Theorem clwwlkn 26295

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.)

Assertion

Ref	Expression
clwwlkn	⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌 ∧ 𝑁 ∈ ℕ0) → ((𝑉 ClWWalksN 𝐸)‘𝑁) = {𝑤 ∈ (𝑉 ClWWalks 𝐸) ∣ (#‘𝑤) = 𝑁})

Distinct variable groups:   𝑤,𝐸   𝑤,𝑉   𝑤,𝑁

Allowed substitution hints:   𝑋(𝑤)   𝑌(𝑤)

Proof of Theorem clwwlkn

Dummy variables 𝑒 𝑛 𝑣 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simp3 1056	. . 3 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌 ∧ 𝑁 ∈ ℕ0) → 𝑁 ∈ ℕ0)
2		ovex 6577	. . . 4 ⊢ (𝑉 ClWWalks 𝐸) ∈ V
3		rabexg 4739	. . . 4 ⊢ ((𝑉 ClWWalks 𝐸) ∈ V → {𝑤 ∈ (𝑉 ClWWalks 𝐸) ∣ (#‘𝑤) = 𝑁} ∈ V)
4	2, 3	mp1i 13	. . 3 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌 ∧ 𝑁 ∈ ℕ0) → {𝑤 ∈ (𝑉 ClWWalks 𝐸) ∣ (#‘𝑤) = 𝑁} ∈ V)
5		eqeq2 2621	. . . . 5 ⊢ (𝑛 = 𝑁 → ((#‘𝑤) = 𝑛 ↔ (#‘𝑤) = 𝑁))
6	5	rabbidv 3164	. . . 4 ⊢ (𝑛 = 𝑁 → {𝑤 ∈ (𝑉 ClWWalks 𝐸) ∣ (#‘𝑤) = 𝑛} = {𝑤 ∈ (𝑉 ClWWalks 𝐸) ∣ (#‘𝑤) = 𝑁})
7		eqid 2610	. . . 4 ⊢ (𝑛 ∈ ℕ0 ↦ {𝑤 ∈ (𝑉 ClWWalks 𝐸) ∣ (#‘𝑤) = 𝑛}) = (𝑛 ∈ ℕ0 ↦ {𝑤 ∈ (𝑉 ClWWalks 𝐸) ∣ (#‘𝑤) = 𝑛})
8	6, 7	fvmptg 6189	. . 3 ⊢ ((𝑁 ∈ ℕ0 ∧ {𝑤 ∈ (𝑉 ClWWalks 𝐸) ∣ (#‘𝑤) = 𝑁} ∈ V) → ((𝑛 ∈ ℕ0 ↦ {𝑤 ∈ (𝑉 ClWWalks 𝐸) ∣ (#‘𝑤) = 𝑛})‘𝑁) = {𝑤 ∈ (𝑉 ClWWalks 𝐸) ∣ (#‘𝑤) = 𝑁})
9	1, 4, 8	syl2anc 691	. 2 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌 ∧ 𝑁 ∈ ℕ0) → ((𝑛 ∈ ℕ0 ↦ {𝑤 ∈ (𝑉 ClWWalks 𝐸) ∣ (#‘𝑤) = 𝑛})‘𝑁) = {𝑤 ∈ (𝑉 ClWWalks 𝐸) ∣ (#‘𝑤) = 𝑁})
10		df-clwwlkn 26280	. . . . . . 7 ⊢ ClWWalksN = (𝑣 ∈ V, 𝑒 ∈ V ↦ (𝑛 ∈ ℕ0 ↦ {𝑤 ∈ (𝑣 ClWWalks 𝑒) ∣ (#‘𝑤) = 𝑛}))
11	10	a1i 11	. . . . . 6 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) → ClWWalksN = (𝑣 ∈ V, 𝑒 ∈ V ↦ (𝑛 ∈ ℕ0 ↦ {𝑤 ∈ (𝑣 ClWWalks 𝑒) ∣ (#‘𝑤) = 𝑛})))
12		oveq12 6558	. . . . . . . . 9 ⊢ ((𝑣 = 𝑉 ∧ 𝑒 = 𝐸) → (𝑣 ClWWalks 𝑒) = (𝑉 ClWWalks 𝐸))
13		rabeq 3166	. . . . . . . . 9 ⊢ ((𝑣 ClWWalks 𝑒) = (𝑉 ClWWalks 𝐸) → {𝑤 ∈ (𝑣 ClWWalks 𝑒) ∣ (#‘𝑤) = 𝑛} = {𝑤 ∈ (𝑉 ClWWalks 𝐸) ∣ (#‘𝑤) = 𝑛})
14	12, 13	syl 17	. . . . . . . 8 ⊢ ((𝑣 = 𝑉 ∧ 𝑒 = 𝐸) → {𝑤 ∈ (𝑣 ClWWalks 𝑒) ∣ (#‘𝑤) = 𝑛} = {𝑤 ∈ (𝑉 ClWWalks 𝐸) ∣ (#‘𝑤) = 𝑛})
15	14	mpteq2dv 4673	. . . . . . 7 ⊢ ((𝑣 = 𝑉 ∧ 𝑒 = 𝐸) → (𝑛 ∈ ℕ0 ↦ {𝑤 ∈ (𝑣 ClWWalks 𝑒) ∣ (#‘𝑤) = 𝑛}) = (𝑛 ∈ ℕ0 ↦ {𝑤 ∈ (𝑉 ClWWalks 𝐸) ∣ (#‘𝑤) = 𝑛}))
16	15	adantl 481	. . . . . 6 ⊢ (((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) ∧ (𝑣 = 𝑉 ∧ 𝑒 = 𝐸)) → (𝑛 ∈ ℕ0 ↦ {𝑤 ∈ (𝑣 ClWWalks 𝑒) ∣ (#‘𝑤) = 𝑛}) = (𝑛 ∈ ℕ0 ↦ {𝑤 ∈ (𝑉 ClWWalks 𝐸) ∣ (#‘𝑤) = 𝑛}))
17		elex 3185	. . . . . . 7 ⊢ (𝑉 ∈ 𝑋 → 𝑉 ∈ V)
18	17	adantr 480	. . . . . 6 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) → 𝑉 ∈ V)
19		elex 3185	. . . . . . 7 ⊢ (𝐸 ∈ 𝑌 → 𝐸 ∈ V)
20	19	adantl 481	. . . . . 6 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) → 𝐸 ∈ V)
21		nn0ex 11175	. . . . . . . 8 ⊢ ℕ0 ∈ V
22	21	mptex 6390	. . . . . . 7 ⊢ (𝑛 ∈ ℕ0 ↦ {𝑤 ∈ (𝑉 ClWWalks 𝐸) ∣ (#‘𝑤) = 𝑛}) ∈ V
23	22	a1i 11	. . . . . 6 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) → (𝑛 ∈ ℕ0 ↦ {𝑤 ∈ (𝑉 ClWWalks 𝐸) ∣ (#‘𝑤) = 𝑛}) ∈ V)
24	11, 16, 18, 20, 23	ovmpt2d 6686	. . . . 5 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) → (𝑉 ClWWalksN 𝐸) = (𝑛 ∈ ℕ0 ↦ {𝑤 ∈ (𝑉 ClWWalks 𝐸) ∣ (#‘𝑤) = 𝑛}))
25	24	fveq1d 6105	. . . 4 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) → ((𝑉 ClWWalksN 𝐸)‘𝑁) = ((𝑛 ∈ ℕ0 ↦ {𝑤 ∈ (𝑉 ClWWalks 𝐸) ∣ (#‘𝑤) = 𝑛})‘𝑁))
26	25	eqeq1d 2612	. . 3 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) → (((𝑉 ClWWalksN 𝐸)‘𝑁) = {𝑤 ∈ (𝑉 ClWWalks 𝐸) ∣ (#‘𝑤) = 𝑁} ↔ ((𝑛 ∈ ℕ0 ↦ {𝑤 ∈ (𝑉 ClWWalks 𝐸) ∣ (#‘𝑤) = 𝑛})‘𝑁) = {𝑤 ∈ (𝑉 ClWWalks 𝐸) ∣ (#‘𝑤) = 𝑁}))
27	26	3adant3 1074	. 2 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌 ∧ 𝑁 ∈ ℕ0) → (((𝑉 ClWWalksN 𝐸)‘𝑁) = {𝑤 ∈ (𝑉 ClWWalks 𝐸) ∣ (#‘𝑤) = 𝑁} ↔ ((𝑛 ∈ ℕ0 ↦ {𝑤 ∈ (𝑉 ClWWalks 𝐸) ∣ (#‘𝑤) = 𝑛})‘𝑁) = {𝑤 ∈ (𝑉 ClWWalks 𝐸) ∣ (#‘𝑤) = 𝑁}))
28	9, 27	mpbird 246	1 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌 ∧ 𝑁 ∈ ℕ0) → ((𝑉 ClWWalksN 𝐸)‘𝑁) = {𝑤 ∈ (𝑉 ClWWalks 𝐸) ∣ (#‘𝑤) = 𝑁})

Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  {crab 2900  Vcvv 3173   ↦ cmpt 4643  ‘cfv 5804  (class class class)co 6549   ↦ cmpt2 6551  ℕ₀cn0 11169  #chash 12979   ClWWalks cclwwlk 26276   ClWWalksN cclwwlkn 26277

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  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-i2m1 9883  ax-1ne0 9884  ax-rrecex 9887  ax-cnre 9888

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  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-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  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-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  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-om 6958  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-nn 10898  df-n0 11170  df-clwwlkn 26280

This theorem is referenced by:  isclwwlkn  26297  clwwlknprop  26300  clwwlkn0  26302  clwwlknfi  26306