ (β™―β€˜π‘“) = 2 ∧ (𝐴 = (π‘β€˜0) ∧ 𝐡 = (π‘β€˜1) ∧ 𝐢 = (π‘β€˜2))))
4746exp32 420 . . . . . . 7 ((𝐺 ∈ UMGraph ∧ {𝐴, 𝐡} ∈ 𝐸 ∧ {𝐡, 𝐢} ∈ 𝐸) β†’ ({𝐴, 𝐡} = ((iEdgβ€˜πΊ)β€˜π‘—) β†’ ({𝐡, 𝐢} = ((iEdgβ€˜πΊ)β€˜π‘–) β†’ βˆƒπ‘“βˆƒπ‘(𝑓(Walksβ€˜πΊ)𝑝 ∧ (β™―β€˜π‘“) = 2 ∧ (𝐴 = (π‘β€˜0) ∧ 𝐡 = (π‘β€˜1) ∧ 𝐢 = (π‘β€˜2))))))
4847com12 32 . . . . . 6 ({𝐴, 𝐡} = ((iEdgβ€˜πΊ)β€˜π‘—) β†’ ((𝐺 ∈ UMGraph ∧ {𝐴, 𝐡} ∈ 𝐸 ∧ {𝐡, 𝐢} ∈ 𝐸) β†’ ({𝐡, 𝐢} = ((iEdgβ€˜πΊ)β€˜π‘–) β†’ βˆƒπ‘“βˆƒπ‘(𝑓(Walksβ€˜πΊ)𝑝 ∧ (β™―β€˜π‘“) = 2 ∧ (𝐴 = (π‘β€˜0) ∧ 𝐡 = (π‘β€˜1) ∧ 𝐢 = (π‘β€˜2))))))
4948rexlimivw 3145 . . . . 5 (βˆƒπ‘— ∈ dom (iEdgβ€˜πΊ){𝐴, 𝐡} = ((iEdgβ€˜πΊ)β€˜π‘—) β†’ ((𝐺 ∈ UMGraph ∧ {𝐴, 𝐡} ∈ 𝐸 ∧ {𝐡, 𝐢} ∈ 𝐸) β†’ ({𝐡, 𝐢} = ((iEdgβ€˜πΊ)β€˜π‘–) β†’ βˆƒπ‘“βˆƒπ‘(𝑓(Walksβ€˜πΊ)𝑝 ∧ (β™―β€˜π‘“) = 2 ∧ (𝐴 = (π‘β€˜0) ∧ 𝐡 = (π‘β€˜1) ∧ 𝐢 = (π‘β€˜2))))))
5049com13 88 . . . 4 ({𝐡, 𝐢} = ((iEdgβ€˜πΊ)β€˜π‘–) β†’ ((𝐺 ∈ UMGraph ∧ {𝐴, 𝐡} ∈ 𝐸 ∧ {𝐡, 𝐢} ∈ 𝐸) β†’ (βˆƒπ‘— ∈ dom (iEdgβ€˜πΊ){𝐴, 𝐡} = ((iEdgβ€˜πΊ)β€˜π‘—) β†’ βˆƒπ‘“βˆƒπ‘(𝑓(Walksβ€˜πΊ)𝑝 ∧ (β™―β€˜π‘“) = 2 ∧ (𝐴 = (π‘β€˜0) ∧ 𝐡 = (π‘β€˜1) ∧ 𝐢 = (π‘β€˜2))))))
5150rexlimivw 3145 . . 3 (βˆƒπ‘– ∈ dom (iEdgβ€˜πΊ){𝐡, 𝐢} = ((iEdgβ€˜πΊ)β€˜π‘–) β†’ ((𝐺 ∈ UMGraph ∧ {𝐴, 𝐡} ∈ 𝐸 ∧ {𝐡, 𝐢} ∈ 𝐸) β†’ (βˆƒπ‘— ∈ dom (iEdgβ€˜πΊ){𝐴, 𝐡} = ((iEdgβ€˜πΊ)β€˜π‘—) β†’ βˆƒπ‘“βˆƒπ‘(𝑓(Walksβ€˜πΊ)𝑝 ∧ (β™―β€˜π‘“) = 2 ∧ (𝐴 = (π‘β€˜0) ∧ 𝐡 = (π‘β€˜1) ∧ 𝐢 = (π‘β€˜2))))))
5251com12 32 . 2 ((𝐺 ∈ UMGraph ∧ {𝐴, 𝐡} ∈ 𝐸 ∧ {𝐡, 𝐢} ∈ 𝐸) β†’ (βˆƒπ‘– ∈ dom (iEdgβ€˜πΊ){𝐡, 𝐢} = ((iEdgβ€˜πΊ)β€˜π‘–) β†’ (βˆƒπ‘— ∈ dom (iEdgβ€˜πΊ){𝐴, 𝐡} = ((iEdgβ€˜πΊ)β€˜π‘—) β†’ βˆƒπ‘“βˆƒπ‘(𝑓(Walksβ€˜πΊ)𝑝 ∧ (β™―β€˜π‘“) = 2 ∧ (𝐴 = (π‘β€˜0) ∧ 𝐡 = (π‘β€˜1) ∧ 𝐢 = (π‘β€˜2))))))
5310, 17, 52mp2d 49 1 ((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))))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 395   ∧ w3a 1084   = wceq 1533  βˆƒwex 1773   ∈ wcel 2098  βˆƒwrex 3064  Vcvv 3468  {cpr 4625   class class class wbr 5141  dom cdm 5669  β€˜cfv 6537  0cc0 11112  1c1 11113  2c2 12271  β™―chash 14295  Word cword 14470  βŸ¨β€œcs2 14798  βŸ¨β€œcs3 14799  iEdgciedg 28765  Edgcedg 28815  UHGraphcuhgr 28824  UMGraphcumgr 28849  Walkscwlks 29362
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7722  ax-cnex 11168  ax-resscn 11169  ax-1cn 11170  ax-icn 11171  ax-addcl 11172  ax-addrcl 11173  ax-mulcl 11174  ax-mulrcl 11175  ax-mulcom 11176  ax-addass 11177  ax-mulass 11178  ax-distr 11179  ax-i2m1 11180  ax-1ne0 11181  ax-1rid 11182  ax-rnegex 11183  ax-rrecex 11184  ax-cnre 11185  ax-pre-lttri 11186  ax-pre-lttrn 11187  ax-pre-ltadd 11188  ax-pre-mulgt0 11189
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-ifp 1060  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-nel 3041  df-ral 3056  df-rex 3065  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-pss 3962  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-tp 4628  df-op 4630  df-uni 4903  df-int 4944  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-tr 5259  df-id 5567  df-eprel 5573  df-po 5581  df-so 5582  df-fr 5624  df-we 5626  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-pred 6294  df-ord 6361  df-on 6362  df-lim 6363  df-suc 6364  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7361  df-ov 7408  df-oprab 7409  df-mpo 7410  df-om 7853  df-1st 7974  df-2nd 7975  df-frecs 8267  df-wrecs 8298  df-recs 8372  df-rdg 8411  df-1o 8467  df-oadd 8471  df-er 8705  df-map 8824  df-en 8942  df-dom 8943  df-sdom 8944  df-fin 8945  df-dju 9898  df-card 9936  df-pnf 11254  df-mnf 11255  df-xr 11256  df-ltxr 11257  df-le 11258  df-sub 11450  df-neg 11451  df-nn 12217  df-2 12279  df-3 12280  df-n0 12477  df-z 12563  df-uz 12827  df-fz 13491  df-fzo 13634  df-hash 14296  df-word 14471  df-concat 14527  df-s1 14552  df-s2 14805  df-s3 14806  df-edg 28816  df-uhgr 28826  df-upgr 28850  df-umgr 28851  df-wlks 29365
This theorem is referenced by:  umgr2wlkon  29713  umgrwwlks2on  29720
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