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Theorem clwlknf1oclwwlkn 29881
Description: There is a one-to-one onto function between the set of closed walks as words of length 𝑁 and the set of closed walks of length 𝑁 in a simple pseudograph. (Contributed by Alexander van der Vekens, 5-Jul-2018.) (Revised by AV, 3-May-2021.) (Revised by AV, 1-Nov-2022.)
Hypotheses
Ref Expression
clwlknf1oclwwlkn.a 𝐴 = (1st β€˜π‘)
clwlknf1oclwwlkn.b 𝐡 = (2nd β€˜π‘)
clwlknf1oclwwlkn.c 𝐢 = {𝑀 ∈ (ClWalksβ€˜πΊ) ∣ (β™―β€˜(1st β€˜π‘€)) = 𝑁}
clwlknf1oclwwlkn.f 𝐹 = (𝑐 ∈ 𝐢 ↦ (𝐡 prefix (β™―β€˜π΄)))
Assertion
Ref Expression
clwlknf1oclwwlkn ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ β„•) β†’ 𝐹:𝐢–1-1-ontoβ†’(𝑁 ClWWalksN 𝐺))
Distinct variable groups:   𝐢,𝑐   𝐺,𝑐,𝑀   𝑀,𝑁,𝑐
Allowed substitution hints:   𝐴(𝑀,𝑐)   𝐡(𝑀,𝑐)   𝐢(𝑀)   𝐹(𝑀,𝑐)

Proof of Theorem clwlknf1oclwwlkn
Dummy variables 𝑑 𝑠 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2727 . . 3 (𝑐 ∈ {𝑀 ∈ (ClWalksβ€˜πΊ) ∣ 1 ≀ (β™―β€˜(1st β€˜π‘€))} ↦ ((2nd β€˜π‘) prefix ((β™―β€˜(2nd β€˜π‘)) βˆ’ 1))) = (𝑐 ∈ {𝑀 ∈ (ClWalksβ€˜πΊ) ∣ 1 ≀ (β™―β€˜(1st β€˜π‘€))} ↦ ((2nd β€˜π‘) prefix ((β™―β€˜(2nd β€˜π‘)) βˆ’ 1)))
2 2fveq3 6896 . . . . . . . 8 (𝑠 = 𝑀 β†’ (β™―β€˜(1st β€˜π‘ )) = (β™―β€˜(1st β€˜π‘€)))
32breq2d 5154 . . . . . . 7 (𝑠 = 𝑀 β†’ (1 ≀ (β™―β€˜(1st β€˜π‘ )) ↔ 1 ≀ (β™―β€˜(1st β€˜π‘€))))
43cbvrabv 3437 . . . . . 6 {𝑠 ∈ (ClWalksβ€˜πΊ) ∣ 1 ≀ (β™―β€˜(1st β€˜π‘ ))} = {𝑀 ∈ (ClWalksβ€˜πΊ) ∣ 1 ≀ (β™―β€˜(1st β€˜π‘€))}
5 fveq2 6891 . . . . . . . 8 (𝑑 = 𝑐 β†’ (2nd β€˜π‘‘) = (2nd β€˜π‘))
6 2fveq3 6896 . . . . . . . . 9 (𝑑 = 𝑐 β†’ (β™―β€˜(2nd β€˜π‘‘)) = (β™―β€˜(2nd β€˜π‘)))
76oveq1d 7429 . . . . . . . 8 (𝑑 = 𝑐 β†’ ((β™―β€˜(2nd β€˜π‘‘)) βˆ’ 1) = ((β™―β€˜(2nd β€˜π‘)) βˆ’ 1))
85, 7oveq12d 7432 . . . . . . 7 (𝑑 = 𝑐 β†’ ((2nd β€˜π‘‘) prefix ((β™―β€˜(2nd β€˜π‘‘)) βˆ’ 1)) = ((2nd β€˜π‘) prefix ((β™―β€˜(2nd β€˜π‘)) βˆ’ 1)))
98cbvmptv 5255 . . . . . 6 (𝑑 ∈ {𝑠 ∈ (ClWalksβ€˜πΊ) ∣ 1 ≀ (β™―β€˜(1st β€˜π‘ ))} ↦ ((2nd β€˜π‘‘) prefix ((β™―β€˜(2nd β€˜π‘‘)) βˆ’ 1))) = (𝑐 ∈ {𝑠 ∈ (ClWalksβ€˜πΊ) ∣ 1 ≀ (β™―β€˜(1st β€˜π‘ ))} ↦ ((2nd β€˜π‘) prefix ((β™―β€˜(2nd β€˜π‘)) βˆ’ 1)))
104, 9clwlkclwwlkf1o 29808 . . . . 5 (𝐺 ∈ USPGraph β†’ (𝑑 ∈ {𝑠 ∈ (ClWalksβ€˜πΊ) ∣ 1 ≀ (β™―β€˜(1st β€˜π‘ ))} ↦ ((2nd β€˜π‘‘) prefix ((β™―β€˜(2nd β€˜π‘‘)) βˆ’ 1))):{𝑠 ∈ (ClWalksβ€˜πΊ) ∣ 1 ≀ (β™―β€˜(1st β€˜π‘ ))}–1-1-ontoβ†’(ClWWalksβ€˜πΊ))
1110adantr 480 . . . 4 ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ β„•) β†’ (𝑑 ∈ {𝑠 ∈ (ClWalksβ€˜πΊ) ∣ 1 ≀ (β™―β€˜(1st β€˜π‘ ))} ↦ ((2nd β€˜π‘‘) prefix ((β™―β€˜(2nd β€˜π‘‘)) βˆ’ 1))):{𝑠 ∈ (ClWalksβ€˜πΊ) ∣ 1 ≀ (β™―β€˜(1st β€˜π‘ ))}–1-1-ontoβ†’(ClWWalksβ€˜πΊ))
12 2fveq3 6896 . . . . . . . . . 10 (𝑀 = 𝑠 β†’ (β™―β€˜(1st β€˜π‘€)) = (β™―β€˜(1st β€˜π‘ )))
1312breq2d 5154 . . . . . . . . 9 (𝑀 = 𝑠 β†’ (1 ≀ (β™―β€˜(1st β€˜π‘€)) ↔ 1 ≀ (β™―β€˜(1st β€˜π‘ ))))
1413cbvrabv 3437 . . . . . . . 8 {𝑀 ∈ (ClWalksβ€˜πΊ) ∣ 1 ≀ (β™―β€˜(1st β€˜π‘€))} = {𝑠 ∈ (ClWalksβ€˜πΊ) ∣ 1 ≀ (β™―β€˜(1st β€˜π‘ ))}
1514mpteq1i 5238 . . . . . . 7 (𝑐 ∈ {𝑀 ∈ (ClWalksβ€˜πΊ) ∣ 1 ≀ (β™―β€˜(1st β€˜π‘€))} ↦ ((2nd β€˜π‘) prefix ((β™―β€˜(2nd β€˜π‘)) βˆ’ 1))) = (𝑐 ∈ {𝑠 ∈ (ClWalksβ€˜πΊ) ∣ 1 ≀ (β™―β€˜(1st β€˜π‘ ))} ↦ ((2nd β€˜π‘) prefix ((β™―β€˜(2nd β€˜π‘)) βˆ’ 1)))
16 fveq2 6891 . . . . . . . . 9 (𝑐 = 𝑑 β†’ (2nd β€˜π‘) = (2nd β€˜π‘‘))
17 2fveq3 6896 . . . . . . . . . 10 (𝑐 = 𝑑 β†’ (β™―β€˜(2nd β€˜π‘)) = (β™―β€˜(2nd β€˜π‘‘)))
1817oveq1d 7429 . . . . . . . . 9 (𝑐 = 𝑑 β†’ ((β™―β€˜(2nd β€˜π‘)) βˆ’ 1) = ((β™―β€˜(2nd β€˜π‘‘)) βˆ’ 1))
1916, 18oveq12d 7432 . . . . . . . 8 (𝑐 = 𝑑 β†’ ((2nd β€˜π‘) prefix ((β™―β€˜(2nd β€˜π‘)) βˆ’ 1)) = ((2nd β€˜π‘‘) prefix ((β™―β€˜(2nd β€˜π‘‘)) βˆ’ 1)))
2019cbvmptv 5255 . . . . . . 7 (𝑐 ∈ {𝑠 ∈ (ClWalksβ€˜πΊ) ∣ 1 ≀ (β™―β€˜(1st β€˜π‘ ))} ↦ ((2nd β€˜π‘) prefix ((β™―β€˜(2nd β€˜π‘)) βˆ’ 1))) = (𝑑 ∈ {𝑠 ∈ (ClWalksβ€˜πΊ) ∣ 1 ≀ (β™―β€˜(1st β€˜π‘ ))} ↦ ((2nd β€˜π‘‘) prefix ((β™―β€˜(2nd β€˜π‘‘)) βˆ’ 1)))
2115, 20eqtri 2755 . . . . . 6 (𝑐 ∈ {𝑀 ∈ (ClWalksβ€˜πΊ) ∣ 1 ≀ (β™―β€˜(1st β€˜π‘€))} ↦ ((2nd β€˜π‘) prefix ((β™―β€˜(2nd β€˜π‘)) βˆ’ 1))) = (𝑑 ∈ {𝑠 ∈ (ClWalksβ€˜πΊ) ∣ 1 ≀ (β™―β€˜(1st β€˜π‘ ))} ↦ ((2nd β€˜π‘‘) prefix ((β™―β€˜(2nd β€˜π‘‘)) βˆ’ 1)))
2221a1i 11 . . . . 5 ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ β„•) β†’ (𝑐 ∈ {𝑀 ∈ (ClWalksβ€˜πΊ) ∣ 1 ≀ (β™―β€˜(1st β€˜π‘€))} ↦ ((2nd β€˜π‘) prefix ((β™―β€˜(2nd β€˜π‘)) βˆ’ 1))) = (𝑑 ∈ {𝑠 ∈ (ClWalksβ€˜πΊ) ∣ 1 ≀ (β™―β€˜(1st β€˜π‘ ))} ↦ ((2nd β€˜π‘‘) prefix ((β™―β€˜(2nd β€˜π‘‘)) βˆ’ 1))))
234eqcomi 2736 . . . . . 6 {𝑀 ∈ (ClWalksβ€˜πΊ) ∣ 1 ≀ (β™―β€˜(1st β€˜π‘€))} = {𝑠 ∈ (ClWalksβ€˜πΊ) ∣ 1 ≀ (β™―β€˜(1st β€˜π‘ ))}
2423a1i 11 . . . . 5 ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ β„•) β†’ {𝑀 ∈ (ClWalksβ€˜πΊ) ∣ 1 ≀ (β™―β€˜(1st β€˜π‘€))} = {𝑠 ∈ (ClWalksβ€˜πΊ) ∣ 1 ≀ (β™―β€˜(1st β€˜π‘ ))})
25 eqidd 2728 . . . . 5 ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ β„•) β†’ (ClWWalksβ€˜πΊ) = (ClWWalksβ€˜πΊ))
2622, 24, 25f1oeq123d 6827 . . . 4 ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ β„•) β†’ ((𝑐 ∈ {𝑀 ∈ (ClWalksβ€˜πΊ) ∣ 1 ≀ (β™―β€˜(1st β€˜π‘€))} ↦ ((2nd β€˜π‘) prefix ((β™―β€˜(2nd β€˜π‘)) βˆ’ 1))):{𝑀 ∈ (ClWalksβ€˜πΊ) ∣ 1 ≀ (β™―β€˜(1st β€˜π‘€))}–1-1-ontoβ†’(ClWWalksβ€˜πΊ) ↔ (𝑑 ∈ {𝑠 ∈ (ClWalksβ€˜πΊ) ∣ 1 ≀ (β™―β€˜(1st β€˜π‘ ))} ↦ ((2nd β€˜π‘‘) prefix ((β™―β€˜(2nd β€˜π‘‘)) βˆ’ 1))):{𝑠 ∈ (ClWalksβ€˜πΊ) ∣ 1 ≀ (β™―β€˜(1st β€˜π‘ ))}–1-1-ontoβ†’(ClWWalksβ€˜πΊ)))
2711, 26mpbird 257 . . 3 ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ β„•) β†’ (𝑐 ∈ {𝑀 ∈ (ClWalksβ€˜πΊ) ∣ 1 ≀ (β™―β€˜(1st β€˜π‘€))} ↦ ((2nd β€˜π‘) prefix ((β™―β€˜(2nd β€˜π‘)) βˆ’ 1))):{𝑀 ∈ (ClWalksβ€˜πΊ) ∣ 1 ≀ (β™―β€˜(1st β€˜π‘€))}–1-1-ontoβ†’(ClWWalksβ€˜πΊ))
28 fveq2 6891 . . . . . 6 (𝑠 = ((2nd β€˜π‘) prefix ((β™―β€˜(2nd β€˜π‘)) βˆ’ 1)) β†’ (β™―β€˜π‘ ) = (β™―β€˜((2nd β€˜π‘) prefix ((β™―β€˜(2nd β€˜π‘)) βˆ’ 1))))
29283ad2ant3 1133 . . . . 5 (((𝐺 ∈ USPGraph ∧ 𝑁 ∈ β„•) ∧ 𝑐 ∈ {𝑀 ∈ (ClWalksβ€˜πΊ) ∣ 1 ≀ (β™―β€˜(1st β€˜π‘€))} ∧ 𝑠 = ((2nd β€˜π‘) prefix ((β™―β€˜(2nd β€˜π‘)) βˆ’ 1))) β†’ (β™―β€˜π‘ ) = (β™―β€˜((2nd β€˜π‘) prefix ((β™―β€˜(2nd β€˜π‘)) βˆ’ 1))))
30 2fveq3 6896 . . . . . . . . 9 (𝑀 = 𝑐 β†’ (β™―β€˜(1st β€˜π‘€)) = (β™―β€˜(1st β€˜π‘)))
3130breq2d 5154 . . . . . . . 8 (𝑀 = 𝑐 β†’ (1 ≀ (β™―β€˜(1st β€˜π‘€)) ↔ 1 ≀ (β™―β€˜(1st β€˜π‘))))
3231elrab 3680 . . . . . . 7 (𝑐 ∈ {𝑀 ∈ (ClWalksβ€˜πΊ) ∣ 1 ≀ (β™―β€˜(1st β€˜π‘€))} ↔ (𝑐 ∈ (ClWalksβ€˜πΊ) ∧ 1 ≀ (β™―β€˜(1st β€˜π‘))))
33 clwlknf1oclwwlknlem1 29878 . . . . . . 7 ((𝑐 ∈ (ClWalksβ€˜πΊ) ∧ 1 ≀ (β™―β€˜(1st β€˜π‘))) β†’ (β™―β€˜((2nd β€˜π‘) prefix ((β™―β€˜(2nd β€˜π‘)) βˆ’ 1))) = (β™―β€˜(1st β€˜π‘)))
3432, 33sylbi 216 . . . . . 6 (𝑐 ∈ {𝑀 ∈ (ClWalksβ€˜πΊ) ∣ 1 ≀ (β™―β€˜(1st β€˜π‘€))} β†’ (β™―β€˜((2nd β€˜π‘) prefix ((β™―β€˜(2nd β€˜π‘)) βˆ’ 1))) = (β™―β€˜(1st β€˜π‘)))
35343ad2ant2 1132 . . . . 5 (((𝐺 ∈ USPGraph ∧ 𝑁 ∈ β„•) ∧ 𝑐 ∈ {𝑀 ∈ (ClWalksβ€˜πΊ) ∣ 1 ≀ (β™―β€˜(1st β€˜π‘€))} ∧ 𝑠 = ((2nd β€˜π‘) prefix ((β™―β€˜(2nd β€˜π‘)) βˆ’ 1))) β†’ (β™―β€˜((2nd β€˜π‘) prefix ((β™―β€˜(2nd β€˜π‘)) βˆ’ 1))) = (β™―β€˜(1st β€˜π‘)))
3629, 35eqtrd 2767 . . . 4 (((𝐺 ∈ USPGraph ∧ 𝑁 ∈ β„•) ∧ 𝑐 ∈ {𝑀 ∈ (ClWalksβ€˜πΊ) ∣ 1 ≀ (β™―β€˜(1st β€˜π‘€))} ∧ 𝑠 = ((2nd β€˜π‘) prefix ((β™―β€˜(2nd β€˜π‘)) βˆ’ 1))) β†’ (β™―β€˜π‘ ) = (β™―β€˜(1st β€˜π‘)))
3736eqeq1d 2729 . . 3 (((𝐺 ∈ USPGraph ∧ 𝑁 ∈ β„•) ∧ 𝑐 ∈ {𝑀 ∈ (ClWalksβ€˜πΊ) ∣ 1 ≀ (β™―β€˜(1st β€˜π‘€))} ∧ 𝑠 = ((2nd β€˜π‘) prefix ((β™―β€˜(2nd β€˜π‘)) βˆ’ 1))) β†’ ((β™―β€˜π‘ ) = 𝑁 ↔ (β™―β€˜(1st β€˜π‘)) = 𝑁))
381, 27, 37f1oresrab 7130 . 2 ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ β„•) β†’ ((𝑐 ∈ {𝑀 ∈ (ClWalksβ€˜πΊ) ∣ 1 ≀ (β™―β€˜(1st β€˜π‘€))} ↦ ((2nd β€˜π‘) prefix ((β™―β€˜(2nd β€˜π‘)) βˆ’ 1))) β†Ύ {𝑐 ∈ {𝑀 ∈ (ClWalksβ€˜πΊ) ∣ 1 ≀ (β™―β€˜(1st β€˜π‘€))} ∣ (β™―β€˜(1st β€˜π‘)) = 𝑁}):{𝑐 ∈ {𝑀 ∈ (ClWalksβ€˜πΊ) ∣ 1 ≀ (β™―β€˜(1st β€˜π‘€))} ∣ (β™―β€˜(1st β€˜π‘)) = 𝑁}–1-1-ontoβ†’{𝑠 ∈ (ClWWalksβ€˜πΊ) ∣ (β™―β€˜π‘ ) = 𝑁})
39 clwlknf1oclwwlkn.a . . . . 5 𝐴 = (1st β€˜π‘)
40 clwlknf1oclwwlkn.b . . . . 5 𝐡 = (2nd β€˜π‘)
41 clwlknf1oclwwlkn.c . . . . 5 𝐢 = {𝑀 ∈ (ClWalksβ€˜πΊ) ∣ (β™―β€˜(1st β€˜π‘€)) = 𝑁}
42 clwlknf1oclwwlkn.f . . . . 5 𝐹 = (𝑐 ∈ 𝐢 ↦ (𝐡 prefix (β™―β€˜π΄)))
4339, 40, 41, 42clwlknf1oclwwlknlem3 29880 . . . 4 ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ β„•) β†’ 𝐹 = ((𝑐 ∈ {𝑀 ∈ (ClWalksβ€˜πΊ) ∣ 1 ≀ (β™―β€˜(1st β€˜π‘€))} ↦ (𝐡 prefix (β™―β€˜π΄))) β†Ύ 𝐢))
4440a1i 11 . . . . . . 7 (((𝐺 ∈ USPGraph ∧ 𝑁 ∈ β„•) ∧ 𝑐 ∈ {𝑀 ∈ (ClWalksβ€˜πΊ) ∣ 1 ≀ (β™―β€˜(1st β€˜π‘€))}) β†’ 𝐡 = (2nd β€˜π‘))
45 clwlkwlk 29576 . . . . . . . . . . 11 (𝑐 ∈ (ClWalksβ€˜πΊ) β†’ 𝑐 ∈ (Walksβ€˜πΊ))
46 wlkcpr 29430 . . . . . . . . . . . 12 (𝑐 ∈ (Walksβ€˜πΊ) ↔ (1st β€˜π‘)(Walksβ€˜πΊ)(2nd β€˜π‘))
4739fveq2i 6894 . . . . . . . . . . . . 13 (β™―β€˜π΄) = (β™―β€˜(1st β€˜π‘))
48 wlklenvm1 29423 . . . . . . . . . . . . 13 ((1st β€˜π‘)(Walksβ€˜πΊ)(2nd β€˜π‘) β†’ (β™―β€˜(1st β€˜π‘)) = ((β™―β€˜(2nd β€˜π‘)) βˆ’ 1))
4947, 48eqtrid 2779 . . . . . . . . . . . 12 ((1st β€˜π‘)(Walksβ€˜πΊ)(2nd β€˜π‘) β†’ (β™―β€˜π΄) = ((β™―β€˜(2nd β€˜π‘)) βˆ’ 1))
5046, 49sylbi 216 . . . . . . . . . . 11 (𝑐 ∈ (Walksβ€˜πΊ) β†’ (β™―β€˜π΄) = ((β™―β€˜(2nd β€˜π‘)) βˆ’ 1))
5145, 50syl 17 . . . . . . . . . 10 (𝑐 ∈ (ClWalksβ€˜πΊ) β†’ (β™―β€˜π΄) = ((β™―β€˜(2nd β€˜π‘)) βˆ’ 1))
5251adantr 480 . . . . . . . . 9 ((𝑐 ∈ (ClWalksβ€˜πΊ) ∧ 1 ≀ (β™―β€˜(1st β€˜π‘))) β†’ (β™―β€˜π΄) = ((β™―β€˜(2nd β€˜π‘)) βˆ’ 1))
5332, 52sylbi 216 . . . . . . . 8 (𝑐 ∈ {𝑀 ∈ (ClWalksβ€˜πΊ) ∣ 1 ≀ (β™―β€˜(1st β€˜π‘€))} β†’ (β™―β€˜π΄) = ((β™―β€˜(2nd β€˜π‘)) βˆ’ 1))
5453adantl 481 . . . . . . 7 (((𝐺 ∈ USPGraph ∧ 𝑁 ∈ β„•) ∧ 𝑐 ∈ {𝑀 ∈ (ClWalksβ€˜πΊ) ∣ 1 ≀ (β™―β€˜(1st β€˜π‘€))}) β†’ (β™―β€˜π΄) = ((β™―β€˜(2nd β€˜π‘)) βˆ’ 1))
5544, 54oveq12d 7432 . . . . . 6 (((𝐺 ∈ USPGraph ∧ 𝑁 ∈ β„•) ∧ 𝑐 ∈ {𝑀 ∈ (ClWalksβ€˜πΊ) ∣ 1 ≀ (β™―β€˜(1st β€˜π‘€))}) β†’ (𝐡 prefix (β™―β€˜π΄)) = ((2nd β€˜π‘) prefix ((β™―β€˜(2nd β€˜π‘)) βˆ’ 1)))
5655mpteq2dva 5242 . . . . 5 ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ β„•) β†’ (𝑐 ∈ {𝑀 ∈ (ClWalksβ€˜πΊ) ∣ 1 ≀ (β™―β€˜(1st β€˜π‘€))} ↦ (𝐡 prefix (β™―β€˜π΄))) = (𝑐 ∈ {𝑀 ∈ (ClWalksβ€˜πΊ) ∣ 1 ≀ (β™―β€˜(1st β€˜π‘€))} ↦ ((2nd β€˜π‘) prefix ((β™―β€˜(2nd β€˜π‘)) βˆ’ 1))))
5730eqeq1d 2729 . . . . . . . 8 (𝑀 = 𝑐 β†’ ((β™―β€˜(1st β€˜π‘€)) = 𝑁 ↔ (β™―β€˜(1st β€˜π‘)) = 𝑁))
5857cbvrabv 3437 . . . . . . 7 {𝑀 ∈ (ClWalksβ€˜πΊ) ∣ (β™―β€˜(1st β€˜π‘€)) = 𝑁} = {𝑐 ∈ (ClWalksβ€˜πΊ) ∣ (β™―β€˜(1st β€˜π‘)) = 𝑁}
59 nnge1 12262 . . . . . . . . . . . 12 (𝑁 ∈ β„• β†’ 1 ≀ 𝑁)
60 breq2 5146 . . . . . . . . . . . 12 ((β™―β€˜(1st β€˜π‘)) = 𝑁 β†’ (1 ≀ (β™―β€˜(1st β€˜π‘)) ↔ 1 ≀ 𝑁))
6159, 60syl5ibrcom 246 . . . . . . . . . . 11 (𝑁 ∈ β„• β†’ ((β™―β€˜(1st β€˜π‘)) = 𝑁 β†’ 1 ≀ (β™―β€˜(1st β€˜π‘))))
6261adantl 481 . . . . . . . . . 10 ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ β„•) β†’ ((β™―β€˜(1st β€˜π‘)) = 𝑁 β†’ 1 ≀ (β™―β€˜(1st β€˜π‘))))
6362adantr 480 . . . . . . . . 9 (((𝐺 ∈ USPGraph ∧ 𝑁 ∈ β„•) ∧ 𝑐 ∈ (ClWalksβ€˜πΊ)) β†’ ((β™―β€˜(1st β€˜π‘)) = 𝑁 β†’ 1 ≀ (β™―β€˜(1st β€˜π‘))))
6463pm4.71rd 562 . . . . . . . 8 (((𝐺 ∈ USPGraph ∧ 𝑁 ∈ β„•) ∧ 𝑐 ∈ (ClWalksβ€˜πΊ)) β†’ ((β™―β€˜(1st β€˜π‘)) = 𝑁 ↔ (1 ≀ (β™―β€˜(1st β€˜π‘)) ∧ (β™―β€˜(1st β€˜π‘)) = 𝑁)))
6564rabbidva 3434 . . . . . . 7 ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ β„•) β†’ {𝑐 ∈ (ClWalksβ€˜πΊ) ∣ (β™―β€˜(1st β€˜π‘)) = 𝑁} = {𝑐 ∈ (ClWalksβ€˜πΊ) ∣ (1 ≀ (β™―β€˜(1st β€˜π‘)) ∧ (β™―β€˜(1st β€˜π‘)) = 𝑁)})
6658, 65eqtrid 2779 . . . . . 6 ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ β„•) β†’ {𝑀 ∈ (ClWalksβ€˜πΊ) ∣ (β™―β€˜(1st β€˜π‘€)) = 𝑁} = {𝑐 ∈ (ClWalksβ€˜πΊ) ∣ (1 ≀ (β™―β€˜(1st β€˜π‘)) ∧ (β™―β€˜(1st β€˜π‘)) = 𝑁)})
6732anbi1i 623 . . . . . . . 8 ((𝑐 ∈ {𝑀 ∈ (ClWalksβ€˜πΊ) ∣ 1 ≀ (β™―β€˜(1st β€˜π‘€))} ∧ (β™―β€˜(1st β€˜π‘)) = 𝑁) ↔ ((𝑐 ∈ (ClWalksβ€˜πΊ) ∧ 1 ≀ (β™―β€˜(1st β€˜π‘))) ∧ (β™―β€˜(1st β€˜π‘)) = 𝑁))
68 anass 468 . . . . . . . 8 (((𝑐 ∈ (ClWalksβ€˜πΊ) ∧ 1 ≀ (β™―β€˜(1st β€˜π‘))) ∧ (β™―β€˜(1st β€˜π‘)) = 𝑁) ↔ (𝑐 ∈ (ClWalksβ€˜πΊ) ∧ (1 ≀ (β™―β€˜(1st β€˜π‘)) ∧ (β™―β€˜(1st β€˜π‘)) = 𝑁)))
6967, 68bitri 275 . . . . . . 7 ((𝑐 ∈ {𝑀 ∈ (ClWalksβ€˜πΊ) ∣ 1 ≀ (β™―β€˜(1st β€˜π‘€))} ∧ (β™―β€˜(1st β€˜π‘)) = 𝑁) ↔ (𝑐 ∈ (ClWalksβ€˜πΊ) ∧ (1 ≀ (β™―β€˜(1st β€˜π‘)) ∧ (β™―β€˜(1st β€˜π‘)) = 𝑁)))
7069rabbia2 3430 . . . . . 6 {𝑐 ∈ {𝑀 ∈ (ClWalksβ€˜πΊ) ∣ 1 ≀ (β™―β€˜(1st β€˜π‘€))} ∣ (β™―β€˜(1st β€˜π‘)) = 𝑁} = {𝑐 ∈ (ClWalksβ€˜πΊ) ∣ (1 ≀ (β™―β€˜(1st β€˜π‘)) ∧ (β™―β€˜(1st β€˜π‘)) = 𝑁)}
7166, 41, 703eqtr4g 2792 . . . . 5 ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ β„•) β†’ 𝐢 = {𝑐 ∈ {𝑀 ∈ (ClWalksβ€˜πΊ) ∣ 1 ≀ (β™―β€˜(1st β€˜π‘€))} ∣ (β™―β€˜(1st β€˜π‘)) = 𝑁})
7256, 71reseq12d 5980 . . . 4 ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ β„•) β†’ ((𝑐 ∈ {𝑀 ∈ (ClWalksβ€˜πΊ) ∣ 1 ≀ (β™―β€˜(1st β€˜π‘€))} ↦ (𝐡 prefix (β™―β€˜π΄))) β†Ύ 𝐢) = ((𝑐 ∈ {𝑀 ∈ (ClWalksβ€˜πΊ) ∣ 1 ≀ (β™―β€˜(1st β€˜π‘€))} ↦ ((2nd β€˜π‘) prefix ((β™―β€˜(2nd β€˜π‘)) βˆ’ 1))) β†Ύ {𝑐 ∈ {𝑀 ∈ (ClWalksβ€˜πΊ) ∣ 1 ≀ (β™―β€˜(1st β€˜π‘€))} ∣ (β™―β€˜(1st β€˜π‘)) = 𝑁}))
7343, 72eqtrd 2767 . . 3 ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ β„•) β†’ 𝐹 = ((𝑐 ∈ {𝑀 ∈ (ClWalksβ€˜πΊ) ∣ 1 ≀ (β™―β€˜(1st β€˜π‘€))} ↦ ((2nd β€˜π‘) prefix ((β™―β€˜(2nd β€˜π‘)) βˆ’ 1))) β†Ύ {𝑐 ∈ {𝑀 ∈ (ClWalksβ€˜πΊ) ∣ 1 ≀ (β™―β€˜(1st β€˜π‘€))} ∣ (β™―β€˜(1st β€˜π‘)) = 𝑁}))
74 clwlknf1oclwwlknlem2 29879 . . . . 5 (𝑁 ∈ β„• β†’ {𝑀 ∈ (ClWalksβ€˜πΊ) ∣ (β™―β€˜(1st β€˜π‘€)) = 𝑁} = {𝑐 ∈ (ClWalksβ€˜πΊ) ∣ (1 ≀ (β™―β€˜(1st β€˜π‘)) ∧ (β™―β€˜(1st β€˜π‘)) = 𝑁)})
7574adantl 481 . . . 4 ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ β„•) β†’ {𝑀 ∈ (ClWalksβ€˜πΊ) ∣ (β™―β€˜(1st β€˜π‘€)) = 𝑁} = {𝑐 ∈ (ClWalksβ€˜πΊ) ∣ (1 ≀ (β™―β€˜(1st β€˜π‘)) ∧ (β™―β€˜(1st β€˜π‘)) = 𝑁)})
7675, 41, 703eqtr4g 2792 . . 3 ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ β„•) β†’ 𝐢 = {𝑐 ∈ {𝑀 ∈ (ClWalksβ€˜πΊ) ∣ 1 ≀ (β™―β€˜(1st β€˜π‘€))} ∣ (β™―β€˜(1st β€˜π‘)) = 𝑁})
77 clwwlkn 29823 . . . 4 (𝑁 ClWWalksN 𝐺) = {𝑠 ∈ (ClWWalksβ€˜πΊ) ∣ (β™―β€˜π‘ ) = 𝑁}
7877a1i 11 . . 3 ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ β„•) β†’ (𝑁 ClWWalksN 𝐺) = {𝑠 ∈ (ClWWalksβ€˜πΊ) ∣ (β™―β€˜π‘ ) = 𝑁})
7973, 76, 78f1oeq123d 6827 . 2 ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ β„•) β†’ (𝐹:𝐢–1-1-ontoβ†’(𝑁 ClWWalksN 𝐺) ↔ ((𝑐 ∈ {𝑀 ∈ (ClWalksβ€˜πΊ) ∣ 1 ≀ (β™―β€˜(1st β€˜π‘€))} ↦ ((2nd β€˜π‘) prefix ((β™―β€˜(2nd β€˜π‘)) βˆ’ 1))) β†Ύ {𝑐 ∈ {𝑀 ∈ (ClWalksβ€˜πΊ) ∣ 1 ≀ (β™―β€˜(1st β€˜π‘€))} ∣ (β™―β€˜(1st β€˜π‘)) = 𝑁}):{𝑐 ∈ {𝑀 ∈ (ClWalksβ€˜πΊ) ∣ 1 ≀ (β™―β€˜(1st β€˜π‘€))} ∣ (β™―β€˜(1st β€˜π‘)) = 𝑁}–1-1-ontoβ†’{𝑠 ∈ (ClWWalksβ€˜πΊ) ∣ (β™―β€˜π‘ ) = 𝑁}))
8038, 79mpbird 257 1 ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ β„•) β†’ 𝐹:𝐢–1-1-ontoβ†’(𝑁 ClWWalksN 𝐺))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 395   ∧ w3a 1085   = wceq 1534   ∈ wcel 2099  {crab 3427   class class class wbr 5142   ↦ cmpt 5225   β†Ύ cres 5674  β€“1-1-ontoβ†’wf1o 6541  β€˜cfv 6542  (class class class)co 7414  1st c1st 7985  2nd c2nd 7986  1c1 11131   ≀ cle 11271   βˆ’ cmin 11466  β„•cn 12234  β™―chash 14313   prefix cpfx 14644  USPGraphcuspgr 28948  Walkscwlks 29397  ClWalkscclwlks 29571  ClWWalkscclwwlk 29778   ClWWalksN cclwwlkn 29821
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-ifp 1062  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-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-pm 8839  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-rp 12999  df-fz 13509  df-fzo 13652  df-hash 14314  df-word 14489  df-lsw 14537  df-concat 14545  df-s1 14570  df-substr 14615  df-pfx 14645  df-edg 28848  df-uhgr 28858  df-upgr 28882  df-uspgr 28950  df-wlks 29400  df-clwlks 29572  df-clwwlk 29779  df-clwwlkn 29822
This theorem is referenced by:  clwlkssizeeq  29882  clwwlknonclwlknonf1o  30159
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