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Theorem umgr2wlk 28958
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 28119 . . . . . 6 (𝐺 ∈ UMGraph β†’ 𝐺 ∈ UHGraph)
2 umgr2wlk.e . . . . . . . 8 𝐸 = (Edgβ€˜πΊ)
32eleq2i 2825 . . . . . . 7 ({𝐡, 𝐢} ∈ 𝐸 ↔ {𝐡, 𝐢} ∈ (Edgβ€˜πΊ))
4 eqid 2732 . . . . . . . 8 (iEdgβ€˜πΊ) = (iEdgβ€˜πΊ)
54uhgredgiedgb 28141 . . . . . . 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 1112 . 2 ((𝐺 ∈ UMGraph ∧ {𝐴, 𝐡} ∈ 𝐸 ∧ {𝐡, 𝐢} ∈ 𝐸) β†’ βˆƒπ‘– ∈ dom (iEdgβ€˜πΊ){𝐡, 𝐢} = ((iEdgβ€˜πΊ)β€˜π‘–))
112eleq2i 2825 . . . . . . 7 ({𝐴, 𝐡} ∈ 𝐸 ↔ {𝐴, 𝐡} ∈ (Edgβ€˜πΊ))
124uhgredgiedgb 28141 . . . . . . 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 1112 . 2 ((𝐺 ∈ UMGraph ∧ {𝐴, 𝐡} ∈ 𝐸 ∧ {𝐡, 𝐢} ∈ 𝐸) β†’ βˆƒπ‘— ∈ dom (iEdgβ€˜πΊ){𝐴, 𝐡} = ((iEdgβ€˜πΊ)β€˜π‘—))
18 s2cli 14782 . . . . . . . . . 10 βŸ¨β€œπ‘—π‘–β€βŸ© ∈ Word V
19 s3cli 14783 . . . . . . . . . 10 βŸ¨β€œπ΄π΅πΆβ€βŸ© ∈ Word V
2018, 19pm3.2i 472 . . . . . . . . 9 (βŸ¨β€œπ‘—π‘–β€βŸ© ∈ Word V ∧ βŸ¨β€œπ΄π΅πΆβ€βŸ© ∈ Word V)
21 eqid 2732 . . . . . . . . . 10 βŸ¨β€œπ‘—π‘–β€βŸ© = βŸ¨β€œπ‘—π‘–β€βŸ©
22 eqid 2732 . . . . . . . . . 10 βŸ¨β€œπ΄π΅πΆβ€βŸ© = βŸ¨β€œπ΄π΅πΆβ€βŸ©
23 simpl1 1192 . . . . . . . . . 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 1151 . . . . . . . . . . 11 ((𝐺 ∈ UMGraph ∧ {𝐴, 𝐡} ∈ 𝐸 ∧ {𝐡, 𝐢} ∈ 𝐸) β†’ ({𝐴, 𝐡} ∈ 𝐸 ∧ {𝐡, 𝐢} ∈ 𝐸))
2524adantr 482 . . . . . . . . . 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 484 . . . . . . . . . . . 12 (({𝐴, 𝐡} = ((iEdgβ€˜πΊ)β€˜π‘—) ∧ {𝐡, 𝐢} = ((iEdgβ€˜πΊ)β€˜π‘–)) β†’ {𝐴, 𝐡} = ((iEdgβ€˜πΊ)β€˜π‘—))
2726eqcomd 2738 . . . . . . . . . . 11 (({𝐴, 𝐡} = ((iEdgβ€˜πΊ)β€˜π‘—) ∧ {𝐡, 𝐢} = ((iEdgβ€˜πΊ)β€˜π‘–)) β†’ ((iEdgβ€˜πΊ)β€˜π‘—) = {𝐴, 𝐡})
2827adantl 483 . . . . . . . . . 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 486 . . . . . . . . . . . 12 (({𝐴, 𝐡} = ((iEdgβ€˜πΊ)β€˜π‘—) ∧ {𝐡, 𝐢} = ((iEdgβ€˜πΊ)β€˜π‘–)) β†’ {𝐡, 𝐢} = ((iEdgβ€˜πΊ)β€˜π‘–))
3029eqcomd 2738 . . . . . . . . . . 11 (({𝐴, 𝐡} = ((iEdgβ€˜πΊ)β€˜π‘—) ∧ {𝐡, 𝐢} = ((iEdgβ€˜πΊ)β€˜π‘–)) β†’ ((iEdgβ€˜πΊ)β€˜π‘–) = {𝐡, 𝐢})
3130adantl 483 . . . . . . . . . 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 28954 . . . . . . . . 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 5116 . . . . . . . . . . 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 6857 . . . . . . . . . . . 12 (𝑓 = βŸ¨β€œπ‘—π‘–β€βŸ© β†’ ((β™―β€˜π‘“) = 2 ↔ (β™―β€˜βŸ¨β€œπ‘—π‘–β€βŸ©) = 2))
3534adantr 482 . . . . . . . . . . 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 6847 . . . . . . . . . . . . . 14 (𝑝 = βŸ¨β€œπ΄π΅πΆβ€βŸ© β†’ (π‘β€˜0) = (βŸ¨β€œπ΄π΅πΆβ€βŸ©β€˜0))
3736eqeq2d 2743 . . . . . . . . . . . . 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 6847 . . . . . . . . . . . . . 14 (𝑝 = βŸ¨β€œπ΄π΅πΆβ€βŸ© β†’ (π‘β€˜1) = (βŸ¨β€œπ΄π΅πΆβ€βŸ©β€˜1))
3938eqeq2d 2743 . . . . . . . . . . . . 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 6847 . . . . . . . . . . . . . 14 (𝑝 = βŸ¨β€œπ΄π΅πΆβ€βŸ© β†’ (π‘β€˜2) = (βŸ¨β€œπ΄π΅πΆβ€βŸ©β€˜2))
4140eqeq2d 2743 . . . . . . . . . . . . 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 1437 . . . . . . . . . . . 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 483 . . . . . . . . . . 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 1437 . . . . . . . . . 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 3560 . . . . . . . . 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 422 . . . . . . 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 397   ∧ w3a 1088   = wceq 1542  βˆƒwex 1782   ∈ wcel 2107  βˆƒwrex 3070  Vcvv 3447  {cpr 4594   class class class wbr 5111  dom cdm 5639  β€˜cfv 6502  0cc0 11061  1c1 11062  2c2 12218  β™―chash 14241  Word cword 14415  βŸ¨β€œcs2 14743  βŸ¨β€œcs3 14744  iEdgciedg 28012  Edgcedg 28062  UHGraphcuhgr 28071  UMGraphcumgr 28096  Walkscwlks 28608
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2703  ax-rep 5248  ax-sep 5262  ax-nul 5269  ax-pow 5326  ax-pr 5390  ax-un 7678  ax-cnex 11117  ax-resscn 11118  ax-1cn 11119  ax-icn 11120  ax-addcl 11121  ax-addrcl 11122  ax-mulcl 11123  ax-mulrcl 11124  ax-mulcom 11125  ax-addass 11126  ax-mulass 11127  ax-distr 11128  ax-i2m1 11129  ax-1ne0 11130  ax-1rid 11131  ax-rnegex 11132  ax-rrecex 11133  ax-cnre 11134  ax-pre-lttri 11135  ax-pre-lttrn 11136  ax-pre-ltadd 11137  ax-pre-mulgt0 11138
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-ifp 1063  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-reu 3353  df-rab 3407  df-v 3449  df-sbc 3744  df-csb 3860  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3933  df-nul 4289  df-if 4493  df-pw 4568  df-sn 4593  df-pr 4595  df-tp 4597  df-op 4599  df-uni 4872  df-int 4914  df-iun 4962  df-br 5112  df-opab 5174  df-mpt 5195  df-tr 5229  df-id 5537  df-eprel 5543  df-po 5551  df-so 5552  df-fr 5594  df-we 5596  df-xp 5645  df-rel 5646  df-cnv 5647  df-co 5648  df-dm 5649  df-rn 5650  df-res 5651  df-ima 5652  df-pred 6259  df-ord 6326  df-on 6327  df-lim 6328  df-suc 6329  df-iota 6454  df-fun 6504  df-fn 6505  df-f 6506  df-f1 6507  df-fo 6508  df-f1o 6509  df-fv 6510  df-riota 7319  df-ov 7366  df-oprab 7367  df-mpo 7368  df-om 7809  df-1st 7927  df-2nd 7928  df-frecs 8218  df-wrecs 8249  df-recs 8323  df-rdg 8362  df-1o 8418  df-oadd 8422  df-er 8656  df-map 8775  df-en 8892  df-dom 8893  df-sdom 8894  df-fin 8895  df-dju 9847  df-card 9885  df-pnf 11201  df-mnf 11202  df-xr 11203  df-ltxr 11204  df-le 11205  df-sub 11397  df-neg 11398  df-nn 12164  df-2 12226  df-3 12227  df-n0 12424  df-z 12510  df-uz 12774  df-fz 13436  df-fzo 13579  df-hash 14242  df-word 14416  df-concat 14472  df-s1 14497  df-s2 14750  df-s3 14751  df-edg 28063  df-uhgr 28073  df-upgr 28097  df-umgr 28098  df-wlks 28611
This theorem is referenced by:  umgr2wlkon  28959  umgrwwlks2on  28966
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