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Theorem wlkiswwlks2 29728
Description: A walk as word corresponds to the sequence of vertices in a walk in a simple pseudograph. (Contributed by Alexander van der Vekens, 21-Jul-2018.) (Revised by AV, 10-Apr-2021.)
Assertion
Ref Expression
wlkiswwlks2 (𝐺 ∈ USPGraph β†’ (𝑃 ∈ (WWalksβ€˜πΊ) β†’ βˆƒπ‘“ 𝑓(Walksβ€˜πΊ)𝑃))
Distinct variable groups:   𝑓,𝐺   𝑃,𝑓
Proof of Theorem wlkiswwlks2
Dummy variables 𝑖 π‘₯ are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2725 . . . 4 (Vtxβ€˜πΊ) = (Vtxβ€˜πΊ)
21wwlkbp 29694 . . 3 (𝑃 ∈ (WWalksβ€˜πΊ) β†’ (𝐺 ∈ V ∧ 𝑃 ∈ Word (Vtxβ€˜πΊ)))
3 eqid 2725 . . . . 5 (Edgβ€˜πΊ) = (Edgβ€˜πΊ)
41, 3iswwlks 29689 . . . 4 (𝑃 ∈ (WWalksβ€˜πΊ) ↔ (𝑃 β‰  βˆ… ∧ 𝑃 ∈ Word (Vtxβ€˜πΊ) ∧ βˆ€π‘– ∈ (0..^((β™―β€˜π‘ƒ) βˆ’ 1)){(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))} ∈ (Edgβ€˜πΊ)))
5 ovex 7448 . . . . . . . . . . . . . . 15 (0..^((β™―β€˜π‘ƒ) βˆ’ 1)) ∈ V
6 mptexg 7228 . . . . . . . . . . . . . . 15 ((0..^((β™―β€˜π‘ƒ) βˆ’ 1)) ∈ V β†’ (π‘₯ ∈ (0..^((β™―β€˜π‘ƒ) βˆ’ 1)) ↦ (β—‘(iEdgβ€˜πΊ)β€˜{(π‘ƒβ€˜π‘₯), (π‘ƒβ€˜(π‘₯ + 1))})) ∈ V)
75, 6mp1i 13 . . . . . . . . . . . . . 14 (((𝑃 β‰  βˆ… ∧ 𝑃 ∈ Word (Vtxβ€˜πΊ)) ∧ ((𝐺 ∈ V ∧ 𝑃 ∈ Word (Vtxβ€˜πΊ)) ∧ 𝐺 ∈ USPGraph)) β†’ (π‘₯ ∈ (0..^((β™―β€˜π‘ƒ) βˆ’ 1)) ↦ (β—‘(iEdgβ€˜πΊ)β€˜{(π‘ƒβ€˜π‘₯), (π‘ƒβ€˜(π‘₯ + 1))})) ∈ V)
8 simprr 771 . . . . . . . . . . . . . . . . . 18 (((𝑃 β‰  βˆ… ∧ 𝑃 ∈ Word (Vtxβ€˜πΊ)) ∧ ((𝐺 ∈ V ∧ 𝑃 ∈ Word (Vtxβ€˜πΊ)) ∧ 𝐺 ∈ USPGraph)) β†’ 𝐺 ∈ USPGraph)
9 simplr 767 . . . . . . . . . . . . . . . . . 18 (((𝑃 β‰  βˆ… ∧ 𝑃 ∈ Word (Vtxβ€˜πΊ)) ∧ ((𝐺 ∈ V ∧ 𝑃 ∈ Word (Vtxβ€˜πΊ)) ∧ 𝐺 ∈ USPGraph)) β†’ 𝑃 ∈ Word (Vtxβ€˜πΊ))
10 hashge1 14378 . . . . . . . . . . . . . . . . . . . 20 ((𝑃 ∈ Word (Vtxβ€˜πΊ) ∧ 𝑃 β‰  βˆ…) β†’ 1 ≀ (β™―β€˜π‘ƒ))
1110ancoms 457 . . . . . . . . . . . . . . . . . . 19 ((𝑃 β‰  βˆ… ∧ 𝑃 ∈ Word (Vtxβ€˜πΊ)) β†’ 1 ≀ (β™―β€˜π‘ƒ))
1211adantr 479 . . . . . . . . . . . . . . . . . 18 (((𝑃 β‰  βˆ… ∧ 𝑃 ∈ Word (Vtxβ€˜πΊ)) ∧ ((𝐺 ∈ V ∧ 𝑃 ∈ Word (Vtxβ€˜πΊ)) ∧ 𝐺 ∈ USPGraph)) β†’ 1 ≀ (β™―β€˜π‘ƒ))
138, 9, 123jca 1125 . . . . . . . . . . . . . . . . 17 (((𝑃 β‰  βˆ… ∧ 𝑃 ∈ Word (Vtxβ€˜πΊ)) ∧ ((𝐺 ∈ V ∧ 𝑃 ∈ Word (Vtxβ€˜πΊ)) ∧ 𝐺 ∈ USPGraph)) β†’ (𝐺 ∈ USPGraph ∧ 𝑃 ∈ Word (Vtxβ€˜πΊ) ∧ 1 ≀ (β™―β€˜π‘ƒ)))
1413adantr 479 . . . . . . . . . . . . . . . 16 ((((𝑃 β‰  βˆ… ∧ 𝑃 ∈ Word (Vtxβ€˜πΊ)) ∧ ((𝐺 ∈ V ∧ 𝑃 ∈ Word (Vtxβ€˜πΊ)) ∧ 𝐺 ∈ USPGraph)) ∧ 𝑓 = (π‘₯ ∈ (0..^((β™―β€˜π‘ƒ) βˆ’ 1)) ↦ (β—‘(iEdgβ€˜πΊ)β€˜{(π‘ƒβ€˜π‘₯), (π‘ƒβ€˜(π‘₯ + 1))}))) β†’ (𝐺 ∈ USPGraph ∧ 𝑃 ∈ Word (Vtxβ€˜πΊ) ∧ 1 ≀ (β™―β€˜π‘ƒ)))
15 edgval 28904 . . . . . . . . . . . . . . . . . . . 20 (Edgβ€˜πΊ) = ran (iEdgβ€˜πΊ)
1615a1i 11 . . . . . . . . . . . . . . . . . . 19 ((((𝑃 β‰  βˆ… ∧ 𝑃 ∈ Word (Vtxβ€˜πΊ)) ∧ ((𝐺 ∈ V ∧ 𝑃 ∈ Word (Vtxβ€˜πΊ)) ∧ 𝐺 ∈ USPGraph)) ∧ 𝑓 = (π‘₯ ∈ (0..^((β™―β€˜π‘ƒ) βˆ’ 1)) ↦ (β—‘(iEdgβ€˜πΊ)β€˜{(π‘ƒβ€˜π‘₯), (π‘ƒβ€˜(π‘₯ + 1))}))) β†’ (Edgβ€˜πΊ) = ran (iEdgβ€˜πΊ))
1716eleq2d 2811 . . . . . . . . . . . . . . . . . 18 ((((𝑃 β‰  βˆ… ∧ 𝑃 ∈ Word (Vtxβ€˜πΊ)) ∧ ((𝐺 ∈ V ∧ 𝑃 ∈ Word (Vtxβ€˜πΊ)) ∧ 𝐺 ∈ USPGraph)) ∧ 𝑓 = (π‘₯ ∈ (0..^((β™―β€˜π‘ƒ) βˆ’ 1)) ↦ (β—‘(iEdgβ€˜πΊ)β€˜{(π‘ƒβ€˜π‘₯), (π‘ƒβ€˜(π‘₯ + 1))}))) β†’ ({(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))} ∈ (Edgβ€˜πΊ) ↔ {(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))} ∈ ran (iEdgβ€˜πΊ)))
1817ralbidv 3168 . . . . . . . . . . . . . . . . 17 ((((𝑃 β‰  βˆ… ∧ 𝑃 ∈ Word (Vtxβ€˜πΊ)) ∧ ((𝐺 ∈ V ∧ 𝑃 ∈ Word (Vtxβ€˜πΊ)) ∧ 𝐺 ∈ USPGraph)) ∧ 𝑓 = (π‘₯ ∈ (0..^((β™―β€˜π‘ƒ) βˆ’ 1)) ↦ (β—‘(iEdgβ€˜πΊ)β€˜{(π‘ƒβ€˜π‘₯), (π‘ƒβ€˜(π‘₯ + 1))}))) β†’ (βˆ€π‘– ∈ (0..^((β™―β€˜π‘ƒ) βˆ’ 1)){(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))} ∈ (Edgβ€˜πΊ) ↔ βˆ€π‘– ∈ (0..^((β™―β€˜π‘ƒ) βˆ’ 1)){(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))} ∈ ran (iEdgβ€˜πΊ)))
1918biimpd 228 . . . . . . . . . . . . . . . 16 ((((𝑃 β‰  βˆ… ∧ 𝑃 ∈ Word (Vtxβ€˜πΊ)) ∧ ((𝐺 ∈ V ∧ 𝑃 ∈ Word (Vtxβ€˜πΊ)) ∧ 𝐺 ∈ USPGraph)) ∧ 𝑓 = (π‘₯ ∈ (0..^((β™―β€˜π‘ƒ) βˆ’ 1)) ↦ (β—‘(iEdgβ€˜πΊ)β€˜{(π‘ƒβ€˜π‘₯), (π‘ƒβ€˜(π‘₯ + 1))}))) β†’ (βˆ€π‘– ∈ (0..^((β™―β€˜π‘ƒ) βˆ’ 1)){(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))} ∈ (Edgβ€˜πΊ) β†’ βˆ€π‘– ∈ (0..^((β™―β€˜π‘ƒ) βˆ’ 1)){(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))} ∈ ran (iEdgβ€˜πΊ)))
20 eqid 2725 . . . . . . . . . . . . . . . . 17 (π‘₯ ∈ (0..^((β™―β€˜π‘ƒ) βˆ’ 1)) ↦ (β—‘(iEdgβ€˜πΊ)β€˜{(π‘ƒβ€˜π‘₯), (π‘ƒβ€˜(π‘₯ + 1))})) = (π‘₯ ∈ (0..^((β™―β€˜π‘ƒ) βˆ’ 1)) ↦ (β—‘(iEdgβ€˜πΊ)β€˜{(π‘ƒβ€˜π‘₯), (π‘ƒβ€˜(π‘₯ + 1))}))
21 eqid 2725 . . . . . . . . . . . . . . . . 17 (iEdgβ€˜πΊ) = (iEdgβ€˜πΊ)
2220, 21wlkiswwlks2lem6 29727 . . . . . . . . . . . . . . . 16 ((𝐺 ∈ USPGraph ∧ 𝑃 ∈ Word (Vtxβ€˜πΊ) ∧ 1 ≀ (β™―β€˜π‘ƒ)) β†’ (βˆ€π‘– ∈ (0..^((β™―β€˜π‘ƒ) βˆ’ 1)){(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))} ∈ ran (iEdgβ€˜πΊ) β†’ ((π‘₯ ∈ (0..^((β™―β€˜π‘ƒ) βˆ’ 1)) ↦ (β—‘(iEdgβ€˜πΊ)β€˜{(π‘ƒβ€˜π‘₯), (π‘ƒβ€˜(π‘₯ + 1))})) ∈ Word dom (iEdgβ€˜πΊ) ∧ 𝑃:(0...(β™―β€˜(π‘₯ ∈ (0..^((β™―β€˜π‘ƒ) βˆ’ 1)) ↦ (β—‘(iEdgβ€˜πΊ)β€˜{(π‘ƒβ€˜π‘₯), (π‘ƒβ€˜(π‘₯ + 1))}))))⟢(Vtxβ€˜πΊ) ∧ βˆ€π‘– ∈ (0..^(β™―β€˜(π‘₯ ∈ (0..^((β™―β€˜π‘ƒ) βˆ’ 1)) ↦ (β—‘(iEdgβ€˜πΊ)β€˜{(π‘ƒβ€˜π‘₯), (π‘ƒβ€˜(π‘₯ + 1))}))))((iEdgβ€˜πΊ)β€˜((π‘₯ ∈ (0..^((β™―β€˜π‘ƒ) βˆ’ 1)) ↦ (β—‘(iEdgβ€˜πΊ)β€˜{(π‘ƒβ€˜π‘₯), (π‘ƒβ€˜(π‘₯ + 1))}))β€˜π‘–)) = {(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))})))
2314, 19, 22sylsyld 61 . . . . . . . . . . . . . . 15 ((((𝑃 β‰  βˆ… ∧ 𝑃 ∈ Word (Vtxβ€˜πΊ)) ∧ ((𝐺 ∈ V ∧ 𝑃 ∈ Word (Vtxβ€˜πΊ)) ∧ 𝐺 ∈ USPGraph)) ∧ 𝑓 = (π‘₯ ∈ (0..^((β™―β€˜π‘ƒ) βˆ’ 1)) ↦ (β—‘(iEdgβ€˜πΊ)β€˜{(π‘ƒβ€˜π‘₯), (π‘ƒβ€˜(π‘₯ + 1))}))) β†’ (βˆ€π‘– ∈ (0..^((β™―β€˜π‘ƒ) βˆ’ 1)){(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))} ∈ (Edgβ€˜πΊ) β†’ ((π‘₯ ∈ (0..^((β™―β€˜π‘ƒ) βˆ’ 1)) ↦ (β—‘(iEdgβ€˜πΊ)β€˜{(π‘ƒβ€˜π‘₯), (π‘ƒβ€˜(π‘₯ + 1))})) ∈ Word dom (iEdgβ€˜πΊ) ∧ 𝑃:(0...(β™―β€˜(π‘₯ ∈ (0..^((β™―β€˜π‘ƒ) βˆ’ 1)) ↦ (β—‘(iEdgβ€˜πΊ)β€˜{(π‘ƒβ€˜π‘₯), (π‘ƒβ€˜(π‘₯ + 1))}))))⟢(Vtxβ€˜πΊ) ∧ βˆ€π‘– ∈ (0..^(β™―β€˜(π‘₯ ∈ (0..^((β™―β€˜π‘ƒ) βˆ’ 1)) ↦ (β—‘(iEdgβ€˜πΊ)β€˜{(π‘ƒβ€˜π‘₯), (π‘ƒβ€˜(π‘₯ + 1))}))))((iEdgβ€˜πΊ)β€˜((π‘₯ ∈ (0..^((β™―β€˜π‘ƒ) βˆ’ 1)) ↦ (β—‘(iEdgβ€˜πΊ)β€˜{(π‘ƒβ€˜π‘₯), (π‘ƒβ€˜(π‘₯ + 1))}))β€˜π‘–)) = {(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))})))
24 eleq1 2813 . . . . . . . . . . . . . . . . . 18 (𝑓 = (π‘₯ ∈ (0..^((β™―β€˜π‘ƒ) βˆ’ 1)) ↦ (β—‘(iEdgβ€˜πΊ)β€˜{(π‘ƒβ€˜π‘₯), (π‘ƒβ€˜(π‘₯ + 1))})) β†’ (𝑓 ∈ Word dom (iEdgβ€˜πΊ) ↔ (π‘₯ ∈ (0..^((β™―β€˜π‘ƒ) βˆ’ 1)) ↦ (β—‘(iEdgβ€˜πΊ)β€˜{(π‘ƒβ€˜π‘₯), (π‘ƒβ€˜(π‘₯ + 1))})) ∈ Word dom (iEdgβ€˜πΊ)))
25 fveq2 6891 . . . . . . . . . . . . . . . . . . . 20 (𝑓 = (π‘₯ ∈ (0..^((β™―β€˜π‘ƒ) βˆ’ 1)) ↦ (β—‘(iEdgβ€˜πΊ)β€˜{(π‘ƒβ€˜π‘₯), (π‘ƒβ€˜(π‘₯ + 1))})) β†’ (β™―β€˜π‘“) = (β™―β€˜(π‘₯ ∈ (0..^((β™―β€˜π‘ƒ) βˆ’ 1)) ↦ (β—‘(iEdgβ€˜πΊ)β€˜{(π‘ƒβ€˜π‘₯), (π‘ƒβ€˜(π‘₯ + 1))}))))
2625oveq2d 7431 . . . . . . . . . . . . . . . . . . 19 (𝑓 = (π‘₯ ∈ (0..^((β™―β€˜π‘ƒ) βˆ’ 1)) ↦ (β—‘(iEdgβ€˜πΊ)β€˜{(π‘ƒβ€˜π‘₯), (π‘ƒβ€˜(π‘₯ + 1))})) β†’ (0...(β™―β€˜π‘“)) = (0...(β™―β€˜(π‘₯ ∈ (0..^((β™―β€˜π‘ƒ) βˆ’ 1)) ↦ (β—‘(iEdgβ€˜πΊ)β€˜{(π‘ƒβ€˜π‘₯), (π‘ƒβ€˜(π‘₯ + 1))})))))
2726feq2d 6702 . . . . . . . . . . . . . . . . . 18 (𝑓 = (π‘₯ ∈ (0..^((β™―β€˜π‘ƒ) βˆ’ 1)) ↦ (β—‘(iEdgβ€˜πΊ)β€˜{(π‘ƒβ€˜π‘₯), (π‘ƒβ€˜(π‘₯ + 1))})) β†’ (𝑃:(0...(β™―β€˜π‘“))⟢(Vtxβ€˜πΊ) ↔ 𝑃:(0...(β™―β€˜(π‘₯ ∈ (0..^((β™―β€˜π‘ƒ) βˆ’ 1)) ↦ (β—‘(iEdgβ€˜πΊ)β€˜{(π‘ƒβ€˜π‘₯), (π‘ƒβ€˜(π‘₯ + 1))}))))⟢(Vtxβ€˜πΊ)))
2825oveq2d 7431 . . . . . . . . . . . . . . . . . . 19 (𝑓 = (π‘₯ ∈ (0..^((β™―β€˜π‘ƒ) βˆ’ 1)) ↦ (β—‘(iEdgβ€˜πΊ)β€˜{(π‘ƒβ€˜π‘₯), (π‘ƒβ€˜(π‘₯ + 1))})) β†’ (0..^(β™―β€˜π‘“)) = (0..^(β™―β€˜(π‘₯ ∈ (0..^((β™―β€˜π‘ƒ) βˆ’ 1)) ↦ (β—‘(iEdgβ€˜πΊ)β€˜{(π‘ƒβ€˜π‘₯), (π‘ƒβ€˜(π‘₯ + 1))})))))
29 fveq1 6890 . . . . . . . . . . . . . . . . . . . 20 (𝑓 = (π‘₯ ∈ (0..^((β™―β€˜π‘ƒ) βˆ’ 1)) ↦ (β—‘(iEdgβ€˜πΊ)β€˜{(π‘ƒβ€˜π‘₯), (π‘ƒβ€˜(π‘₯ + 1))})) β†’ (π‘“β€˜π‘–) = ((π‘₯ ∈ (0..^((β™―β€˜π‘ƒ) βˆ’ 1)) ↦ (β—‘(iEdgβ€˜πΊ)β€˜{(π‘ƒβ€˜π‘₯), (π‘ƒβ€˜(π‘₯ + 1))}))β€˜π‘–))
3029fveqeq2d 6899 . . . . . . . . . . . . . . . . . . 19 (𝑓 = (π‘₯ ∈ (0..^((β™―β€˜π‘ƒ) βˆ’ 1)) ↦ (β—‘(iEdgβ€˜πΊ)β€˜{(π‘ƒβ€˜π‘₯), (π‘ƒβ€˜(π‘₯ + 1))})) β†’ (((iEdgβ€˜πΊ)β€˜(π‘“β€˜π‘–)) = {(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))} ↔ ((iEdgβ€˜πΊ)β€˜((π‘₯ ∈ (0..^((β™―β€˜π‘ƒ) βˆ’ 1)) ↦ (β—‘(iEdgβ€˜πΊ)β€˜{(π‘ƒβ€˜π‘₯), (π‘ƒβ€˜(π‘₯ + 1))}))β€˜π‘–)) = {(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))}))
3128, 30raleqbidv 3330 . . . . . . . . . . . . . . . . . 18 (𝑓 = (π‘₯ ∈ (0..^((β™―β€˜π‘ƒ) βˆ’ 1)) ↦ (β—‘(iEdgβ€˜πΊ)β€˜{(π‘ƒβ€˜π‘₯), (π‘ƒβ€˜(π‘₯ + 1))})) β†’ (βˆ€π‘– ∈ (0..^(β™―β€˜π‘“))((iEdgβ€˜πΊ)β€˜(π‘“β€˜π‘–)) = {(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))} ↔ βˆ€π‘– ∈ (0..^(β™―β€˜(π‘₯ ∈ (0..^((β™―β€˜π‘ƒ) βˆ’ 1)) ↦ (β—‘(iEdgβ€˜πΊ)β€˜{(π‘ƒβ€˜π‘₯), (π‘ƒβ€˜(π‘₯ + 1))}))))((iEdgβ€˜πΊ)β€˜((π‘₯ ∈ (0..^((β™―β€˜π‘ƒ) βˆ’ 1)) ↦ (β—‘(iEdgβ€˜πΊ)β€˜{(π‘ƒβ€˜π‘₯), (π‘ƒβ€˜(π‘₯ + 1))}))β€˜π‘–)) = {(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))}))
3224, 27, 313anbi123d 1432 . . . . . . . . . . . . . . . . 17 (𝑓 = (π‘₯ ∈ (0..^((β™―β€˜π‘ƒ) βˆ’ 1)) ↦ (β—‘(iEdgβ€˜πΊ)β€˜{(π‘ƒβ€˜π‘₯), (π‘ƒβ€˜(π‘₯ + 1))})) β†’ ((𝑓 ∈ Word dom (iEdgβ€˜πΊ) ∧ 𝑃:(0...(β™―β€˜π‘“))⟢(Vtxβ€˜πΊ) ∧ βˆ€π‘– ∈ (0..^(β™―β€˜π‘“))((iEdgβ€˜πΊ)β€˜(π‘“β€˜π‘–)) = {(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))}) ↔ ((π‘₯ ∈ (0..^((β™―β€˜π‘ƒ) βˆ’ 1)) ↦ (β—‘(iEdgβ€˜πΊ)β€˜{(π‘ƒβ€˜π‘₯), (π‘ƒβ€˜(π‘₯ + 1))})) ∈ Word dom (iEdgβ€˜πΊ) ∧ 𝑃:(0...(β™―β€˜(π‘₯ ∈ (0..^((β™―β€˜π‘ƒ) βˆ’ 1)) ↦ (β—‘(iEdgβ€˜πΊ)β€˜{(π‘ƒβ€˜π‘₯), (π‘ƒβ€˜(π‘₯ + 1))}))))⟢(Vtxβ€˜πΊ) ∧ βˆ€π‘– ∈ (0..^(β™―β€˜(π‘₯ ∈ (0..^((β™―β€˜π‘ƒ) βˆ’ 1)) ↦ (β—‘(iEdgβ€˜πΊ)β€˜{(π‘ƒβ€˜π‘₯), (π‘ƒβ€˜(π‘₯ + 1))}))))((iEdgβ€˜πΊ)β€˜((π‘₯ ∈ (0..^((β™―β€˜π‘ƒ) βˆ’ 1)) ↦ (β—‘(iEdgβ€˜πΊ)β€˜{(π‘ƒβ€˜π‘₯), (π‘ƒβ€˜(π‘₯ + 1))}))β€˜π‘–)) = {(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))})))
3332imbi2d 339 . . . . . . . . . . . . . . . 16 (𝑓 = (π‘₯ ∈ (0..^((β™―β€˜π‘ƒ) βˆ’ 1)) ↦ (β—‘(iEdgβ€˜πΊ)β€˜{(π‘ƒβ€˜π‘₯), (π‘ƒβ€˜(π‘₯ + 1))})) β†’ ((βˆ€π‘– ∈ (0..^((β™―β€˜π‘ƒ) βˆ’ 1)){(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))} ∈ (Edgβ€˜πΊ) β†’ (𝑓 ∈ Word dom (iEdgβ€˜πΊ) ∧ 𝑃:(0...(β™―β€˜π‘“))⟢(Vtxβ€˜πΊ) ∧ βˆ€π‘– ∈ (0..^(β™―β€˜π‘“))((iEdgβ€˜πΊ)β€˜(π‘“β€˜π‘–)) = {(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))})) ↔ (βˆ€π‘– ∈ (0..^((β™―β€˜π‘ƒ) βˆ’ 1)){(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))} ∈ (Edgβ€˜πΊ) β†’ ((π‘₯ ∈ (0..^((β™―β€˜π‘ƒ) βˆ’ 1)) ↦ (β—‘(iEdgβ€˜πΊ)β€˜{(π‘ƒβ€˜π‘₯), (π‘ƒβ€˜(π‘₯ + 1))})) ∈ Word dom (iEdgβ€˜πΊ) ∧ 𝑃:(0...(β™―β€˜(π‘₯ ∈ (0..^((β™―β€˜π‘ƒ) βˆ’ 1)) ↦ (β—‘(iEdgβ€˜πΊ)β€˜{(π‘ƒβ€˜π‘₯), (π‘ƒβ€˜(π‘₯ + 1))}))))⟢(Vtxβ€˜πΊ) ∧ βˆ€π‘– ∈ (0..^(β™―β€˜(π‘₯ ∈ (0..^((β™―β€˜π‘ƒ) βˆ’ 1)) ↦ (β—‘(iEdgβ€˜πΊ)β€˜{(π‘ƒβ€˜π‘₯), (π‘ƒβ€˜(π‘₯ + 1))}))))((iEdgβ€˜πΊ)β€˜((π‘₯ ∈ (0..^((β™―β€˜π‘ƒ) βˆ’ 1)) ↦ (β—‘(iEdgβ€˜πΊ)β€˜{(π‘ƒβ€˜π‘₯), (π‘ƒβ€˜(π‘₯ + 1))}))β€˜π‘–)) = {(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))}))))
3433adantl 480 . . . . . . . . . . . . . . 15 ((((𝑃 β‰  βˆ… ∧ 𝑃 ∈ Word (Vtxβ€˜πΊ)) ∧ ((𝐺 ∈ V ∧ 𝑃 ∈ Word (Vtxβ€˜πΊ)) ∧ 𝐺 ∈ USPGraph)) ∧ 𝑓 = (π‘₯ ∈ (0..^((β™―β€˜π‘ƒ) βˆ’ 1)) ↦ (β—‘(iEdgβ€˜πΊ)β€˜{(π‘ƒβ€˜π‘₯), (π‘ƒβ€˜(π‘₯ + 1))}))) β†’ ((βˆ€π‘– ∈ (0..^((β™―β€˜π‘ƒ) βˆ’ 1)){(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))} ∈ (Edgβ€˜πΊ) β†’ (𝑓 ∈ Word dom (iEdgβ€˜πΊ) ∧ 𝑃:(0...(β™―β€˜π‘“))⟢(Vtxβ€˜πΊ) ∧ βˆ€π‘– ∈ (0..^(β™―β€˜π‘“))((iEdgβ€˜πΊ)β€˜(π‘“β€˜π‘–)) = {(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))})) ↔ (βˆ€π‘– ∈ (0..^((β™―β€˜π‘ƒ) βˆ’ 1)){(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))} ∈ (Edgβ€˜πΊ) β†’ ((π‘₯ ∈ (0..^((β™―β€˜π‘ƒ) βˆ’ 1)) ↦ (β—‘(iEdgβ€˜πΊ)β€˜{(π‘ƒβ€˜π‘₯), (π‘ƒβ€˜(π‘₯ + 1))})) ∈ Word dom (iEdgβ€˜πΊ) ∧ 𝑃:(0...(β™―β€˜(π‘₯ ∈ (0..^((β™―β€˜π‘ƒ) βˆ’ 1)) ↦ (β—‘(iEdgβ€˜πΊ)β€˜{(π‘ƒβ€˜π‘₯), (π‘ƒβ€˜(π‘₯ + 1))}))))⟢(Vtxβ€˜πΊ) ∧ βˆ€π‘– ∈ (0..^(β™―β€˜(π‘₯ ∈ (0..^((β™―β€˜π‘ƒ) βˆ’ 1)) ↦ (β—‘(iEdgβ€˜πΊ)β€˜{(π‘ƒβ€˜π‘₯), (π‘ƒβ€˜(π‘₯ + 1))}))))((iEdgβ€˜πΊ)β€˜((π‘₯ ∈ (0..^((β™―β€˜π‘ƒ) βˆ’ 1)) ↦ (β—‘(iEdgβ€˜πΊ)β€˜{(π‘ƒβ€˜π‘₯), (π‘ƒβ€˜(π‘₯ + 1))}))β€˜π‘–)) = {(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))}))))
3523, 34mpbird 256 . . . . . . . . . . . . . 14 ((((𝑃 β‰  βˆ… ∧ 𝑃 ∈ Word (Vtxβ€˜πΊ)) ∧ ((𝐺 ∈ V ∧ 𝑃 ∈ Word (Vtxβ€˜πΊ)) ∧ 𝐺 ∈ USPGraph)) ∧ 𝑓 = (π‘₯ ∈ (0..^((β™―β€˜π‘ƒ) βˆ’ 1)) ↦ (β—‘(iEdgβ€˜πΊ)β€˜{(π‘ƒβ€˜π‘₯), (π‘ƒβ€˜(π‘₯ + 1))}))) β†’ (βˆ€π‘– ∈ (0..^((β™―β€˜π‘ƒ) βˆ’ 1)){(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))} ∈ (Edgβ€˜πΊ) β†’ (𝑓 ∈ Word dom (iEdgβ€˜πΊ) ∧ 𝑃:(0...(β™―β€˜π‘“))⟢(Vtxβ€˜πΊ) ∧ βˆ€π‘– ∈ (0..^(β™―β€˜π‘“))((iEdgβ€˜πΊ)β€˜(π‘“β€˜π‘–)) = {(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))})))
367, 35spcimedv 3575 . . . . . . . . . . . . 13 (((𝑃 β‰  βˆ… ∧ 𝑃 ∈ Word (Vtxβ€˜πΊ)) ∧ ((𝐺 ∈ V ∧ 𝑃 ∈ Word (Vtxβ€˜πΊ)) ∧ 𝐺 ∈ USPGraph)) β†’ (βˆ€π‘– ∈ (0..^((β™―β€˜π‘ƒ) βˆ’ 1)){(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))} ∈ (Edgβ€˜πΊ) β†’ βˆƒπ‘“(𝑓 ∈ Word dom (iEdgβ€˜πΊ) ∧ 𝑃:(0...(β™―β€˜π‘“))⟢(Vtxβ€˜πΊ) ∧ βˆ€π‘– ∈ (0..^(β™―β€˜π‘“))((iEdgβ€˜πΊ)β€˜(π‘“β€˜π‘–)) = {(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))})))
3736ex 411 . . . . . . . . . . . 12 ((𝑃 β‰  βˆ… ∧ 𝑃 ∈ Word (Vtxβ€˜πΊ)) β†’ (((𝐺 ∈ V ∧ 𝑃 ∈ Word (Vtxβ€˜πΊ)) ∧ 𝐺 ∈ USPGraph) β†’ (βˆ€π‘– ∈ (0..^((β™―β€˜π‘ƒ) βˆ’ 1)){(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))} ∈ (Edgβ€˜πΊ) β†’ βˆƒπ‘“(𝑓 ∈ Word dom (iEdgβ€˜πΊ) ∧ 𝑃:(0...(β™―β€˜π‘“))⟢(Vtxβ€˜πΊ) ∧ βˆ€π‘– ∈ (0..^(β™―β€˜π‘“))((iEdgβ€˜πΊ)β€˜(π‘“β€˜π‘–)) = {(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))}))))
3837com23 86 . . . . . . . . . . 11 ((𝑃 β‰  βˆ… ∧ 𝑃 ∈ Word (Vtxβ€˜πΊ)) β†’ (βˆ€π‘– ∈ (0..^((β™―β€˜π‘ƒ) βˆ’ 1)){(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))} ∈ (Edgβ€˜πΊ) β†’ (((𝐺 ∈ V ∧ 𝑃 ∈ Word (Vtxβ€˜πΊ)) ∧ 𝐺 ∈ USPGraph) β†’ βˆƒπ‘“(𝑓 ∈ Word dom (iEdgβ€˜πΊ) ∧ 𝑃:(0...(β™―β€˜π‘“))⟢(Vtxβ€˜πΊ) ∧ βˆ€π‘– ∈ (0..^(β™―β€˜π‘“))((iEdgβ€˜πΊ)β€˜(π‘“β€˜π‘–)) = {(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))}))))
39383impia 1114 . . . . . . . . . 10 ((𝑃 β‰  βˆ… ∧ 𝑃 ∈ Word (Vtxβ€˜πΊ) ∧ βˆ€π‘– ∈ (0..^((β™―β€˜π‘ƒ) βˆ’ 1)){(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))} ∈ (Edgβ€˜πΊ)) β†’ (((𝐺 ∈ V ∧ 𝑃 ∈ Word (Vtxβ€˜πΊ)) ∧ 𝐺 ∈ USPGraph) β†’ βˆƒπ‘“(𝑓 ∈ Word dom (iEdgβ€˜πΊ) ∧ 𝑃:(0...(β™―β€˜π‘“))⟢(Vtxβ€˜πΊ) ∧ βˆ€π‘– ∈ (0..^(β™―β€˜π‘“))((iEdgβ€˜πΊ)β€˜(π‘“β€˜π‘–)) = {(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))})))
4039expd 414 . . . . . . . . 9 ((𝑃 β‰  βˆ… ∧ 𝑃 ∈ Word (Vtxβ€˜πΊ) ∧ βˆ€π‘– ∈ (0..^((β™―β€˜π‘ƒ) βˆ’ 1)){(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))} ∈ (Edgβ€˜πΊ)) β†’ ((𝐺 ∈ V ∧ 𝑃 ∈ Word (Vtxβ€˜πΊ)) β†’ (𝐺 ∈ USPGraph β†’ βˆƒπ‘“(𝑓 ∈ Word dom (iEdgβ€˜πΊ) ∧ 𝑃:(0...(β™―β€˜π‘“))⟢(Vtxβ€˜πΊ) ∧ βˆ€π‘– ∈ (0..^(β™―β€˜π‘“))((iEdgβ€˜πΊ)β€˜(π‘“β€˜π‘–)) = {(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))}))))
4140impcom 406 . . . . . . . 8 (((𝐺 ∈ V ∧ 𝑃 ∈ Word (Vtxβ€˜πΊ)) ∧ (𝑃 β‰  βˆ… ∧ 𝑃 ∈ Word (Vtxβ€˜πΊ) ∧ βˆ€π‘– ∈ (0..^((β™―β€˜π‘ƒ) βˆ’ 1)){(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))} ∈ (Edgβ€˜πΊ))) β†’ (𝐺 ∈ USPGraph β†’ βˆƒπ‘“(𝑓 ∈ Word dom (iEdgβ€˜πΊ) ∧ 𝑃:(0...(β™―β€˜π‘“))⟢(Vtxβ€˜πΊ) ∧ βˆ€π‘– ∈ (0..^(β™―β€˜π‘“))((iEdgβ€˜πΊ)β€˜(π‘“β€˜π‘–)) = {(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))})))
4241imp 405 . . . . . . 7 ((((𝐺 ∈ V ∧ 𝑃 ∈ Word (Vtxβ€˜πΊ)) ∧ (𝑃 β‰  βˆ… ∧ 𝑃 ∈ Word (Vtxβ€˜πΊ) ∧ βˆ€π‘– ∈ (0..^((β™―β€˜π‘ƒ) βˆ’ 1)){(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))} ∈ (Edgβ€˜πΊ))) ∧ 𝐺 ∈ USPGraph) β†’ βˆƒπ‘“(𝑓 ∈ Word dom (iEdgβ€˜πΊ) ∧ 𝑃:(0...(β™―β€˜π‘“))⟢(Vtxβ€˜πΊ) ∧ βˆ€π‘– ∈ (0..^(β™―β€˜π‘“))((iEdgβ€˜πΊ)β€˜(π‘“β€˜π‘–)) = {(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))}))
43 uspgrupgr 29033 . . . . . . . . . 10 (𝐺 ∈ USPGraph β†’ 𝐺 ∈ UPGraph)
441, 21upgriswlk 29497 . . . . . . . . . 10 (𝐺 ∈ UPGraph β†’ (𝑓(Walksβ€˜πΊ)𝑃 ↔ (𝑓 ∈ Word dom (iEdgβ€˜πΊ) ∧ 𝑃:(0...(β™―β€˜π‘“))⟢(Vtxβ€˜πΊ) ∧ βˆ€π‘– ∈ (0..^(β™―β€˜π‘“))((iEdgβ€˜πΊ)β€˜(π‘“β€˜π‘–)) = {(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))})))
4543, 44syl 17 . . . . . . . . 9 (𝐺 ∈ USPGraph β†’ (𝑓(Walksβ€˜πΊ)𝑃 ↔ (𝑓 ∈ Word dom (iEdgβ€˜πΊ) ∧ 𝑃:(0...(β™―β€˜π‘“))⟢(Vtxβ€˜πΊ) ∧ βˆ€π‘– ∈ (0..^(β™―β€˜π‘“))((iEdgβ€˜πΊ)β€˜(π‘“β€˜π‘–)) = {(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))})))
4645adantl 480 . . . . . . . 8 ((((𝐺 ∈ V ∧ 𝑃 ∈ Word (Vtxβ€˜πΊ)) ∧ (𝑃 β‰  βˆ… ∧ 𝑃 ∈ Word (Vtxβ€˜πΊ) ∧ βˆ€π‘– ∈ (0..^((β™―β€˜π‘ƒ) βˆ’ 1)){(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))} ∈ (Edgβ€˜πΊ))) ∧ 𝐺 ∈ USPGraph) β†’ (𝑓(Walksβ€˜πΊ)𝑃 ↔ (𝑓 ∈ Word dom (iEdgβ€˜πΊ) ∧ 𝑃:(0...(β™―β€˜π‘“))⟢(Vtxβ€˜πΊ) ∧ βˆ€π‘– ∈ (0..^(β™―β€˜π‘“))((iEdgβ€˜πΊ)β€˜(π‘“β€˜π‘–)) = {(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))})))
4746exbidv 1916 . . . . . . 7 ((((𝐺 ∈ V ∧ 𝑃 ∈ Word (Vtxβ€˜πΊ)) ∧ (𝑃 β‰  βˆ… ∧ 𝑃 ∈ Word (Vtxβ€˜πΊ) ∧ βˆ€π‘– ∈ (0..^((β™―β€˜π‘ƒ) βˆ’ 1)){(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))} ∈ (Edgβ€˜πΊ))) ∧ 𝐺 ∈ USPGraph) β†’ (βˆƒπ‘“ 𝑓(Walksβ€˜πΊ)𝑃 ↔ βˆƒπ‘“(𝑓 ∈ Word dom (iEdgβ€˜πΊ) ∧ 𝑃:(0...(β™―β€˜π‘“))⟢(Vtxβ€˜πΊ) ∧ βˆ€π‘– ∈ (0..^(β™―β€˜π‘“))((iEdgβ€˜πΊ)β€˜(π‘“β€˜π‘–)) = {(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))})))
4842, 47mpbird 256 . . . . . 6 ((((𝐺 ∈ V ∧ 𝑃 ∈ Word (Vtxβ€˜πΊ)) ∧ (𝑃 β‰  βˆ… ∧ 𝑃 ∈ Word (Vtxβ€˜πΊ) ∧ βˆ€π‘– ∈ (0..^((β™―β€˜π‘ƒ) βˆ’ 1)){(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))} ∈ (Edgβ€˜πΊ))) ∧ 𝐺 ∈ USPGraph) β†’ βˆƒπ‘“ 𝑓(Walksβ€˜πΊ)𝑃)
4948ex 411 . . . . 5 (((𝐺 ∈ V ∧ 𝑃 ∈ Word (Vtxβ€˜πΊ)) ∧ (𝑃 β‰  βˆ… ∧ 𝑃 ∈ Word (Vtxβ€˜πΊ) ∧ βˆ€π‘– ∈ (0..^((β™―β€˜π‘ƒ) βˆ’ 1)){(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))} ∈ (Edgβ€˜πΊ))) β†’ (𝐺 ∈ USPGraph β†’ βˆƒπ‘“ 𝑓(Walksβ€˜πΊ)𝑃))
5049ex 411 . . . 4 ((𝐺 ∈ V ∧ 𝑃 ∈ Word (Vtxβ€˜πΊ)) β†’ ((𝑃 β‰  βˆ… ∧ 𝑃 ∈ Word (Vtxβ€˜πΊ) ∧ βˆ€π‘– ∈ (0..^((β™―β€˜π‘ƒ) βˆ’ 1)){(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))} ∈ (Edgβ€˜πΊ)) β†’ (𝐺 ∈ USPGraph β†’ βˆƒπ‘“ 𝑓(Walksβ€˜πΊ)𝑃)))
514, 50biimtrid 241 . . 3 ((𝐺 ∈ V ∧ 𝑃 ∈ Word (Vtxβ€˜πΊ)) β†’ (𝑃 ∈ (WWalksβ€˜πΊ) β†’ (𝐺 ∈ USPGraph β†’ βˆƒπ‘“ 𝑓(Walksβ€˜πΊ)𝑃)))
522, 51mpcom 38 . 2 (𝑃 ∈ (WWalksβ€˜πΊ) β†’ (𝐺 ∈ USPGraph β†’ βˆƒπ‘“ 𝑓(Walksβ€˜πΊ)𝑃))
5352com12 32 1 (𝐺 ∈ USPGraph β†’ (𝑃 ∈ (WWalksβ€˜πΊ) β†’ βˆƒπ‘“ 𝑓(Walksβ€˜πΊ)𝑃))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 394   ∧ w3a 1084   = wceq 1533  βˆƒwex 1773   ∈ wcel 2098   β‰  wne 2930  βˆ€wral 3051  Vcvv 3463  βˆ…c0 4318  {cpr 4626   class class class wbr 5143   ↦ cmpt 5226  β—‘ccnv 5671  dom cdm 5672  ran crn 5673  βŸΆwf 6538  β€˜cfv 6542  (class class class)co 7415  0cc0 11136  1c1 11137   + caddc 11139   ≀ cle 11277   βˆ’ cmin 11472  ...cfz 13514  ..^cfzo 13657  β™―chash 14319  Word cword 14494  Vtxcvtx 28851  iEdgciedg 28852  Edgcedg 28902  UPGraphcupgr 28935  USPGraphcuspgr 29003  Walkscwlks 29452  WWalkscwwlks 29678
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 2696  ax-rep 5280  ax-sep 5294  ax-nul 5301  ax-pow 5359  ax-pr 5423  ax-un 7737  ax-cnex 11192  ax-resscn 11193  ax-1cn 11194  ax-icn 11195  ax-addcl 11196  ax-addrcl 11197  ax-mulcl 11198  ax-mulrcl 11199  ax-mulcom 11200  ax-addass 11201  ax-mulass 11202  ax-distr 11203  ax-i2m1 11204  ax-1ne0 11205  ax-1rid 11206  ax-rnegex 11207  ax-rrecex 11208  ax-cnre 11209  ax-pre-lttri 11210  ax-pre-lttrn 11211  ax-pre-ltadd 11212  ax-pre-mulgt0 11213
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 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-nel 3037  df-ral 3052  df-rex 3061  df-reu 3365  df-rab 3420  df-v 3465  df-sbc 3770  df-csb 3886  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-pss 3960  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-int 4945  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5227  df-tr 5261  df-id 5570  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  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 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  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-riota 7371  df-ov 7418  df-oprab 7419  df-mpo 7420  df-om 7868  df-1st 7989  df-2nd 7990  df-frecs 8283  df-wrecs 8314  df-recs 8388  df-rdg 8427  df-1o 8483  df-2o 8484  df-oadd 8487  df-er 8721  df-map 8843  df-pm 8844  df-en 8961  df-dom 8962  df-sdom 8963  df-fin 8964  df-dju 9922  df-card 9960  df-pnf 11278  df-mnf 11279  df-xr 11280  df-ltxr 11281  df-le 11282  df-sub 11474  df-neg 11475  df-nn 12241  df-2 12303  df-n0 12501  df-xnn0 12573  df-z 12587  df-uz 12851  df-fz 13515  df-fzo 13658  df-hash 14320  df-word 14495  df-edg 28903  df-uhgr 28913  df-upgr 28937  df-uspgr 29005  df-wlks 29455  df-wwlks 29683
This theorem is referenced by:  wlkiswwlks  29729  wlklnwwlkln2  29736
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