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Theorem rusgrnumwwlkl1 29766
Description: In a k-regular graph, there are k walks (as word) of length 1 starting at each vertex. (Contributed by Alexander van der Vekens, 28-Jul-2018.) (Revised by AV, 7-May-2021.)
Hypothesis
Ref Expression
rusgrnumwwlkl1.v 𝑉 = (Vtxβ€˜πΊ)
Assertion
Ref Expression
rusgrnumwwlkl1 ((𝐺 RegUSGraph 𝐾 ∧ 𝑃 ∈ 𝑉) β†’ (β™―β€˜{𝑀 ∈ (1 WWalksN 𝐺) ∣ (π‘€β€˜0) = 𝑃}) = 𝐾)
Distinct variable groups:   𝑀,𝐺   𝑀,𝐾   𝑀,𝑃   𝑀,𝑉

Proof of Theorem rusgrnumwwlkl1
Dummy variable 𝑖 is distinct from all other variables.
StepHypRef Expression
1 1nn0 12510 . . . . . . . . 9 1 ∈ β„•0
2 iswwlksn 29636 . . . . . . . . 9 (1 ∈ β„•0 β†’ (𝑀 ∈ (1 WWalksN 𝐺) ↔ (𝑀 ∈ (WWalksβ€˜πΊ) ∧ (β™―β€˜π‘€) = (1 + 1))))
31, 2ax-mp 5 . . . . . . . 8 (𝑀 ∈ (1 WWalksN 𝐺) ↔ (𝑀 ∈ (WWalksβ€˜πΊ) ∧ (β™―β€˜π‘€) = (1 + 1)))
4 rusgrnumwwlkl1.v . . . . . . . . . 10 𝑉 = (Vtxβ€˜πΊ)
5 eqid 2727 . . . . . . . . . 10 (Edgβ€˜πΊ) = (Edgβ€˜πΊ)
64, 5iswwlks 29634 . . . . . . . . 9 (𝑀 ∈ (WWalksβ€˜πΊ) ↔ (𝑀 β‰  βˆ… ∧ 𝑀 ∈ Word 𝑉 ∧ βˆ€π‘– ∈ (0..^((β™―β€˜π‘€) βˆ’ 1)){(π‘€β€˜π‘–), (π‘€β€˜(𝑖 + 1))} ∈ (Edgβ€˜πΊ)))
76anbi1i 623 . . . . . . . 8 ((𝑀 ∈ (WWalksβ€˜πΊ) ∧ (β™―β€˜π‘€) = (1 + 1)) ↔ ((𝑀 β‰  βˆ… ∧ 𝑀 ∈ Word 𝑉 ∧ βˆ€π‘– ∈ (0..^((β™―β€˜π‘€) βˆ’ 1)){(π‘€β€˜π‘–), (π‘€β€˜(𝑖 + 1))} ∈ (Edgβ€˜πΊ)) ∧ (β™―β€˜π‘€) = (1 + 1)))
83, 7bitri 275 . . . . . . 7 (𝑀 ∈ (1 WWalksN 𝐺) ↔ ((𝑀 β‰  βˆ… ∧ 𝑀 ∈ Word 𝑉 ∧ βˆ€π‘– ∈ (0..^((β™―β€˜π‘€) βˆ’ 1)){(π‘€β€˜π‘–), (π‘€β€˜(𝑖 + 1))} ∈ (Edgβ€˜πΊ)) ∧ (β™―β€˜π‘€) = (1 + 1)))
98a1i 11 . . . . . 6 ((𝐺 RegUSGraph 𝐾 ∧ 𝑃 ∈ 𝑉) β†’ (𝑀 ∈ (1 WWalksN 𝐺) ↔ ((𝑀 β‰  βˆ… ∧ 𝑀 ∈ Word 𝑉 ∧ βˆ€π‘– ∈ (0..^((β™―β€˜π‘€) βˆ’ 1)){(π‘€β€˜π‘–), (π‘€β€˜(𝑖 + 1))} ∈ (Edgβ€˜πΊ)) ∧ (β™―β€˜π‘€) = (1 + 1))))
109anbi1d 629 . . . . 5 ((𝐺 RegUSGraph 𝐾 ∧ 𝑃 ∈ 𝑉) β†’ ((𝑀 ∈ (1 WWalksN 𝐺) ∧ (π‘€β€˜0) = 𝑃) ↔ (((𝑀 β‰  βˆ… ∧ 𝑀 ∈ Word 𝑉 ∧ βˆ€π‘– ∈ (0..^((β™―β€˜π‘€) βˆ’ 1)){(π‘€β€˜π‘–), (π‘€β€˜(𝑖 + 1))} ∈ (Edgβ€˜πΊ)) ∧ (β™―β€˜π‘€) = (1 + 1)) ∧ (π‘€β€˜0) = 𝑃)))
11 1p1e2 12359 . . . . . . . . . . 11 (1 + 1) = 2
1211eqeq2i 2740 . . . . . . . . . 10 ((β™―β€˜π‘€) = (1 + 1) ↔ (β™―β€˜π‘€) = 2)
1312a1i 11 . . . . . . . . 9 ((𝐺 RegUSGraph 𝐾 ∧ 𝑃 ∈ 𝑉) β†’ ((β™―β€˜π‘€) = (1 + 1) ↔ (β™―β€˜π‘€) = 2))
1413anbi2d 628 . . . . . . . 8 ((𝐺 RegUSGraph 𝐾 ∧ 𝑃 ∈ 𝑉) β†’ (((𝑀 β‰  βˆ… ∧ 𝑀 ∈ Word 𝑉 ∧ βˆ€π‘– ∈ (0..^((β™―β€˜π‘€) βˆ’ 1)){(π‘€β€˜π‘–), (π‘€β€˜(𝑖 + 1))} ∈ (Edgβ€˜πΊ)) ∧ (β™―β€˜π‘€) = (1 + 1)) ↔ ((𝑀 β‰  βˆ… ∧ 𝑀 ∈ Word 𝑉 ∧ βˆ€π‘– ∈ (0..^((β™―β€˜π‘€) βˆ’ 1)){(π‘€β€˜π‘–), (π‘€β€˜(𝑖 + 1))} ∈ (Edgβ€˜πΊ)) ∧ (β™―β€˜π‘€) = 2)))
15 3anass 1093 . . . . . . . . . . . 12 ((𝑀 β‰  βˆ… ∧ 𝑀 ∈ Word 𝑉 ∧ βˆ€π‘– ∈ (0..^((β™―β€˜π‘€) βˆ’ 1)){(π‘€β€˜π‘–), (π‘€β€˜(𝑖 + 1))} ∈ (Edgβ€˜πΊ)) ↔ (𝑀 β‰  βˆ… ∧ (𝑀 ∈ Word 𝑉 ∧ βˆ€π‘– ∈ (0..^((β™―β€˜π‘€) βˆ’ 1)){(π‘€β€˜π‘–), (π‘€β€˜(𝑖 + 1))} ∈ (Edgβ€˜πΊ))))
1615a1i 11 . . . . . . . . . . 11 (((𝐺 RegUSGraph 𝐾 ∧ 𝑃 ∈ 𝑉) ∧ (β™―β€˜π‘€) = 2) β†’ ((𝑀 β‰  βˆ… ∧ 𝑀 ∈ Word 𝑉 ∧ βˆ€π‘– ∈ (0..^((β™―β€˜π‘€) βˆ’ 1)){(π‘€β€˜π‘–), (π‘€β€˜(𝑖 + 1))} ∈ (Edgβ€˜πΊ)) ↔ (𝑀 β‰  βˆ… ∧ (𝑀 ∈ Word 𝑉 ∧ βˆ€π‘– ∈ (0..^((β™―β€˜π‘€) βˆ’ 1)){(π‘€β€˜π‘–), (π‘€β€˜(𝑖 + 1))} ∈ (Edgβ€˜πΊ)))))
17 fveq2 6891 . . . . . . . . . . . . . . . 16 (𝑀 = βˆ… β†’ (β™―β€˜π‘€) = (β™―β€˜βˆ…))
18 hash0 14350 . . . . . . . . . . . . . . . 16 (β™―β€˜βˆ…) = 0
1917, 18eqtrdi 2783 . . . . . . . . . . . . . . 15 (𝑀 = βˆ… β†’ (β™―β€˜π‘€) = 0)
20 2ne0 12338 . . . . . . . . . . . . . . . . 17 2 β‰  0
2120nesymi 2993 . . . . . . . . . . . . . . . 16 Β¬ 0 = 2
22 eqeq1 2731 . . . . . . . . . . . . . . . 16 ((β™―β€˜π‘€) = 0 β†’ ((β™―β€˜π‘€) = 2 ↔ 0 = 2))
2321, 22mtbiri 327 . . . . . . . . . . . . . . 15 ((β™―β€˜π‘€) = 0 β†’ Β¬ (β™―β€˜π‘€) = 2)
2419, 23syl 17 . . . . . . . . . . . . . 14 (𝑀 = βˆ… β†’ Β¬ (β™―β€˜π‘€) = 2)
2524necon2ai 2965 . . . . . . . . . . . . 13 ((β™―β€˜π‘€) = 2 β†’ 𝑀 β‰  βˆ…)
2625adantl 481 . . . . . . . . . . . 12 (((𝐺 RegUSGraph 𝐾 ∧ 𝑃 ∈ 𝑉) ∧ (β™―β€˜π‘€) = 2) β†’ 𝑀 β‰  βˆ…)
2726biantrurd 532 . . . . . . . . . . 11 (((𝐺 RegUSGraph 𝐾 ∧ 𝑃 ∈ 𝑉) ∧ (β™―β€˜π‘€) = 2) β†’ ((𝑀 ∈ Word 𝑉 ∧ βˆ€π‘– ∈ (0..^((β™―β€˜π‘€) βˆ’ 1)){(π‘€β€˜π‘–), (π‘€β€˜(𝑖 + 1))} ∈ (Edgβ€˜πΊ)) ↔ (𝑀 β‰  βˆ… ∧ (𝑀 ∈ Word 𝑉 ∧ βˆ€π‘– ∈ (0..^((β™―β€˜π‘€) βˆ’ 1)){(π‘€β€˜π‘–), (π‘€β€˜(𝑖 + 1))} ∈ (Edgβ€˜πΊ)))))
28 oveq1 7421 . . . . . . . . . . . . . . . . 17 ((β™―β€˜π‘€) = 2 β†’ ((β™―β€˜π‘€) βˆ’ 1) = (2 βˆ’ 1))
29 2m1e1 12360 . . . . . . . . . . . . . . . . 17 (2 βˆ’ 1) = 1
3028, 29eqtrdi 2783 . . . . . . . . . . . . . . . 16 ((β™―β€˜π‘€) = 2 β†’ ((β™―β€˜π‘€) βˆ’ 1) = 1)
3130oveq2d 7430 . . . . . . . . . . . . . . 15 ((β™―β€˜π‘€) = 2 β†’ (0..^((β™―β€˜π‘€) βˆ’ 1)) = (0..^1))
3231adantl 481 . . . . . . . . . . . . . 14 (((𝐺 RegUSGraph 𝐾 ∧ 𝑃 ∈ 𝑉) ∧ (β™―β€˜π‘€) = 2) β†’ (0..^((β™―β€˜π‘€) βˆ’ 1)) = (0..^1))
3332raleqdv 3320 . . . . . . . . . . . . 13 (((𝐺 RegUSGraph 𝐾 ∧ 𝑃 ∈ 𝑉) ∧ (β™―β€˜π‘€) = 2) β†’ (βˆ€π‘– ∈ (0..^((β™―β€˜π‘€) βˆ’ 1)){(π‘€β€˜π‘–), (π‘€β€˜(𝑖 + 1))} ∈ (Edgβ€˜πΊ) ↔ βˆ€π‘– ∈ (0..^1){(π‘€β€˜π‘–), (π‘€β€˜(𝑖 + 1))} ∈ (Edgβ€˜πΊ)))
34 fzo01 13738 . . . . . . . . . . . . . . 15 (0..^1) = {0}
3534raleqi 3318 . . . . . . . . . . . . . 14 (βˆ€π‘– ∈ (0..^1){(π‘€β€˜π‘–), (π‘€β€˜(𝑖 + 1))} ∈ (Edgβ€˜πΊ) ↔ βˆ€π‘– ∈ {0} {(π‘€β€˜π‘–), (π‘€β€˜(𝑖 + 1))} ∈ (Edgβ€˜πΊ))
36 c0ex 11230 . . . . . . . . . . . . . . 15 0 ∈ V
37 fveq2 6891 . . . . . . . . . . . . . . . . 17 (𝑖 = 0 β†’ (π‘€β€˜π‘–) = (π‘€β€˜0))
38 fv0p1e1 12357 . . . . . . . . . . . . . . . . 17 (𝑖 = 0 β†’ (π‘€β€˜(𝑖 + 1)) = (π‘€β€˜1))
3937, 38preq12d 4741 . . . . . . . . . . . . . . . 16 (𝑖 = 0 β†’ {(π‘€β€˜π‘–), (π‘€β€˜(𝑖 + 1))} = {(π‘€β€˜0), (π‘€β€˜1)})
4039eleq1d 2813 . . . . . . . . . . . . . . 15 (𝑖 = 0 β†’ ({(π‘€β€˜π‘–), (π‘€β€˜(𝑖 + 1))} ∈ (Edgβ€˜πΊ) ↔ {(π‘€β€˜0), (π‘€β€˜1)} ∈ (Edgβ€˜πΊ)))
4136, 40ralsn 4681 . . . . . . . . . . . . . 14 (βˆ€π‘– ∈ {0} {(π‘€β€˜π‘–), (π‘€β€˜(𝑖 + 1))} ∈ (Edgβ€˜πΊ) ↔ {(π‘€β€˜0), (π‘€β€˜1)} ∈ (Edgβ€˜πΊ))
4235, 41bitri 275 . . . . . . . . . . . . 13 (βˆ€π‘– ∈ (0..^1){(π‘€β€˜π‘–), (π‘€β€˜(𝑖 + 1))} ∈ (Edgβ€˜πΊ) ↔ {(π‘€β€˜0), (π‘€β€˜1)} ∈ (Edgβ€˜πΊ))
4333, 42bitrdi 287 . . . . . . . . . . . 12 (((𝐺 RegUSGraph 𝐾 ∧ 𝑃 ∈ 𝑉) ∧ (β™―β€˜π‘€) = 2) β†’ (βˆ€π‘– ∈ (0..^((β™―β€˜π‘€) βˆ’ 1)){(π‘€β€˜π‘–), (π‘€β€˜(𝑖 + 1))} ∈ (Edgβ€˜πΊ) ↔ {(π‘€β€˜0), (π‘€β€˜1)} ∈ (Edgβ€˜πΊ)))
4443anbi2d 628 . . . . . . . . . . 11 (((𝐺 RegUSGraph 𝐾 ∧ 𝑃 ∈ 𝑉) ∧ (β™―β€˜π‘€) = 2) β†’ ((𝑀 ∈ Word 𝑉 ∧ βˆ€π‘– ∈ (0..^((β™―β€˜π‘€) βˆ’ 1)){(π‘€β€˜π‘–), (π‘€β€˜(𝑖 + 1))} ∈ (Edgβ€˜πΊ)) ↔ (𝑀 ∈ Word 𝑉 ∧ {(π‘€β€˜0), (π‘€β€˜1)} ∈ (Edgβ€˜πΊ))))
4516, 27, 443bitr2d 307 . . . . . . . . . 10 (((𝐺 RegUSGraph 𝐾 ∧ 𝑃 ∈ 𝑉) ∧ (β™―β€˜π‘€) = 2) β†’ ((𝑀 β‰  βˆ… ∧ 𝑀 ∈ Word 𝑉 ∧ βˆ€π‘– ∈ (0..^((β™―β€˜π‘€) βˆ’ 1)){(π‘€β€˜π‘–), (π‘€β€˜(𝑖 + 1))} ∈ (Edgβ€˜πΊ)) ↔ (𝑀 ∈ Word 𝑉 ∧ {(π‘€β€˜0), (π‘€β€˜1)} ∈ (Edgβ€˜πΊ))))
4645ex 412 . . . . . . . . 9 ((𝐺 RegUSGraph 𝐾 ∧ 𝑃 ∈ 𝑉) β†’ ((β™―β€˜π‘€) = 2 β†’ ((𝑀 β‰  βˆ… ∧ 𝑀 ∈ Word 𝑉 ∧ βˆ€π‘– ∈ (0..^((β™―β€˜π‘€) βˆ’ 1)){(π‘€β€˜π‘–), (π‘€β€˜(𝑖 + 1))} ∈ (Edgβ€˜πΊ)) ↔ (𝑀 ∈ Word 𝑉 ∧ {(π‘€β€˜0), (π‘€β€˜1)} ∈ (Edgβ€˜πΊ)))))
4746pm5.32rd 577 . . . . . . . 8 ((𝐺 RegUSGraph 𝐾 ∧ 𝑃 ∈ 𝑉) β†’ (((𝑀 β‰  βˆ… ∧ 𝑀 ∈ Word 𝑉 ∧ βˆ€π‘– ∈ (0..^((β™―β€˜π‘€) βˆ’ 1)){(π‘€β€˜π‘–), (π‘€β€˜(𝑖 + 1))} ∈ (Edgβ€˜πΊ)) ∧ (β™―β€˜π‘€) = 2) ↔ ((𝑀 ∈ Word 𝑉 ∧ {(π‘€β€˜0), (π‘€β€˜1)} ∈ (Edgβ€˜πΊ)) ∧ (β™―β€˜π‘€) = 2)))
4814, 47bitrd 279 . . . . . . 7 ((𝐺 RegUSGraph 𝐾 ∧ 𝑃 ∈ 𝑉) β†’ (((𝑀 β‰  βˆ… ∧ 𝑀 ∈ Word 𝑉 ∧ βˆ€π‘– ∈ (0..^((β™―β€˜π‘€) βˆ’ 1)){(π‘€β€˜π‘–), (π‘€β€˜(𝑖 + 1))} ∈ (Edgβ€˜πΊ)) ∧ (β™―β€˜π‘€) = (1 + 1)) ↔ ((𝑀 ∈ Word 𝑉 ∧ {(π‘€β€˜0), (π‘€β€˜1)} ∈ (Edgβ€˜πΊ)) ∧ (β™―β€˜π‘€) = 2)))
4948anbi1d 629 . . . . . 6 ((𝐺 RegUSGraph 𝐾 ∧ 𝑃 ∈ 𝑉) β†’ ((((𝑀 β‰  βˆ… ∧ 𝑀 ∈ Word 𝑉 ∧ βˆ€π‘– ∈ (0..^((β™―β€˜π‘€) βˆ’ 1)){(π‘€β€˜π‘–), (π‘€β€˜(𝑖 + 1))} ∈ (Edgβ€˜πΊ)) ∧ (β™―β€˜π‘€) = (1 + 1)) ∧ (π‘€β€˜0) = 𝑃) ↔ (((𝑀 ∈ Word 𝑉 ∧ {(π‘€β€˜0), (π‘€β€˜1)} ∈ (Edgβ€˜πΊ)) ∧ (β™―β€˜π‘€) = 2) ∧ (π‘€β€˜0) = 𝑃)))
50 anass 468 . . . . . 6 ((((𝑀 ∈ Word 𝑉 ∧ {(π‘€β€˜0), (π‘€β€˜1)} ∈ (Edgβ€˜πΊ)) ∧ (β™―β€˜π‘€) = 2) ∧ (π‘€β€˜0) = 𝑃) ↔ ((𝑀 ∈ Word 𝑉 ∧ {(π‘€β€˜0), (π‘€β€˜1)} ∈ (Edgβ€˜πΊ)) ∧ ((β™―β€˜π‘€) = 2 ∧ (π‘€β€˜0) = 𝑃)))
5149, 50bitrdi 287 . . . . 5 ((𝐺 RegUSGraph 𝐾 ∧ 𝑃 ∈ 𝑉) β†’ ((((𝑀 β‰  βˆ… ∧ 𝑀 ∈ Word 𝑉 ∧ βˆ€π‘– ∈ (0..^((β™―β€˜π‘€) βˆ’ 1)){(π‘€β€˜π‘–), (π‘€β€˜(𝑖 + 1))} ∈ (Edgβ€˜πΊ)) ∧ (β™―β€˜π‘€) = (1 + 1)) ∧ (π‘€β€˜0) = 𝑃) ↔ ((𝑀 ∈ Word 𝑉 ∧ {(π‘€β€˜0), (π‘€β€˜1)} ∈ (Edgβ€˜πΊ)) ∧ ((β™―β€˜π‘€) = 2 ∧ (π‘€β€˜0) = 𝑃))))
52 anass 468 . . . . . . 7 (((𝑀 ∈ Word 𝑉 ∧ {(π‘€β€˜0), (π‘€β€˜1)} ∈ (Edgβ€˜πΊ)) ∧ ((β™―β€˜π‘€) = 2 ∧ (π‘€β€˜0) = 𝑃)) ↔ (𝑀 ∈ Word 𝑉 ∧ ({(π‘€β€˜0), (π‘€β€˜1)} ∈ (Edgβ€˜πΊ) ∧ ((β™―β€˜π‘€) = 2 ∧ (π‘€β€˜0) = 𝑃))))
53 ancom 460 . . . . . . . . 9 (({(π‘€β€˜0), (π‘€β€˜1)} ∈ (Edgβ€˜πΊ) ∧ ((β™―β€˜π‘€) = 2 ∧ (π‘€β€˜0) = 𝑃)) ↔ (((β™―β€˜π‘€) = 2 ∧ (π‘€β€˜0) = 𝑃) ∧ {(π‘€β€˜0), (π‘€β€˜1)} ∈ (Edgβ€˜πΊ)))
54 df-3an 1087 . . . . . . . . 9 (((β™―β€˜π‘€) = 2 ∧ (π‘€β€˜0) = 𝑃 ∧ {(π‘€β€˜0), (π‘€β€˜1)} ∈ (Edgβ€˜πΊ)) ↔ (((β™―β€˜π‘€) = 2 ∧ (π‘€β€˜0) = 𝑃) ∧ {(π‘€β€˜0), (π‘€β€˜1)} ∈ (Edgβ€˜πΊ)))
5553, 54bitr4i 278 . . . . . . . 8 (({(π‘€β€˜0), (π‘€β€˜1)} ∈ (Edgβ€˜πΊ) ∧ ((β™―β€˜π‘€) = 2 ∧ (π‘€β€˜0) = 𝑃)) ↔ ((β™―β€˜π‘€) = 2 ∧ (π‘€β€˜0) = 𝑃 ∧ {(π‘€β€˜0), (π‘€β€˜1)} ∈ (Edgβ€˜πΊ)))
5655anbi2i 622 . . . . . . 7 ((𝑀 ∈ Word 𝑉 ∧ ({(π‘€β€˜0), (π‘€β€˜1)} ∈ (Edgβ€˜πΊ) ∧ ((β™―β€˜π‘€) = 2 ∧ (π‘€β€˜0) = 𝑃))) ↔ (𝑀 ∈ Word 𝑉 ∧ ((β™―β€˜π‘€) = 2 ∧ (π‘€β€˜0) = 𝑃 ∧ {(π‘€β€˜0), (π‘€β€˜1)} ∈ (Edgβ€˜πΊ))))
5752, 56bitri 275 . . . . . 6 (((𝑀 ∈ Word 𝑉 ∧ {(π‘€β€˜0), (π‘€β€˜1)} ∈ (Edgβ€˜πΊ)) ∧ ((β™―β€˜π‘€) = 2 ∧ (π‘€β€˜0) = 𝑃)) ↔ (𝑀 ∈ Word 𝑉 ∧ ((β™―β€˜π‘€) = 2 ∧ (π‘€β€˜0) = 𝑃 ∧ {(π‘€β€˜0), (π‘€β€˜1)} ∈ (Edgβ€˜πΊ))))
5857a1i 11 . . . . 5 ((𝐺 RegUSGraph 𝐾 ∧ 𝑃 ∈ 𝑉) β†’ (((𝑀 ∈ Word 𝑉 ∧ {(π‘€β€˜0), (π‘€β€˜1)} ∈ (Edgβ€˜πΊ)) ∧ ((β™―β€˜π‘€) = 2 ∧ (π‘€β€˜0) = 𝑃)) ↔ (𝑀 ∈ Word 𝑉 ∧ ((β™―β€˜π‘€) = 2 ∧ (π‘€β€˜0) = 𝑃 ∧ {(π‘€β€˜0), (π‘€β€˜1)} ∈ (Edgβ€˜πΊ)))))
5910, 51, 583bitrd 305 . . . 4 ((𝐺 RegUSGraph 𝐾 ∧ 𝑃 ∈ 𝑉) β†’ ((𝑀 ∈ (1 WWalksN 𝐺) ∧ (π‘€β€˜0) = 𝑃) ↔ (𝑀 ∈ Word 𝑉 ∧ ((β™―β€˜π‘€) = 2 ∧ (π‘€β€˜0) = 𝑃 ∧ {(π‘€β€˜0), (π‘€β€˜1)} ∈ (Edgβ€˜πΊ)))))
6059rabbidva2 3429 . . 3 ((𝐺 RegUSGraph 𝐾 ∧ 𝑃 ∈ 𝑉) β†’ {𝑀 ∈ (1 WWalksN 𝐺) ∣ (π‘€β€˜0) = 𝑃} = {𝑀 ∈ Word 𝑉 ∣ ((β™―β€˜π‘€) = 2 ∧ (π‘€β€˜0) = 𝑃 ∧ {(π‘€β€˜0), (π‘€β€˜1)} ∈ (Edgβ€˜πΊ))})
6160fveq2d 6895 . 2 ((𝐺 RegUSGraph 𝐾 ∧ 𝑃 ∈ 𝑉) β†’ (β™―β€˜{𝑀 ∈ (1 WWalksN 𝐺) ∣ (π‘€β€˜0) = 𝑃}) = (β™―β€˜{𝑀 ∈ Word 𝑉 ∣ ((β™―β€˜π‘€) = 2 ∧ (π‘€β€˜0) = 𝑃 ∧ {(π‘€β€˜0), (π‘€β€˜1)} ∈ (Edgβ€˜πΊ))}))
624rusgrnumwrdl2 29387 . 2 ((𝐺 RegUSGraph 𝐾 ∧ 𝑃 ∈ 𝑉) β†’ (β™―β€˜{𝑀 ∈ Word 𝑉 ∣ ((β™―β€˜π‘€) = 2 ∧ (π‘€β€˜0) = 𝑃 ∧ {(π‘€β€˜0), (π‘€β€˜1)} ∈ (Edgβ€˜πΊ))}) = 𝐾)
6361, 62eqtrd 2767 1 ((𝐺 RegUSGraph 𝐾 ∧ 𝑃 ∈ 𝑉) β†’ (β™―β€˜{𝑀 ∈ (1 WWalksN 𝐺) ∣ (π‘€β€˜0) = 𝑃}) = 𝐾)
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ↔ wb 205   ∧ wa 395   ∧ w3a 1085   = wceq 1534   ∈ wcel 2099   β‰  wne 2935  βˆ€wral 3056  {crab 3427  βˆ…c0 4318  {csn 4624  {cpr 4626   class class class wbr 5142  β€˜cfv 6542  (class class class)co 7414  0cc0 11130  1c1 11131   + caddc 11133   βˆ’ cmin 11466  2c2 12289  β„•0cn0 12494  ..^cfzo 13651  β™―chash 14313  Word cword 14488  Vtxcvtx 28796  Edgcedg 28847   RegUSGraph crusgr 29357  WWalkscwwlks 29623   WWalksN cwwlksn 29624
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 7734  ax-cnex 11186  ax-resscn 11187  ax-1cn 11188  ax-icn 11189  ax-addcl 11190  ax-addrcl 11191  ax-mulcl 11192  ax-mulrcl 11193  ax-mulcom 11194  ax-addass 11195  ax-mulass 11196  ax-distr 11197  ax-i2m1 11198  ax-1ne0 11199  ax-1rid 11200  ax-rnegex 11201  ax-rrecex 11202  ax-cnre 11203  ax-pre-lttri 11204  ax-pre-lttrn 11205  ax-pre-ltadd 11206  ax-pre-mulgt0 11207
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  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-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-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-riota 7370  df-ov 7417  df-oprab 7418  df-mpo 7419  df-om 7865  df-1st 7987  df-2nd 7988  df-frecs 8280  df-wrecs 8311  df-recs 8385  df-rdg 8424  df-1o 8480  df-2o 8481  df-oadd 8484  df-er 8718  df-map 8838  df-en 8956  df-dom 8957  df-sdom 8958  df-fin 8959  df-dju 9916  df-card 9954  df-pnf 11272  df-mnf 11273  df-xr 11274  df-ltxr 11275  df-le 11276  df-sub 11468  df-neg 11469  df-nn 12235  df-2 12297  df-n0 12495  df-xnn0 12567  df-z 12581  df-uz 12845  df-xadd 13117  df-fz 13509  df-fzo 13652  df-hash 14314  df-word 14489  df-edg 28848  df-uhgr 28858  df-ushgr 28859  df-upgr 28882  df-umgr 28883  df-uspgr 28950  df-usgr 28951  df-nbgr 29133  df-vtxdg 29267  df-rgr 29358  df-rusgr 29359  df-wwlks 29628  df-wwlksn 29629
This theorem is referenced by:  rusgrnumwwlkb1  29770
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