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Theorem wwlksn 29356
Description: The set of walks (in an undirected graph) of a fixed length as words over the set of vertices. (Contributed by Alexander van der Vekens, 15-Jul-2018.) (Revised by AV, 8-Apr-2021.)
Assertion
Ref Expression
wwlksn (𝑁 ∈ β„•0 β†’ (𝑁 WWalksN 𝐺) = {𝑀 ∈ (WWalksβ€˜πΊ) ∣ (β™―β€˜π‘€) = (𝑁 + 1)})
Distinct variable groups:   𝑀,𝐺   𝑀,𝑁

Proof of Theorem wwlksn
Dummy variables 𝑔 𝑛 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fveq2 6892 . . . . . 6 (𝑔 = 𝐺 β†’ (WWalksβ€˜π‘”) = (WWalksβ€˜πΊ))
21adantl 480 . . . . 5 ((𝑛 = 𝑁 ∧ 𝑔 = 𝐺) β†’ (WWalksβ€˜π‘”) = (WWalksβ€˜πΊ))
3 oveq1 7420 . . . . . . 7 (𝑛 = 𝑁 β†’ (𝑛 + 1) = (𝑁 + 1))
43eqeq2d 2741 . . . . . 6 (𝑛 = 𝑁 β†’ ((β™―β€˜π‘€) = (𝑛 + 1) ↔ (β™―β€˜π‘€) = (𝑁 + 1)))
54adantr 479 . . . . 5 ((𝑛 = 𝑁 ∧ 𝑔 = 𝐺) β†’ ((β™―β€˜π‘€) = (𝑛 + 1) ↔ (β™―β€˜π‘€) = (𝑁 + 1)))
62, 5rabeqbidv 3447 . . . 4 ((𝑛 = 𝑁 ∧ 𝑔 = 𝐺) β†’ {𝑀 ∈ (WWalksβ€˜π‘”) ∣ (β™―β€˜π‘€) = (𝑛 + 1)} = {𝑀 ∈ (WWalksβ€˜πΊ) ∣ (β™―β€˜π‘€) = (𝑁 + 1)})
7 df-wwlksn 29350 . . . 4 WWalksN = (𝑛 ∈ β„•0, 𝑔 ∈ V ↦ {𝑀 ∈ (WWalksβ€˜π‘”) ∣ (β™―β€˜π‘€) = (𝑛 + 1)})
8 fvex 6905 . . . . 5 (WWalksβ€˜πΊ) ∈ V
98rabex 5333 . . . 4 {𝑀 ∈ (WWalksβ€˜πΊ) ∣ (β™―β€˜π‘€) = (𝑁 + 1)} ∈ V
106, 7, 9ovmpoa 7567 . . 3 ((𝑁 ∈ β„•0 ∧ 𝐺 ∈ V) β†’ (𝑁 WWalksN 𝐺) = {𝑀 ∈ (WWalksβ€˜πΊ) ∣ (β™―β€˜π‘€) = (𝑁 + 1)})
1110expcom 412 . 2 (𝐺 ∈ V β†’ (𝑁 ∈ β„•0 β†’ (𝑁 WWalksN 𝐺) = {𝑀 ∈ (WWalksβ€˜πΊ) ∣ (β™―β€˜π‘€) = (𝑁 + 1)}))
127reldmmpo 7547 . . . . 5 Rel dom WWalksN
1312ovprc2 7453 . . . 4 (Β¬ 𝐺 ∈ V β†’ (𝑁 WWalksN 𝐺) = βˆ…)
14 fvprc 6884 . . . . . 6 (Β¬ 𝐺 ∈ V β†’ (WWalksβ€˜πΊ) = βˆ…)
1514rabeqdv 3445 . . . . 5 (Β¬ 𝐺 ∈ V β†’ {𝑀 ∈ (WWalksβ€˜πΊ) ∣ (β™―β€˜π‘€) = (𝑁 + 1)} = {𝑀 ∈ βˆ… ∣ (β™―β€˜π‘€) = (𝑁 + 1)})
16 rab0 4383 . . . . 5 {𝑀 ∈ βˆ… ∣ (β™―β€˜π‘€) = (𝑁 + 1)} = βˆ…
1715, 16eqtrdi 2786 . . . 4 (Β¬ 𝐺 ∈ V β†’ {𝑀 ∈ (WWalksβ€˜πΊ) ∣ (β™―β€˜π‘€) = (𝑁 + 1)} = βˆ…)
1813, 17eqtr4d 2773 . . 3 (Β¬ 𝐺 ∈ V β†’ (𝑁 WWalksN 𝐺) = {𝑀 ∈ (WWalksβ€˜πΊ) ∣ (β™―β€˜π‘€) = (𝑁 + 1)})
1918a1d 25 . 2 (Β¬ 𝐺 ∈ V β†’ (𝑁 ∈ β„•0 β†’ (𝑁 WWalksN 𝐺) = {𝑀 ∈ (WWalksβ€˜πΊ) ∣ (β™―β€˜π‘€) = (𝑁 + 1)}))
2011, 19pm2.61i 182 1 (𝑁 ∈ β„•0 β†’ (𝑁 WWalksN 𝐺) = {𝑀 ∈ (WWalksβ€˜πΊ) ∣ (β™―β€˜π‘€) = (𝑁 + 1)})
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ↔ wb 205   ∧ wa 394   = wceq 1539   ∈ wcel 2104  {crab 3430  Vcvv 3472  βˆ…c0 4323  β€˜cfv 6544  (class class class)co 7413  1c1 11115   + caddc 11117  β„•0cn0 12478  β™―chash 14296  WWalkscwwlks 29344   WWalksN cwwlksn 29345
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-sep 5300  ax-nul 5307  ax-pr 5428
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3431  df-v 3474  df-sbc 3779  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-iota 6496  df-fun 6546  df-fv 6552  df-ov 7416  df-oprab 7417  df-mpo 7418  df-wwlksn 29350
This theorem is referenced by:  iswwlksn  29357  wwlksn0s  29380  0enwwlksnge1  29383  wlknwwlksnbij  29407  wwlksnfi  29425
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