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Theorem wwlksnon 41049
 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 ∈ ℕ0𝐺𝑈) → (𝑁 WWalksNOn 𝐺) = (𝑎𝑉, 𝑏𝑉 ↦ {𝑤 ∈ (𝑁 WWalkSN 𝐺) ∣ ((𝑤‘0) = 𝑎 ∧ (𝑤𝑁) = 𝑏)}))
Distinct variable groups:   𝐺,𝑎,𝑏,𝑤   𝑁,𝑎,𝑏,𝑤   𝑉,𝑎,𝑏
Allowed substitution hints:   𝑈(𝑤,𝑎,𝑏)   𝑉(𝑤)

Proof of Theorem wwlksnon
Dummy variables 𝑔 𝑛 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-wwlksnon 41035 . . 3 WWalksNOn = (𝑛 ∈ ℕ0, 𝑔 ∈ V ↦ (𝑎 ∈ (Vtx‘𝑔), 𝑏 ∈ (Vtx‘𝑔) ↦ {𝑤 ∈ (𝑛 WWalkSN 𝑔) ∣ ((𝑤‘0) = 𝑎 ∧ (𝑤𝑛) = 𝑏)}))
21a1i 11 . 2 ((𝑁 ∈ ℕ0𝐺𝑈) → WWalksNOn = (𝑛 ∈ ℕ0, 𝑔 ∈ V ↦ (𝑎 ∈ (Vtx‘𝑔), 𝑏 ∈ (Vtx‘𝑔) ↦ {𝑤 ∈ (𝑛 WWalkSN 𝑔) ∣ ((𝑤‘0) = 𝑎 ∧ (𝑤𝑛) = 𝑏)})))
3 fveq2 6103 . . . . . 6 (𝑔 = 𝐺 → (Vtx‘𝑔) = (Vtx‘𝐺))
4 wwlksnon.v . . . . . 6 𝑉 = (Vtx‘𝐺)
53, 4syl6eqr 2662 . . . . 5 (𝑔 = 𝐺 → (Vtx‘𝑔) = 𝑉)
65adantl 481 . . . 4 ((𝑛 = 𝑁𝑔 = 𝐺) → (Vtx‘𝑔) = 𝑉)
7 oveq12 6558 . . . . 5 ((𝑛 = 𝑁𝑔 = 𝐺) → (𝑛 WWalkSN 𝑔) = (𝑁 WWalkSN 𝐺))
8 fveq2 6103 . . . . . . . 8 (𝑛 = 𝑁 → (𝑤𝑛) = (𝑤𝑁))
98eqeq1d 2612 . . . . . . 7 (𝑛 = 𝑁 → ((𝑤𝑛) = 𝑏 ↔ (𝑤𝑁) = 𝑏))
109anbi2d 736 . . . . . 6 (𝑛 = 𝑁 → (((𝑤‘0) = 𝑎 ∧ (𝑤𝑛) = 𝑏) ↔ ((𝑤‘0) = 𝑎 ∧ (𝑤𝑁) = 𝑏)))
1110adantr 480 . . . . 5 ((𝑛 = 𝑁𝑔 = 𝐺) → (((𝑤‘0) = 𝑎 ∧ (𝑤𝑛) = 𝑏) ↔ ((𝑤‘0) = 𝑎 ∧ (𝑤𝑁) = 𝑏)))
127, 11rabeqbidv 3168 . . . 4 ((𝑛 = 𝑁𝑔 = 𝐺) → {𝑤 ∈ (𝑛 WWalkSN 𝑔) ∣ ((𝑤‘0) = 𝑎 ∧ (𝑤𝑛) = 𝑏)} = {𝑤 ∈ (𝑁 WWalkSN 𝐺) ∣ ((𝑤‘0) = 𝑎 ∧ (𝑤𝑁) = 𝑏)})
136, 6, 12mpt2eq123dv 6615 . . 3 ((𝑛 = 𝑁𝑔 = 𝐺) → (𝑎 ∈ (Vtx‘𝑔), 𝑏 ∈ (Vtx‘𝑔) ↦ {𝑤 ∈ (𝑛 WWalkSN 𝑔) ∣ ((𝑤‘0) = 𝑎 ∧ (𝑤𝑛) = 𝑏)}) = (𝑎𝑉, 𝑏𝑉 ↦ {𝑤 ∈ (𝑁 WWalkSN 𝐺) ∣ ((𝑤‘0) = 𝑎 ∧ (𝑤𝑁) = 𝑏)}))
1413adantl 481 . 2 (((𝑁 ∈ ℕ0𝐺𝑈) ∧ (𝑛 = 𝑁𝑔 = 𝐺)) → (𝑎 ∈ (Vtx‘𝑔), 𝑏 ∈ (Vtx‘𝑔) ↦ {𝑤 ∈ (𝑛 WWalkSN 𝑔) ∣ ((𝑤‘0) = 𝑎 ∧ (𝑤𝑛) = 𝑏)}) = (𝑎𝑉, 𝑏𝑉 ↦ {𝑤 ∈ (𝑁 WWalkSN 𝐺) ∣ ((𝑤‘0) = 𝑎 ∧ (𝑤𝑁) = 𝑏)}))
15 simpl 472 . 2 ((𝑁 ∈ ℕ0𝐺𝑈) → 𝑁 ∈ ℕ0)
16 elex 3185 . . 3 (𝐺𝑈𝐺 ∈ V)
1716adantl 481 . 2 ((𝑁 ∈ ℕ0𝐺𝑈) → 𝐺 ∈ V)
18 fvex 6113 . . . . 5 (Vtx‘𝐺) ∈ V
194, 18eqeltri 2684 . . . 4 𝑉 ∈ V
2019, 19mpt2ex 7136 . . 3 (𝑎𝑉, 𝑏𝑉 ↦ {𝑤 ∈ (𝑁 WWalkSN 𝐺) ∣ ((𝑤‘0) = 𝑎 ∧ (𝑤𝑁) = 𝑏)}) ∈ V
2120a1i 11 . 2 ((𝑁 ∈ ℕ0𝐺𝑈) → (𝑎𝑉, 𝑏𝑉 ↦ {𝑤 ∈ (𝑁 WWalkSN 𝐺) ∣ ((𝑤‘0) = 𝑎 ∧ (𝑤𝑁) = 𝑏)}) ∈ V)
222, 14, 15, 17, 21ovmpt2d 6686 1 ((𝑁 ∈ ℕ0𝐺𝑈) → (𝑁 WWalksNOn 𝐺) = (𝑎𝑉, 𝑏𝑉 ↦ {𝑤 ∈ (𝑁 WWalkSN 𝐺) ∣ ((𝑤‘0) = 𝑎 ∧ (𝑤𝑁) = 𝑏)}))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  {crab 2900  Vcvv 3173  ‘cfv 5804  (class class class)co 6549   ↦ cmpt2 6551  0cc0 9815  ℕ0cn0 11169  Vtxcvtx 25673   WWalkSN cwwlksn 41029   WWalksNOn cwwlksnon 41030 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-1st 7059  df-2nd 7060  df-wwlksnon 41035 This theorem is referenced by:  iswwlksnon  41051  wwlksnon0  41123  wwlks2onv  41158
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