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Theorem umgr2wlk 29788
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 28945 . . . . . 6 (𝐺 ∈ UMGraph β†’ 𝐺 ∈ UHGraph)
2 umgr2wlk.e . . . . . . . 8 𝐸 = (Edgβ€˜πΊ)
32eleq2i 2821 . . . . . . 7 ({𝐡, 𝐢} ∈ 𝐸 ↔ {𝐡, 𝐢} ∈ (Edgβ€˜πΊ))
4 eqid 2728 . . . . . . . 8 (iEdgβ€˜πΊ) = (iEdgβ€˜πΊ)
54uhgredgiedgb 28967 . . . . . . 7 (𝐺 ∈ UHGraph β†’ ({𝐡, 𝐢} ∈ (Edgβ€˜πΊ) ↔ βˆƒπ‘– ∈ dom (iEdgβ€˜πΊ){𝐡, 𝐢} = ((iEdgβ€˜πΊ)β€˜π‘–)))
63, 5bitrid 282 . . . . . 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 2821 . . . . . . 7 ({𝐴, 𝐡} ∈ 𝐸 ↔ {𝐴, 𝐡} ∈ (Edgβ€˜πΊ))
124uhgredgiedgb 28967 . . . . . . 7 (𝐺 ∈ UHGraph β†’ ({𝐴, 𝐡} ∈ (Edgβ€˜πΊ) ↔ βˆƒπ‘— ∈ dom (iEdgβ€˜πΊ){𝐴, 𝐡} = ((iEdgβ€˜πΊ)β€˜π‘—)))
1311, 12bitrid 282 . . . . . 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 14873 . . . . . . . . . 10 βŸ¨β€œπ‘—π‘–β€βŸ© ∈ Word V
19 s3cli 14874 . . . . . . . . . 10 βŸ¨β€œπ΄π΅πΆβ€βŸ© ∈ Word V
2018, 19pm3.2i 469 . . . . . . . . 9 (βŸ¨β€œπ‘—π‘–β€βŸ© ∈ Word V ∧ βŸ¨β€œπ΄π΅πΆβ€βŸ© ∈ Word V)
21 eqid 2728 . . . . . . . . . 10 βŸ¨β€œπ‘—π‘–β€βŸ© = βŸ¨β€œπ‘—π‘–β€βŸ©
22 eqid 2728 . . . . . . . . . 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 479 . . . . . . . . . 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 481 . . . . . . . . . . . 12 (({𝐴, 𝐡} = ((iEdgβ€˜πΊ)β€˜π‘—) ∧ {𝐡, 𝐢} = ((iEdgβ€˜πΊ)β€˜π‘–)) β†’ {𝐴, 𝐡} = ((iEdgβ€˜πΊ)β€˜π‘—))
2726eqcomd 2734 . . . . . . . . . . 11 (({𝐴, 𝐡} = ((iEdgβ€˜πΊ)β€˜π‘—) ∧ {𝐡, 𝐢} = ((iEdgβ€˜πΊ)β€˜π‘–)) β†’ ((iEdgβ€˜πΊ)β€˜π‘—) = {𝐴, 𝐡})
2827adantl 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†’ ((iEdgβ€˜πΊ)β€˜π‘—) = {𝐴, 𝐡})
29 simpr 483 . . . . . . . . . . . 12 (({𝐴, 𝐡} = ((iEdgβ€˜πΊ)β€˜π‘—) ∧ {𝐡, 𝐢} = ((iEdgβ€˜πΊ)β€˜π‘–)) β†’ {𝐡, 𝐢} = ((iEdgβ€˜πΊ)β€˜π‘–))
3029eqcomd 2734 . . . . . . . . . . 11 (({𝐴, 𝐡} = ((iEdgβ€˜πΊ)β€˜π‘—) ∧ {𝐡, 𝐢} = ((iEdgβ€˜πΊ)β€˜π‘–)) β†’ ((iEdgβ€˜πΊ)β€˜π‘–) = {𝐡, 𝐢})
3130adantl 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†’ ((iEdgβ€˜πΊ)β€˜π‘–) = {𝐡, 𝐢})
322, 4, 21, 22, 23, 25, 28, 31umgr2adedgwlk 29784 . . . . . . . . 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 5157 . . . . . . . . . . 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 6911 . . . . . . . . . . . 12 (𝑓 = βŸ¨β€œπ‘—π‘–β€βŸ© β†’ ((β™―β€˜π‘“) = 2 ↔ (β™―β€˜βŸ¨β€œπ‘—π‘–β€βŸ©) = 2))
3534adantr 479 . . . . . . . . . . 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 6901 . . . . . . . . . . . . . 14 (𝑝 = βŸ¨β€œπ΄π΅πΆβ€βŸ© β†’ (π‘β€˜0) = (βŸ¨β€œπ΄π΅πΆβ€βŸ©β€˜0))
3736eqeq2d 2739 . . . . . . . . . . . . 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 6901 . . . . . . . . . . . . . 14 (𝑝 = βŸ¨β€œπ΄π΅πΆβ€βŸ© β†’ (π‘β€˜1) = (βŸ¨β€œπ΄π΅πΆβ€βŸ©β€˜1))
3938eqeq2d 2739 . . . . . . . . . . . . 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 6901 . . . . . . . . . . . . . 14 (𝑝 = βŸ¨β€œπ΄π΅πΆβ€βŸ© β†’ (π‘β€˜2) = (βŸ¨β€œπ΄π΅πΆβ€βŸ©β€˜2))
4140eqeq2d 2739 . . . . . . . . . . . . 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 480 . . . . . . . . . . 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 3588 . . . . . . . . 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 419 . . . . . . 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 3148 . . . . 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 3148 . . 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 394   ∧ w3a 1084   = wceq 1533  βˆƒwex 1773   ∈ wcel 2098  βˆƒwrex 3067  Vcvv 3473  {cpr 4634   class class class wbr 5152  dom cdm 5682  β€˜cfv 6553  0cc0 11148  1c1 11149  2c2 12307  β™―chash 14331  Word cword 14506  βŸ¨β€œcs2 14834  βŸ¨β€œcs3 14835  iEdgciedg 28838  Edgcedg 28888  UHGraphcuhgr 28897  UMGraphcumgr 28922  Walkscwlks 29438
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 2166  ax-ext 2699  ax-rep 5289  ax-sep 5303  ax-nul 5310  ax-pow 5369  ax-pr 5433  ax-un 7748  ax-cnex 11204  ax-resscn 11205  ax-1cn 11206  ax-icn 11207  ax-addcl 11208  ax-addrcl 11209  ax-mulcl 11210  ax-mulrcl 11211  ax-mulcom 11212  ax-addass 11213  ax-mulass 11214  ax-distr 11215  ax-i2m1 11216  ax-1ne0 11217  ax-1rid 11218  ax-rnegex 11219  ax-rrecex 11220  ax-cnre 11221  ax-pre-lttri 11222  ax-pre-lttrn 11223  ax-pre-ltadd 11224  ax-pre-mulgt0 11225
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-ifp 1061  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-reu 3375  df-rab 3431  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-tp 4637  df-op 4639  df-uni 4913  df-int 4954  df-iun 5002  df-br 5153  df-opab 5215  df-mpt 5236  df-tr 5270  df-id 5580  df-eprel 5586  df-po 5594  df-so 5595  df-fr 5637  df-we 5639  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-pred 6310  df-ord 6377  df-on 6378  df-lim 6379  df-suc 6380  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-fo 6559  df-f1o 6560  df-fv 6561  df-riota 7382  df-ov 7429  df-oprab 7430  df-mpo 7431  df-om 7879  df-1st 8001  df-2nd 8002  df-frecs 8295  df-wrecs 8326  df-recs 8400  df-rdg 8439  df-1o 8495  df-oadd 8499  df-er 8733  df-map 8855  df-en 8973  df-dom 8974  df-sdom 8975  df-fin 8976  df-dju 9934  df-card 9972  df-pnf 11290  df-mnf 11291  df-xr 11292  df-ltxr 11293  df-le 11294  df-sub 11486  df-neg 11487  df-nn 12253  df-2 12315  df-3 12316  df-n0 12513  df-z 12599  df-uz 12863  df-fz 13527  df-fzo 13670  df-hash 14332  df-word 14507  df-concat 14563  df-s1 14588  df-s2 14841  df-s3 14842  df-edg 28889  df-uhgr 28899  df-upgr 28923  df-umgr 28924  df-wlks 29441
This theorem is referenced by:  umgr2wlkon  29789  umgrwwlks2on  29796
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