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Theorem numclwwlkovq 26626
 Description: Value of operation Q, mapping a vertex v and a nonnegative integer n to the not closed walks v(0) ... v(n) of length n from a fixed vertex v = v(0). "Not closed" means v(n) =/= v(0). (Contributed by Alexander van der Vekens, 27-Sep-2018.)
Hypotheses
Ref Expression
numclwwlk.c 𝐶 = (𝑛 ∈ ℕ0 ↦ ((𝑉 ClWWalksN 𝐸)‘𝑛))
numclwwlk.f 𝐹 = (𝑣𝑉, 𝑛 ∈ ℕ0 ↦ {𝑤 ∈ (𝐶𝑛) ∣ (𝑤‘0) = 𝑣})
numclwwlk.g 𝐺 = (𝑣𝑉, 𝑛 ∈ (ℤ‘2) ↦ {𝑤 ∈ (𝐶𝑛) ∣ ((𝑤‘0) = 𝑣 ∧ (𝑤‘(𝑛 − 2)) = (𝑤‘0))})
numclwwlk.q 𝑄 = (𝑣𝑉, 𝑛 ∈ ℕ0 ↦ {𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑛) ∣ ((𝑤‘0) = 𝑣 ∧ ( lastS ‘𝑤) ≠ 𝑣)})
Assertion
Ref Expression
numclwwlkovq ((𝑋𝑉𝑁 ∈ ℕ0) → (𝑋𝑄𝑁) = {𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ ((𝑤‘0) = 𝑋 ∧ ( lastS ‘𝑤) ≠ 𝑋)})
Distinct variable groups:   𝑛,𝐸   𝑛,𝑁   𝑛,𝑉   𝑤,𝐶   𝑤,𝑁   𝐶,𝑛,𝑣,𝑤   𝑣,𝑁   𝑛,𝑋,𝑣,𝑤   𝑣,𝑉   𝑤,𝐸   𝑤,𝑉   𝑤,𝐹   𝑤,𝑄   𝑤,𝐺   𝑣,𝐸
Allowed substitution hints:   𝑄(𝑣,𝑛)   𝐹(𝑣,𝑛)   𝐺(𝑣,𝑛)

Proof of Theorem numclwwlkovq
StepHypRef Expression
1 fveq2 6103 . . . 4 (𝑛 = 𝑁 → ((𝑉 WWalksN 𝐸)‘𝑛) = ((𝑉 WWalksN 𝐸)‘𝑁))
21adantl 481 . . 3 ((𝑣 = 𝑋𝑛 = 𝑁) → ((𝑉 WWalksN 𝐸)‘𝑛) = ((𝑉 WWalksN 𝐸)‘𝑁))
3 eqeq2 2621 . . . . 5 (𝑣 = 𝑋 → ((𝑤‘0) = 𝑣 ↔ (𝑤‘0) = 𝑋))
43adantr 480 . . . 4 ((𝑣 = 𝑋𝑛 = 𝑁) → ((𝑤‘0) = 𝑣 ↔ (𝑤‘0) = 𝑋))
5 simpl 472 . . . . 5 ((𝑣 = 𝑋𝑛 = 𝑁) → 𝑣 = 𝑋)
65neeq2d 2842 . . . 4 ((𝑣 = 𝑋𝑛 = 𝑁) → (( lastS ‘𝑤) ≠ 𝑣 ↔ ( lastS ‘𝑤) ≠ 𝑋))
74, 6anbi12d 743 . . 3 ((𝑣 = 𝑋𝑛 = 𝑁) → (((𝑤‘0) = 𝑣 ∧ ( lastS ‘𝑤) ≠ 𝑣) ↔ ((𝑤‘0) = 𝑋 ∧ ( lastS ‘𝑤) ≠ 𝑋)))
82, 7rabeqbidv 3168 . 2 ((𝑣 = 𝑋𝑛 = 𝑁) → {𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑛) ∣ ((𝑤‘0) = 𝑣 ∧ ( lastS ‘𝑤) ≠ 𝑣)} = {𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ ((𝑤‘0) = 𝑋 ∧ ( lastS ‘𝑤) ≠ 𝑋)})
9 numclwwlk.q . 2 𝑄 = (𝑣𝑉, 𝑛 ∈ ℕ0 ↦ {𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑛) ∣ ((𝑤‘0) = 𝑣 ∧ ( lastS ‘𝑤) ≠ 𝑣)})
10 fvex 6113 . . 3 ((𝑉 WWalksN 𝐸)‘𝑁) ∈ V
1110rabex 4740 . 2 {𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ ((𝑤‘0) = 𝑋 ∧ ( lastS ‘𝑤) ≠ 𝑋)} ∈ V
128, 9, 11ovmpt2a 6689 1 ((𝑋𝑉𝑁 ∈ ℕ0) → (𝑋𝑄𝑁) = {𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ ((𝑤‘0) = 𝑋 ∧ ( lastS ‘𝑤) ≠ 𝑋)})
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  {crab 2900   ↦ cmpt 4643  ‘cfv 5804  (class class class)co 6549   ↦ cmpt2 6551  0cc0 9815   − cmin 10145  2c2 10947  ℕ0cn0 11169  ℤ≥cuz 11563   lastS clsw 13147   WWalksN cwwlkn 26206   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-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 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-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-iota 5768  df-fun 5806  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554 This theorem is referenced by:  numclwwlkqhash  26627  numclwwlk2lem1  26629  numclwlk2lem2f  26630  numclwlk2lem2f1o  26632
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