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Theorem numclwwlk1lem2fv 29342
Description: Value of the function 𝑇. (Contributed by Alexander van der Vekens, 20-Sep-2018.) (Revised by AV, 29-May-2021.) (Revised by AV, 31-Oct-2022.)
Hypotheses
Ref Expression
extwwlkfab.v 𝑉 = (Vtxβ€˜πΊ)
extwwlkfab.c 𝐢 = (𝑣 ∈ 𝑉, 𝑛 ∈ (β„€β‰₯β€˜2) ↦ {𝑀 ∈ (𝑣(ClWWalksNOnβ€˜πΊ)𝑛) ∣ (π‘€β€˜(𝑛 βˆ’ 2)) = 𝑣})
extwwlkfab.f 𝐹 = (𝑋(ClWWalksNOnβ€˜πΊ)(𝑁 βˆ’ 2))
numclwwlk.t 𝑇 = (𝑒 ∈ (𝑋𝐢𝑁) ↦ ⟨(𝑒 prefix (𝑁 βˆ’ 2)), (π‘’β€˜(𝑁 βˆ’ 1))⟩)
Assertion
Ref Expression
numclwwlk1lem2fv (π‘Š ∈ (𝑋𝐢𝑁) β†’ (π‘‡β€˜π‘Š) = ⟨(π‘Š prefix (𝑁 βˆ’ 2)), (π‘Šβ€˜(𝑁 βˆ’ 1))⟩)
Distinct variable groups:   𝑛,𝐺,𝑣,𝑀   𝑛,𝑁,𝑣,𝑀   𝑛,𝑉,𝑣,𝑀   𝑛,𝑋,𝑣,𝑀   𝑀,𝐹   𝑀,π‘Š   𝑒,𝐢   𝑒,𝐹   𝑒,𝐺,𝑀   𝑒,𝑁   𝑒,𝑉   𝑒,𝑋   𝑒,π‘Š
Allowed substitution hints:   𝐢(𝑀,𝑣,𝑛)   𝑇(𝑀,𝑣,𝑒,𝑛)   𝐹(𝑣,𝑛)   π‘Š(𝑣,𝑛)

Proof of Theorem numclwwlk1lem2fv
StepHypRef Expression
1 oveq1 7369 . . 3 (𝑒 = π‘Š β†’ (𝑒 prefix (𝑁 βˆ’ 2)) = (π‘Š prefix (𝑁 βˆ’ 2)))
2 fveq1 6846 . . 3 (𝑒 = π‘Š β†’ (π‘’β€˜(𝑁 βˆ’ 1)) = (π‘Šβ€˜(𝑁 βˆ’ 1)))
31, 2opeq12d 4843 . 2 (𝑒 = π‘Š β†’ ⟨(𝑒 prefix (𝑁 βˆ’ 2)), (π‘’β€˜(𝑁 βˆ’ 1))⟩ = ⟨(π‘Š prefix (𝑁 βˆ’ 2)), (π‘Šβ€˜(𝑁 βˆ’ 1))⟩)
4 numclwwlk.t . 2 𝑇 = (𝑒 ∈ (𝑋𝐢𝑁) ↦ ⟨(𝑒 prefix (𝑁 βˆ’ 2)), (π‘’β€˜(𝑁 βˆ’ 1))⟩)
5 opex 5426 . 2 ⟨(π‘Š prefix (𝑁 βˆ’ 2)), (π‘Šβ€˜(𝑁 βˆ’ 1))⟩ ∈ V
63, 4, 5fvmpt 6953 1 (π‘Š ∈ (𝑋𝐢𝑁) β†’ (π‘‡β€˜π‘Š) = ⟨(π‘Š prefix (𝑁 βˆ’ 2)), (π‘Šβ€˜(𝑁 βˆ’ 1))⟩)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   = wceq 1542   ∈ wcel 2107  {crab 3410  βŸ¨cop 4597   ↦ cmpt 5193  β€˜cfv 6501  (class class class)co 7362   ∈ cmpo 7364  1c1 11059   βˆ’ cmin 11392  2c2 12215  β„€β‰₯cuz 12770   prefix cpfx 14565  Vtxcvtx 27989  ClWWalksNOncclwwlknon 29073
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-sep 5261  ax-nul 5268  ax-pr 5389
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ral 3066  df-rex 3075  df-rab 3411  df-v 3450  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-nul 4288  df-if 4492  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-br 5111  df-opab 5173  df-mpt 5194  df-id 5536  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-iota 6453  df-fun 6503  df-fv 6509  df-ov 7365
This theorem is referenced by:  numclwwlk1lem2f1  29343  numclwwlk1lem2fo  29344
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