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Theorem wwlksnon 29094
Description: The set of walks of a fixed length between two vertices as word. (Contributed by Alexander van der Vekens, 15-Feb-2018.) (Revised by AV, 11-May-2021.)
Hypothesis
Ref Expression
wwlksnon.v 𝑉 = (Vtx‘𝐺)
Assertion
Ref Expression
wwlksnon ((𝑁 ∈ ℕ0 ∧ 𝐺 ∈ 𝑈) → (𝑁 WWalksNOn 𝐺) = (𝑎 ∈ 𝑉, 𝑏 ∈ 𝑉 ↩ {𝑀 ∈ (𝑁 WWalksN 𝐺) ∣ ((𝑀‘0) = 𝑎 ∧ (𝑀‘𝑁) = 𝑏)}))
Distinct variable groups:   𝐺,𝑎,𝑏,𝑀   𝑁,𝑎,𝑏,𝑀   𝑉,𝑎,𝑏
Allowed substitution hints:   𝑈(𝑀,𝑎,𝑏)   𝑉(𝑀)

Proof of Theorem wwlksnon
Dummy variables 𝑔 𝑛 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-wwlksnon 29075 . . 3 WWalksNOn = (𝑛 ∈ ℕ0, 𝑔 ∈ V ↩ (𝑎 ∈ (Vtx‘𝑔), 𝑏 ∈ (Vtx‘𝑔) ↩ {𝑀 ∈ (𝑛 WWalksN 𝑔) ∣ ((𝑀‘0) = 𝑎 ∧ (𝑀‘𝑛) = 𝑏)}))
21a1i 11 . 2 ((𝑁 ∈ ℕ0 ∧ 𝐺 ∈ 𝑈) → WWalksNOn = (𝑛 ∈ ℕ0, 𝑔 ∈ V ↩ (𝑎 ∈ (Vtx‘𝑔), 𝑏 ∈ (Vtx‘𝑔) ↩ {𝑀 ∈ (𝑛 WWalksN 𝑔) ∣ ((𝑀‘0) = 𝑎 ∧ (𝑀‘𝑛) = 𝑏)})))
3 fveq2 6888 . . . . . 6 (𝑔 = 𝐺 → (Vtx‘𝑔) = (Vtx‘𝐺))
4 wwlksnon.v . . . . . 6 𝑉 = (Vtx‘𝐺)
53, 4eqtr4di 2790 . . . . 5 (𝑔 = 𝐺 → (Vtx‘𝑔) = 𝑉)
65adantl 482 . . . 4 ((𝑛 = 𝑁 ∧ 𝑔 = 𝐺) → (Vtx‘𝑔) = 𝑉)
7 oveq12 7414 . . . . 5 ((𝑛 = 𝑁 ∧ 𝑔 = 𝐺) → (𝑛 WWalksN 𝑔) = (𝑁 WWalksN 𝐺))
8 fveqeq2 6897 . . . . . . 7 (𝑛 = 𝑁 → ((𝑀‘𝑛) = 𝑏 ↔ (𝑀‘𝑁) = 𝑏))
98anbi2d 629 . . . . . 6 (𝑛 = 𝑁 → (((𝑀‘0) = 𝑎 ∧ (𝑀‘𝑛) = 𝑏) ↔ ((𝑀‘0) = 𝑎 ∧ (𝑀‘𝑁) = 𝑏)))
109adantr 481 . . . . 5 ((𝑛 = 𝑁 ∧ 𝑔 = 𝐺) → (((𝑀‘0) = 𝑎 ∧ (𝑀‘𝑛) = 𝑏) ↔ ((𝑀‘0) = 𝑎 ∧ (𝑀‘𝑁) = 𝑏)))
117, 10rabeqbidv 3449 . . . 4 ((𝑛 = 𝑁 ∧ 𝑔 = 𝐺) → {𝑀 ∈ (𝑛 WWalksN 𝑔) ∣ ((𝑀‘0) = 𝑎 ∧ (𝑀‘𝑛) = 𝑏)} = {𝑀 ∈ (𝑁 WWalksN 𝐺) ∣ ((𝑀‘0) = 𝑎 ∧ (𝑀‘𝑁) = 𝑏)})
126, 6, 11mpoeq123dv 7480 . . 3 ((𝑛 = 𝑁 ∧ 𝑔 = 𝐺) → (𝑎 ∈ (Vtx‘𝑔), 𝑏 ∈ (Vtx‘𝑔) ↩ {𝑀 ∈ (𝑛 WWalksN 𝑔) ∣ ((𝑀‘0) = 𝑎 ∧ (𝑀‘𝑛) = 𝑏)}) = (𝑎 ∈ 𝑉, 𝑏 ∈ 𝑉 ↩ {𝑀 ∈ (𝑁 WWalksN 𝐺) ∣ ((𝑀‘0) = 𝑎 ∧ (𝑀‘𝑁) = 𝑏)}))
1312adantl 482 . 2 (((𝑁 ∈ ℕ0 ∧ 𝐺 ∈ 𝑈) ∧ (𝑛 = 𝑁 ∧ 𝑔 = 𝐺)) → (𝑎 ∈ (Vtx‘𝑔), 𝑏 ∈ (Vtx‘𝑔) ↩ {𝑀 ∈ (𝑛 WWalksN 𝑔) ∣ ((𝑀‘0) = 𝑎 ∧ (𝑀‘𝑛) = 𝑏)}) = (𝑎 ∈ 𝑉, 𝑏 ∈ 𝑉 ↩ {𝑀 ∈ (𝑁 WWalksN 𝐺) ∣ ((𝑀‘0) = 𝑎 ∧ (𝑀‘𝑁) = 𝑏)}))
14 simpl 483 . 2 ((𝑁 ∈ ℕ0 ∧ 𝐺 ∈ 𝑈) → 𝑁 ∈ ℕ0)
15 elex 3492 . . 3 (𝐺 ∈ 𝑈 → 𝐺 ∈ V)
1615adantl 482 . 2 ((𝑁 ∈ ℕ0 ∧ 𝐺 ∈ 𝑈) → 𝐺 ∈ V)
174fvexi 6902 . . . 4 𝑉 ∈ V
1817, 17mpoex 8062 . . 3 (𝑎 ∈ 𝑉, 𝑏 ∈ 𝑉 ↩ {𝑀 ∈ (𝑁 WWalksN 𝐺) ∣ ((𝑀‘0) = 𝑎 ∧ (𝑀‘𝑁) = 𝑏)}) ∈ V
1918a1i 11 . 2 ((𝑁 ∈ ℕ0 ∧ 𝐺 ∈ 𝑈) → (𝑎 ∈ 𝑉, 𝑏 ∈ 𝑉 ↩ {𝑀 ∈ (𝑁 WWalksN 𝐺) ∣ ((𝑀‘0) = 𝑎 ∧ (𝑀‘𝑁) = 𝑏)}) ∈ V)
202, 13, 14, 16, 19ovmpod 7556 1 ((𝑁 ∈ ℕ0 ∧ 𝐺 ∈ 𝑈) → (𝑁 WWalksNOn 𝐺) = (𝑎 ∈ 𝑉, 𝑏 ∈ 𝑉 ↩ {𝑀 ∈ (𝑁 WWalksN 𝐺) ∣ ((𝑀‘0) = 𝑎 ∧ (𝑀‘𝑁) = 𝑏)}))
Colors of variables: wff setvar class
Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1541   ∈ wcel 2106  {crab 3432  Vcvv 3474  â€˜cfv 6540  (class class class)co 7405   ∈ cmpo 7407  0cc0 11106  â„•0cn0 12468  Vtxcvtx 28245   WWalksN cwwlksn 29069   WWalksNOn cwwlksnon 29070
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7721
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-ov 7408  df-oprab 7409  df-mpo 7410  df-1st 7971  df-2nd 7972  df-wwlksnon 29075
This theorem is referenced by:  iswwlksnon  29096  wwlksnon0  29097
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