Theorem wwlksn 26729

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 ∈ ℕ0 → (𝑁 WWalksN 𝐺) = {𝑤 ∈ (WWalks‘𝐺) ∣ (#‘𝑤) = (𝑁 + 1)})

Distinct variable groups:   𝑤,𝐺   𝑤,𝑁

Proof of Theorem wwlksn

Dummy variables 𝑔 𝑛 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-wwlksn 26723	. . . . 5 ⊢ WWalksN = (𝑛 ∈ ℕ0, 𝑔 ∈ V ↦ {𝑤 ∈ (WWalks‘𝑔) ∣ (#‘𝑤) = (𝑛 + 1)})
2	1	a1i 11	. . . 4 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐺 ∈ V) → WWalksN = (𝑛 ∈ ℕ0, 𝑔 ∈ V ↦ {𝑤 ∈ (WWalks‘𝑔) ∣ (#‘𝑤) = (𝑛 + 1)}))
3		fveq2 6191	. . . . . . 7 ⊢ (𝑔 = 𝐺 → (WWalks‘𝑔) = (WWalks‘𝐺))
4	3	adantl 482	. . . . . 6 ⊢ ((𝑛 = 𝑁 ∧ 𝑔 = 𝐺) → (WWalks‘𝑔) = (WWalks‘𝐺))
5		oveq1 6657	. . . . . . . 8 ⊢ (𝑛 = 𝑁 → (𝑛 + 1) = (𝑁 + 1))
6	5	eqeq2d 2632	. . . . . . 7 ⊢ (𝑛 = 𝑁 → ((#‘𝑤) = (𝑛 + 1) ↔ (#‘𝑤) = (𝑁 + 1)))
7	6	adantr 481	. . . . . 6 ⊢ ((𝑛 = 𝑁 ∧ 𝑔 = 𝐺) → ((#‘𝑤) = (𝑛 + 1) ↔ (#‘𝑤) = (𝑁 + 1)))
8	4, 7	rabeqbidv 3195	. . . . 5 ⊢ ((𝑛 = 𝑁 ∧ 𝑔 = 𝐺) → {𝑤 ∈ (WWalks‘𝑔) ∣ (#‘𝑤) = (𝑛 + 1)} = {𝑤 ∈ (WWalks‘𝐺) ∣ (#‘𝑤) = (𝑁 + 1)})
9	8	adantl 482	. . . 4 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝐺 ∈ V) ∧ (𝑛 = 𝑁 ∧ 𝑔 = 𝐺)) → {𝑤 ∈ (WWalks‘𝑔) ∣ (#‘𝑤) = (𝑛 + 1)} = {𝑤 ∈ (WWalks‘𝐺) ∣ (#‘𝑤) = (𝑁 + 1)})
10		simpl 473	. . . 4 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐺 ∈ V) → 𝑁 ∈ ℕ0)
11		simpr 477	. . . 4 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐺 ∈ V) → 𝐺 ∈ V)
12		fvex 6201	. . . . . 6 ⊢ (WWalks‘𝐺) ∈ V
13	12	rabex 4813	. . . . 5 ⊢ {𝑤 ∈ (WWalks‘𝐺) ∣ (#‘𝑤) = (𝑁 + 1)} ∈ V
14	13	a1i 11	. . . 4 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐺 ∈ V) → {𝑤 ∈ (WWalks‘𝐺) ∣ (#‘𝑤) = (𝑁 + 1)} ∈ V)
15	2, 9, 10, 11, 14	ovmpt2d 6788	. . 3 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐺 ∈ V) → (𝑁 WWalksN 𝐺) = {𝑤 ∈ (WWalks‘𝐺) ∣ (#‘𝑤) = (𝑁 + 1)})
16	15	expcom 451	. 2 ⊢ (𝐺 ∈ V → (𝑁 ∈ ℕ0 → (𝑁 WWalksN 𝐺) = {𝑤 ∈ (WWalks‘𝐺) ∣ (#‘𝑤) = (𝑁 + 1)}))
17	1	reldmmpt2 6771	. . . . 5 ⊢ Rel dom WWalksN
18	17	ovprc2 6685	. . . 4 ⊢ (¬ 𝐺 ∈ V → (𝑁 WWalksN 𝐺) = ∅)
19		fvprc 6185	. . . . . 6 ⊢ (¬ 𝐺 ∈ V → (WWalks‘𝐺) = ∅)
20	19	rabeqdv 3194	. . . . 5 ⊢ (¬ 𝐺 ∈ V → {𝑤 ∈ (WWalks‘𝐺) ∣ (#‘𝑤) = (𝑁 + 1)} = {𝑤 ∈ ∅ ∣ (#‘𝑤) = (𝑁 + 1)})
21		rab0 3955	. . . . 5 ⊢ {𝑤 ∈ ∅ ∣ (#‘𝑤) = (𝑁 + 1)} = ∅
22	20, 21	syl6eq 2672	. . . 4 ⊢ (¬ 𝐺 ∈ V → {𝑤 ∈ (WWalks‘𝐺) ∣ (#‘𝑤) = (𝑁 + 1)} = ∅)
23	18, 22	eqtr4d 2659	. . 3 ⊢ (¬ 𝐺 ∈ V → (𝑁 WWalksN 𝐺) = {𝑤 ∈ (WWalks‘𝐺) ∣ (#‘𝑤) = (𝑁 + 1)})
24	23	a1d 25	. 2 ⊢ (¬ 𝐺 ∈ V → (𝑁 ∈ ℕ0 → (𝑁 WWalksN 𝐺) = {𝑤 ∈ (WWalks‘𝐺) ∣ (#‘𝑤) = (𝑁 + 1)}))
25	16, 24	pm2.61i 176	1 ⊢ (𝑁 ∈ ℕ0 → (𝑁 WWalksN 𝐺) = {𝑤 ∈ (WWalks‘𝐺) ∣ (#‘𝑤) = (𝑁 + 1)})

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∧ wa 384   = wceq 1483   ∈ wcel 1990  {crab 2916  Vcvv 3200  ∅c0 3915  ‘cfv 5888  (class class class)co 6650   ↦ cmpt2 6652  1c1 9937   + caddc 9939  ℕ₀cn0 11292  #chash 13117  WWalkscwwlks 26717   WWalksN cwwlksn 26718

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-iota 5851  df-fun 5890  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-wwlksn 26723

This theorem is referenced by:  iswwlksn  26730  wwlksn0s  26746  0enwwlksnge1  26749  wwlksnfi  26801