MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  numclwlk2lem2fv Structured version   Visualization version   GIF version

Theorem numclwlk2lem2fv 30175
Description: Value of the function 𝑅. (Contributed by Alexander van der Vekens, 6-Oct-2018.) (Revised by AV, 31-May-2021.) (Revised by AV, 1-Nov-2022.)
Hypotheses
Ref Expression
numclwwlk.v 𝑉 = (Vtxβ€˜πΊ)
numclwwlk.q 𝑄 = (𝑣 ∈ 𝑉, 𝑛 ∈ β„• ↦ {𝑀 ∈ (𝑛 WWalksN 𝐺) ∣ ((π‘€β€˜0) = 𝑣 ∧ (lastSβ€˜π‘€) β‰  𝑣)})
numclwwlk.h 𝐻 = (𝑣 ∈ 𝑉, 𝑛 ∈ (β„€β‰₯β€˜2) ↦ {𝑀 ∈ (𝑣(ClWWalksNOnβ€˜πΊ)𝑛) ∣ (π‘€β€˜(𝑛 βˆ’ 2)) β‰  𝑣})
numclwwlk.r 𝑅 = (π‘₯ ∈ (𝑋𝐻(𝑁 + 2)) ↦ (π‘₯ prefix (𝑁 + 1)))
Assertion
Ref Expression
numclwlk2lem2fv ((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ β„•) β†’ (π‘Š ∈ (𝑋𝐻(𝑁 + 2)) β†’ (π‘…β€˜π‘Š) = (π‘Š prefix (𝑁 + 1))))
Distinct variable groups:   𝑛,𝐺,𝑣,𝑀   𝑛,𝑁,𝑣,𝑀   𝑛,𝑉,𝑣   𝑛,𝑋,𝑣,𝑀   𝑀,𝑉   𝑣,π‘Š,𝑀   π‘₯,𝐺,𝑀   π‘₯,𝐻   π‘₯,𝑁   π‘₯,𝑄   π‘₯,𝑉   π‘₯,𝑋   π‘₯,π‘Š
Allowed substitution hints:   𝑄(𝑀,𝑣,𝑛)   𝑅(π‘₯,𝑀,𝑣,𝑛)   𝐻(𝑀,𝑣,𝑛)   π‘Š(𝑛)

Proof of Theorem numclwlk2lem2fv
StepHypRef Expression
1 numclwwlk.r . . 3 𝑅 = (π‘₯ ∈ (𝑋𝐻(𝑁 + 2)) ↦ (π‘₯ prefix (𝑁 + 1)))
2 oveq1 7421 . . 3 (π‘₯ = π‘Š β†’ (π‘₯ prefix (𝑁 + 1)) = (π‘Š prefix (𝑁 + 1)))
3 simpr 484 . . 3 (((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ β„•) ∧ π‘Š ∈ (𝑋𝐻(𝑁 + 2))) β†’ π‘Š ∈ (𝑋𝐻(𝑁 + 2)))
4 ovexd 7449 . . 3 (((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ β„•) ∧ π‘Š ∈ (𝑋𝐻(𝑁 + 2))) β†’ (π‘Š prefix (𝑁 + 1)) ∈ V)
51, 2, 3, 4fvmptd3 7022 . 2 (((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ β„•) ∧ π‘Š ∈ (𝑋𝐻(𝑁 + 2))) β†’ (π‘…β€˜π‘Š) = (π‘Š prefix (𝑁 + 1)))
65ex 412 1 ((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ β„•) β†’ (π‘Š ∈ (𝑋𝐻(𝑁 + 2)) β†’ (π‘…β€˜π‘Š) = (π‘Š prefix (𝑁 + 1))))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 395   = wceq 1534   ∈ wcel 2099   β‰  wne 2935  {crab 3427  Vcvv 3469   ↦ cmpt 5225  β€˜cfv 6542  (class class class)co 7414   ∈ cmpo 7416  0cc0 11130  1c1 11131   + caddc 11133   βˆ’ cmin 11466  β„•cn 12234  2c2 12289  β„€β‰₯cuz 12844  lastSclsw 14536   prefix cpfx 14644  Vtxcvtx 28796   WWalksN cwwlksn 29624  ClWWalksNOncclwwlknon 29884
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-sep 5293  ax-nul 5300  ax-pr 5423
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2936  df-ral 3057  df-rex 3066  df-rab 3428  df-v 3471  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-br 5143  df-opab 5205  df-mpt 5226  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-iota 6494  df-fun 6544  df-fv 6550  df-ov 7417
This theorem is referenced by:  numclwlk2lem2f1o  30176
  Copyright terms: Public domain W3C validator