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Theorem wwlksnon 26976
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 26956 . . 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 6353 . . . . . 6 (𝑔 = 𝐺 → (Vtx‘𝑔) = (Vtx‘𝐺))
4 wwlksnon.v . . . . . 6 𝑉 = (Vtx‘𝐺)
53, 4syl6eqr 2812 . . . . 5 (𝑔 = 𝐺 → (Vtx‘𝑔) = 𝑉)
65adantl 473 . . . 4 ((𝑛 = 𝑁𝑔 = 𝐺) → (Vtx‘𝑔) = 𝑉)
7 oveq12 6823 . . . . 5 ((𝑛 = 𝑁𝑔 = 𝐺) → (𝑛 WWalksN 𝑔) = (𝑁 WWalksN 𝐺))
8 fveq2 6353 . . . . . . . 8 (𝑛 = 𝑁 → (𝑤𝑛) = (𝑤𝑁))
98eqeq1d 2762 . . . . . . 7 (𝑛 = 𝑁 → ((𝑤𝑛) = 𝑏 ↔ (𝑤𝑁) = 𝑏))
109anbi2d 742 . . . . . 6 (𝑛 = 𝑁 → (((𝑤‘0) = 𝑎 ∧ (𝑤𝑛) = 𝑏) ↔ ((𝑤‘0) = 𝑎 ∧ (𝑤𝑁) = 𝑏)))
1110adantr 472 . . . . 5 ((𝑛 = 𝑁𝑔 = 𝐺) → (((𝑤‘0) = 𝑎 ∧ (𝑤𝑛) = 𝑏) ↔ ((𝑤‘0) = 𝑎 ∧ (𝑤𝑁) = 𝑏)))
127, 11rabeqbidv 3335 . . . 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 6883 . . 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 473 . 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 474 . 2 ((𝑁 ∈ ℕ0𝐺𝑈) → 𝑁 ∈ ℕ0)
16 elex 3352 . . 3 (𝐺𝑈𝐺 ∈ V)
1716adantl 473 . 2 ((𝑁 ∈ ℕ0𝐺𝑈) → 𝐺 ∈ V)
18 fvex 6363 . . . . 5 (Vtx‘𝐺) ∈ V
194, 18eqeltri 2835 . . . 4 𝑉 ∈ V
2019, 19mpt2ex 7416 . . 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 6954 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 196  wa 383   = wceq 1632  wcel 2139  {crab 3054  Vcvv 3340  cfv 6049  (class class class)co 6814  cmpt2 6816  0cc0 10148  0cn0 11504  Vtxcvtx 26094   WWalksN cwwlksn 26950   WWalksNOn cwwlksnon 26951
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7115
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-ov 6817  df-oprab 6818  df-mpt2 6819  df-1st 7334  df-2nd 7335  df-wwlksnon 26956
This theorem is referenced by:  iswwlksnon  26978  iswwlksnonOLD  26979  wwlksnon0  26980
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