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Theorem av-numclwwlkovq 41531
Description: Value of operation 𝑄, mapping a vertex 𝑣 and a positive 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.) (Revised by AV, 30-May-2021.)
Hypotheses
Ref Expression
av-numclwwlk.v 𝑉 = (Vtx‘𝐺)
av-numclwwlk.q 𝑄 = (𝑣𝑉, 𝑛 ∈ ℕ ↦ {𝑤 ∈ (𝑛 WWalkSN 𝐺) ∣ ((𝑤‘0) = 𝑣 ∧ ( lastS ‘𝑤) ≠ 𝑣)})
Assertion
Ref Expression
av-numclwwlkovq ((𝑋𝑉𝑁 ∈ ℕ) → (𝑋𝑄𝑁) = {𝑤 ∈ (𝑁 WWalkSN 𝐺) ∣ ((𝑤‘0) = 𝑋 ∧ ( lastS ‘𝑤) ≠ 𝑋)})
Distinct variable groups:   𝑛,𝐺,𝑣,𝑤   𝑛,𝑁,𝑣,𝑤   𝑛,𝑉,𝑣   𝑛,𝑋,𝑣,𝑤
Allowed substitution hints:   𝑄(𝑤,𝑣,𝑛)   𝑉(𝑤)

Proof of Theorem av-numclwwlkovq
StepHypRef Expression
1 oveq1 6534 . . . 4 (𝑛 = 𝑁 → (𝑛 WWalkSN 𝐺) = (𝑁 WWalkSN 𝐺))
21adantl 480 . . 3 ((𝑣 = 𝑋𝑛 = 𝑁) → (𝑛 WWalkSN 𝐺) = (𝑁 WWalkSN 𝐺))
3 eqeq2 2620 . . . . 5 (𝑣 = 𝑋 → ((𝑤‘0) = 𝑣 ↔ (𝑤‘0) = 𝑋))
4 neeq2 2844 . . . . 5 (𝑣 = 𝑋 → (( lastS ‘𝑤) ≠ 𝑣 ↔ ( lastS ‘𝑤) ≠ 𝑋))
53, 4anbi12d 742 . . . 4 (𝑣 = 𝑋 → (((𝑤‘0) = 𝑣 ∧ ( lastS ‘𝑤) ≠ 𝑣) ↔ ((𝑤‘0) = 𝑋 ∧ ( lastS ‘𝑤) ≠ 𝑋)))
65adantr 479 . . 3 ((𝑣 = 𝑋𝑛 = 𝑁) → (((𝑤‘0) = 𝑣 ∧ ( lastS ‘𝑤) ≠ 𝑣) ↔ ((𝑤‘0) = 𝑋 ∧ ( lastS ‘𝑤) ≠ 𝑋)))
72, 6rabeqbidv 3167 . 2 ((𝑣 = 𝑋𝑛 = 𝑁) → {𝑤 ∈ (𝑛 WWalkSN 𝐺) ∣ ((𝑤‘0) = 𝑣 ∧ ( lastS ‘𝑤) ≠ 𝑣)} = {𝑤 ∈ (𝑁 WWalkSN 𝐺) ∣ ((𝑤‘0) = 𝑋 ∧ ( lastS ‘𝑤) ≠ 𝑋)})
8 av-numclwwlk.q . 2 𝑄 = (𝑣𝑉, 𝑛 ∈ ℕ ↦ {𝑤 ∈ (𝑛 WWalkSN 𝐺) ∣ ((𝑤‘0) = 𝑣 ∧ ( lastS ‘𝑤) ≠ 𝑣)})
9 ovex 6555 . . 3 (𝑁 WWalkSN 𝐺) ∈ V
109rabex 4735 . 2 {𝑤 ∈ (𝑁 WWalkSN 𝐺) ∣ ((𝑤‘0) = 𝑋 ∧ ( lastS ‘𝑤) ≠ 𝑋)} ∈ V
117, 8, 10ovmpt2a 6667 1 ((𝑋𝑉𝑁 ∈ ℕ) → (𝑋𝑄𝑁) = {𝑤 ∈ (𝑁 WWalkSN 𝐺) ∣ ((𝑤‘0) = 𝑋 ∧ ( lastS ‘𝑤) ≠ 𝑋)})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 194  wa 382   = wceq 1474  wcel 1976  wne 2779  {crab 2899  cfv 5790  (class class class)co 6527  cmpt2 6529  0cc0 9792  cn 10867   lastS clsw 13093  Vtxcvtx 40231   WWalkSN cwwlksn 41031
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2032  ax-13 2232  ax-ext 2589  ax-sep 4703  ax-nul 4712  ax-pr 4828
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-ral 2900  df-rex 2901  df-rab 2904  df-v 3174  df-sbc 3402  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-sn 4125  df-pr 4127  df-op 4131  df-uni 4367  df-br 4578  df-opab 4638  df-id 4943  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-iota 5754  df-fun 5792  df-fv 5798  df-ov 6530  df-oprab 6531  df-mpt2 6532
This theorem is referenced by:  av-numclwwlkqhash  41532  av-numclwwlk2lem1  41534  av-numclwlk2lem2f  41535  av-numclwlk2lem2f1o  41537
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