Theorem clwwlksn 26881

Description: The set of closed walks (in an undirected graph) of a fixed length as words over the set of vertices. (Contributed by Alexander van der Vekens, 20-Mar-2018.) (Revised by AV, 24-Apr-2021.)

Assertion

Ref	Expression
clwwlksn	⊢ (𝑁 ∈ ℕ → (𝑁 ClWWalksN 𝐺) = {𝑤 ∈ (ClWWalks‘𝐺) ∣ (#‘𝑤) = 𝑁})

Distinct variable groups:   𝑤,𝐺   𝑤,𝑁

Proof of Theorem clwwlksn

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

Step	Hyp	Ref	Expression
1		df-clwwlksn 26878	. . . . 5 ⊢ ClWWalksN = (𝑛 ∈ ℕ, 𝑔 ∈ V ↦ {𝑤 ∈ (ClWWalks‘𝑔) ∣ (#‘𝑤) = 𝑛})
2	1	a1i 11	. . . 4 ⊢ ((𝑁 ∈ ℕ ∧ 𝐺 ∈ V) → ClWWalksN = (𝑛 ∈ ℕ, 𝑔 ∈ V ↦ {𝑤 ∈ (ClWWalks‘𝑔) ∣ (#‘𝑤) = 𝑛}))
3		fveq2 6191	. . . . . . 7 ⊢ (𝑔 = 𝐺 → (ClWWalks‘𝑔) = (ClWWalks‘𝐺))
4	3	adantl 482	. . . . . 6 ⊢ ((𝑛 = 𝑁 ∧ 𝑔 = 𝐺) → (ClWWalks‘𝑔) = (ClWWalks‘𝐺))
5		eqeq2 2633	. . . . . . 7 ⊢ (𝑛 = 𝑁 → ((#‘𝑤) = 𝑛 ↔ (#‘𝑤) = 𝑁))
6	5	adantr 481	. . . . . 6 ⊢ ((𝑛 = 𝑁 ∧ 𝑔 = 𝐺) → ((#‘𝑤) = 𝑛 ↔ (#‘𝑤) = 𝑁))
7	4, 6	rabeqbidv 3195	. . . . 5 ⊢ ((𝑛 = 𝑁 ∧ 𝑔 = 𝐺) → {𝑤 ∈ (ClWWalks‘𝑔) ∣ (#‘𝑤) = 𝑛} = {𝑤 ∈ (ClWWalks‘𝐺) ∣ (#‘𝑤) = 𝑁})
8	7	adantl 482	. . . 4 ⊢ (((𝑁 ∈ ℕ ∧ 𝐺 ∈ V) ∧ (𝑛 = 𝑁 ∧ 𝑔 = 𝐺)) → {𝑤 ∈ (ClWWalks‘𝑔) ∣ (#‘𝑤) = 𝑛} = {𝑤 ∈ (ClWWalks‘𝐺) ∣ (#‘𝑤) = 𝑁})
9		simpl 473	. . . 4 ⊢ ((𝑁 ∈ ℕ ∧ 𝐺 ∈ V) → 𝑁 ∈ ℕ)
10		simpr 477	. . . 4 ⊢ ((𝑁 ∈ ℕ ∧ 𝐺 ∈ V) → 𝐺 ∈ V)
11		fvex 6201	. . . . . 6 ⊢ (ClWWalks‘𝐺) ∈ V
12	11	rabex 4813	. . . . 5 ⊢ {𝑤 ∈ (ClWWalks‘𝐺) ∣ (#‘𝑤) = 𝑁} ∈ V
13	12	a1i 11	. . . 4 ⊢ ((𝑁 ∈ ℕ ∧ 𝐺 ∈ V) → {𝑤 ∈ (ClWWalks‘𝐺) ∣ (#‘𝑤) = 𝑁} ∈ V)
14	2, 8, 9, 10, 13	ovmpt2d 6788	. . 3 ⊢ ((𝑁 ∈ ℕ ∧ 𝐺 ∈ V) → (𝑁 ClWWalksN 𝐺) = {𝑤 ∈ (ClWWalks‘𝐺) ∣ (#‘𝑤) = 𝑁})
15	14	expcom 451	. 2 ⊢ (𝐺 ∈ V → (𝑁 ∈ ℕ → (𝑁 ClWWalksN 𝐺) = {𝑤 ∈ (ClWWalks‘𝐺) ∣ (#‘𝑤) = 𝑁}))
16	1	reldmmpt2 6771	. . . . 5 ⊢ Rel dom ClWWalksN
17	16	ovprc2 6685	. . . 4 ⊢ (¬ 𝐺 ∈ V → (𝑁 ClWWalksN 𝐺) = ∅)
18		fvprc 6185	. . . . . 6 ⊢ (¬ 𝐺 ∈ V → (ClWWalks‘𝐺) = ∅)
19	18	rabeqdv 3194	. . . . 5 ⊢ (¬ 𝐺 ∈ V → {𝑤 ∈ (ClWWalks‘𝐺) ∣ (#‘𝑤) = 𝑁} = {𝑤 ∈ ∅ ∣ (#‘𝑤) = 𝑁})
20		rab0 3955	. . . . 5 ⊢ {𝑤 ∈ ∅ ∣ (#‘𝑤) = 𝑁} = ∅
21	19, 20	syl6eq 2672	. . . 4 ⊢ (¬ 𝐺 ∈ V → {𝑤 ∈ (ClWWalks‘𝐺) ∣ (#‘𝑤) = 𝑁} = ∅)
22	17, 21	eqtr4d 2659	. . 3 ⊢ (¬ 𝐺 ∈ V → (𝑁 ClWWalksN 𝐺) = {𝑤 ∈ (ClWWalks‘𝐺) ∣ (#‘𝑤) = 𝑁})
23	22	a1d 25	. 2 ⊢ (¬ 𝐺 ∈ V → (𝑁 ∈ ℕ → (𝑁 ClWWalksN 𝐺) = {𝑤 ∈ (ClWWalks‘𝐺) ∣ (#‘𝑤) = 𝑁}))
24	15, 23	pm2.61i 176	1 ⊢ (𝑁 ∈ ℕ → (𝑁 ClWWalksN 𝐺) = {𝑤 ∈ (ClWWalks‘𝐺) ∣ (#‘𝑤) = 𝑁})

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  ℕcn 11020  #chash 13117  ClWWalkscclwwlks 26875   ClWWalksN cclwwlksn 26876

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-clwwlksn 26878

This theorem is referenced by:  isclwwlksn  26882  clwwlksnfi  26913